Quantum Magnetism

Quantum Magnetism NATO Science for Peace and Security Series This Series presents the results of scientific meetings ...

Author: B. Barbara | B. Barbara | Y. Imry | G. Sawatzky | P.C.E. Stamp

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Quantum Magnetism

NATO Science for Peace and Security Series This Series presents the results of scientific meetings supported under the NATO Programme: Science for Peace and Security (SPS). The NATO SPS Programme supports meetings in the following Key Priority areas: (1) Defence Against Terrorism; (2) Countering other Threats to Security and (3) NATO, Partner and Mediterranean Dialogue Country Priorities. The types of meeting supported are generally "Advanced Study Institutes" and "Advanced Research Workshops". The NATO SPS Series collects together the results of these meetings. The meetings are coorganized by scientists from NATO countries and scientists from NATO's "Partner" or "Mediterranean Dialogue" countries. The observations and recommendations made at the meetings, as well as the contents of the volumes in the Series, reflect those of participants and contributors only; they should not necessarily be regarded as reflecting NATO views or policy. Advanced Study Institutes (ASI) are high-level tutorial courses intended to convey the latest developments in a subject to an advanced-level audience Advanced Research Workshops (ARW) are expert meetings where an intense but informal exchange of views at the frontiers of a subject aims at identifying directions for future action Following a transformation of the programme in 2006 the Series has been re-named and re-organised. Recent volumes on topics not related to security, which result from meetings supported under the programme earlier, may be found in the NATO Science Series. The Series is published by IOS Press, Amsterdam, and Springer, Dordrecht, in conjunction with the NATO Public Diplomacy Division. Sub-Series A. B. C. D. E.

Chemistry and Biology Physics and Biophysics Environmental Security Information and Communication Security Human and Societal Dynamics

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Series B: Physics and Biophysics

Springer Springer Springer IOS Press IOS Press

Quantum Magnetism

edited by

Bernard Barbara CNRS, Laboratoire Louis Néel, Grenoble, France

Yosef Imry The Wei z mann Institute of Science, Rehovot, Israel

G. Sawatzky University of British Columbia, Vancouver, BC, Canada and

P.C.E. Stamp University of British Columbia, Vancouver, BC, Canada

123 Published in cooperation with NATO Public Diplomacy Division

Proceedings of the NATO Advanced Study Institute on Quantum Magnetism Les Houches, France 6–23 June 2006

Library of Congress Control Number: 2008928505

ISBN 978-1-4020-8511-6 (PB) ISBN 978-1-4020-8510-9 (HB) ISBN 978-1-4020-8512-3 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

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All Rights Reserved © 2008 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Preface

This book is based on some of the lectures during the Paciﬁc Institute of Theoretical Physics (PITP) summer school on “Quantum Magnetism”, held during June 2006 in Les Houches, in the French Alps. The school was funded jointly by NATO, the CNRS, and PITP, and entirely organized by PITP. Magnetism is a somewhat peculiar research ﬁeld. It clearly has a quantummechanical basis – the microscopic exchange interactions arise entirely from the exclusion principle, in conjunction with repulsive interactions between electrons. And yet until recently the vast majority of magnetism researchers and users of magnetic phenomena around the world paid no attention to these quantum-mechanical roots. Thus, e.g., the huge ($400 billion per annum) industry which manufactures hard discs, and other components in the information technology sector, depends entirely on room-temperature properties of magnets – yet at the macroscopic or mesoscopic scales of interest to this industry, room-temperature magnets behave entirely classically. This situation has now begun to change, and the quantum collective properties of magnetic systems, for so long of interest only to a few, have begun to move to centre stage. There are several reasons for this. One is the increasing use of low temperatures in industrial and applied research labs, and the recognition that the low-T properties of many new magnetic materials are of great potential use in future devices. Another is the emergence of nanoscience, and its offshoot nanotechnology, as an important new discipline – the majority of high-tech applications of nanotechnology so far envisaged will also be low-temperature ones. All this has meant that collective quantum phenomena, occurring at low T in quantum dots, magnetic molecules, and nanoscopic conductors, have suddenly become interesting to more than just pure physics researchers. The design of new magnetic quantum materials, of spin-based quantum devices in low-dimensional geometries, and quantum nanomagnetic systems, using either physical or chemical techniques, has now become the concern of many applied physicists and chemists. An interesting by-product of this broader interest in collective quantum spin phenomena has been the increasing focus, by applied physicists and even startup companies, on some of the more exotic theoretical ideas current in quantum

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magnetism. These draw upon various branches of quantum ﬁeld theory (including topological ﬁeld theory, the theory of decoherence, and string theory), quantum computation, and upon older ﬁelds like spin glass theory and the theory of quantum phase transitions – all of which are fairly esoteric. Some of these ideas have found more application in cosmology or string theory than in condensed matter physics – so that their use in designing circuit arrays for quantum information processors, or spintronic devices, may seem disconcerting to some. Yet from another point of view it simply conﬁrms the broadly uniﬁed nature of the principles and techniques used in theoretical physics. For the PITP/Les Houches school a number of topics of prime interest were selected, and these reﬂect the interests of a broad community. They were as follows: 1. MAGNETISM AT THE MICROSCOPIC SCALE: This topic concerns principally the microscopic basis of magnetic phenomena, including the hierarchies of effective Hamiltonians in strongly correlated systems. Best known are models like the Anderson and Kondo Hamiltonians, the Hubbard Hamiltonian, and reﬁnements of these. There is increasing focus on the chemistry of interesting magnetic systems, including magnetic molecules, and on new ‘quantum materials’ showing magnetic properties. The convergence of ‘top-down’ nanofabrication techniques and ‘bottom-up’ chemical techniques has meant that there is no longer a sharp line between quantum chemistry and physics when it comes to magnetic systems. A key new tool in this ﬁeld, notably for strongly correlated systems, is the development of powerful new analytic and numerical methods, and the results of some of these latter methods were discussed in some detail, notably density functional, dynamical cluster, and dynamical mean ﬁeld theories, and the application of remarkable new Quantum Monte Carlo methods. 2. EXOTIC ORDER IN QUANTUM MAGNETS: The standard classical magnetic ordering theory fails to describe ordering in genuine quantum magnets. This is particularly clear in lower dimensions, where one can get many exotic kinds of ordering, both local and non-local. This even happens in 3 dimensions, with systems of particular interest being He-3 (solid and superﬂuid) and quantum spin glasses like the LiHoYF system. A key feature of many of the novel magnetic states is their non-trivial topological properties. New kinds of quantum liquids range from simple spin liquids to more exotic systems like the Quantum Hall ferromagnets or spin Bose-Einstein condensates – of key interest are spin and charge fractionalisation, and exotic quasiparticle statistics. Some of the most interesting states occur in 1-dimensional spin systems, which are also of great current interest in the context of quantum computation. 3. DISORDERED MAGNETS: Many remarkable critical phenomena (including the clearest examples of quantum critical phenomena) occur in disordered magnetic systems. In recent years novel features of these have been discovered in the low-T quantum regime of these systems, including quantum spin glass phases, as well as novel ordering in systems having random ﬁelds and/or positional disorder. There are also very interesting connections to phenomena in other systems, such as dipolar glasses at very low T.

Preface

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4. QUANTUM NANOMAGNETISM: At very small scales, or near surfaces, magnetic systems can behave very differently from at macroscopic scales. Three are of particular interest, viz., (i) magnetic molecules, which show a variety of quantum tunneling phenomena, and which have been the object of many studies in a search for coherent tunneling in the search for spin qubit systems; (ii) mesoscopic and nanoscopic conductors, which not only show interesting classical ‘spintronic’ phenomena, but also are predicted to show a variety of interesting quantum effects, including ‘spin Hall’ effects, which often depend on interesting topological properties of the underlying quantum states; and (iii) spins on metallic surfaces, which show fascinating many-body effects, and which can be investigated directly using STM imaging techniques. 5. LARGE-SCALE QUANTUM PHENOMENA IN MAGNETS: Tunneling of large-spin ions and molecules, and of magnetic solitons, has been seen already at the nanoscopic and mesoscopic scale, and further more exotic phenomena of this kind are predicted. Perhaps the most dramatic part of this ﬁeld is the search for large-scale entanglement, and the application of this to quantum computation. The key problem here, as in other kinds of quantum computation, is the understanding of both the mechanisms of decoherence, how to suppress decoherence, and how to understand its dynamics – most decoherence appears to come from a ‘spin bath’ of either two-level systems or nuclear spins. Magnetic systems are offering a rather unique window on these very general questions. One exciting new possibility has appeared in the idea of topological quantum computation, where the computation is embodied in the topological properties of a spin wave-function, and is almost immune to decoherence. A variety of different magnetic systems are being explored in this context, ranging from rare earth spins in simple compounds, large-spin magnetic molecules, and quantum dots, to more complicated nanofabricated spin architectures. All of these ﬁelds are evolving rapidly – as this volume was going to press, new discoveries were being made on graphene and on the behaviour of dipolar quantum spin glasses, and physicists were digesting the discovery of room-temperature Bose-Einstein condensation of magnons in YIG ﬁlms. However it is clear that the questions, issues, and techniques in these areas will be of central interest for many years to come.

Contents

A Gentle Introduction to the Functional Renormalization Group: The Kondo Effect in Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sabine Andergassen, Tilman Enss, Christoph Karrasch, and Volker Meden 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The fRG Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Green and Vertex Functions . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Functional Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Flow Equation Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Application to Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Single-Impurity Anderson Model . . . . . . . . . . . . . . . . 3.2 Flow Equation for the Self-Energy . . . . . . . . . . . . . . . . . . . 3.3 Flow Equation for the Fermion Interaction . . . . . . . . . . . . 3.4 Effect of Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Determining the Kondo Scale . . . . . . . . . . . . . . . . . . . . . . . 4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Simple View on the Quantum Hall System . . . . . . . . . . . . . . . . . . . . . . . . . Emil J. Bergholtz and Anders Karlhede 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1D Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Thin Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Gapped Fractions and Fractional Charge . . . . . . . . . . . . . . 3.2 The Non-Abelian Pfafﬁan State . . . . . . . . . . . . . . . . . . . . . . 3.3 The Half-Filled Landau Level . . . . . . . . . . . . . . . . . . . . . . . 4 Bulk Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Abelian States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Non-Abelian States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Gapless State at ν = 1/2 . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 4 6 8 8 10 12 13 13 16 16 19 19 21 22 23 24 25 28 28 31 31 32 32

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Halfvortices in Flat Nanomagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gia-Wei Chern, David Clarke, Hyun Youk, and Oleg Tchernyshyov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Exchange Limit of Flat Nanomagnets . . . . . . . . . . . . . . . . . . . . . . . . 3 Dipolar Limit of Flat Nanomagnets . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Halfvortices in Smectic Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin Structure and Dynamical Magnetic Response of Spin-Orbital Polarons in Lightly Doped Cobaltates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Daghofer, P. Horsch, and G. Khaliullin 1 Spin-Orbital Polarons in Nax CoO . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Corrections to the Ising Interactions in LiY1−x Hox F4 . . . . . . . . . A. Chin and P.R. Eastham 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Derivation of the Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 4 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Dipolar Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hyperﬁne Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Mixed Hyperﬁne-Dipolar Processes . . . . . . . . . . . . . . . . . . 5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Orbital-Lattice Physics in Ca-Based Ruthenates . . . . . . . . . . . . . . . . . Mario Cuoco, Filomena Forte, and Canio Noce 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Coulomb Interaction Versus c-axis Octahedral Distortions . . . . . . . 3.1 Fully Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Evolution from Flatten to Elongated Octahedra . . . . . . . . . 4 Interplay Between Spin-Orbit Coupling and Octahedral Compressive Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Thermal Evolution of Correlation Functions . . . . . . . . . . . . . . . . . . . 6 Relevance for Ruthenate Oxides Physics and Conclusions . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

Local Moment Approach to Multi-Orbital Anderson and Hubbard Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anna Kauch and Krzysztof Byczuk 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Local Moment Method in One Orbital SIAM . . . . . . . . . . . . . . . . . . 2.1 Mean Field Solution of the Single Impurity Anderson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Two Self-Energy Description . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Symmetry Restoration Ansatz . . . . . . . . . . . . . . . . . . . . . . . 2.4 Determining the Value of Local Moment . . . . . . . . . . . . . . 2.5 Ground State Energy in the Anderson Impurity Model . . . 2.6 LMA as a Conserving Approximation . . . . . . . . . . . . . . . . 3 Local Moment Approach for the Multi-Orbital SIAM . . . . . . . . . . . 3.1 LMA Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Symmetry Restoration and Determining the Local Moment Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Application to Multilevel Quantum Dots . . . . . . . . . . . . . . . . . . . . . . 4.1 V-LMA in Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Application to the Multi-Orbital Hubbard Model . . . . . . . . . . . . . . . 5.1 V-LMA Method in DMFT . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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High Field Level Crossing Studies on Spin Dimers in the Low Dimensional Quantum Spin System Na2 T2 (C2 O4 )3 (H2 O)2 with T = Ni, Co, Fe, Mn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 C. Mennerich, H.-H. Klauss, A.U.B. Wolter, S. S¨ullow, F.J. Litterst, C. Golze, R. Klingeler, V. Kataev, B. B¨uchner, M. Goiran, H. Rakoto, J.-M. Broto, O. Kataeva, and D.J. Price 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2 Synthesis and Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4 Na2 Ni2 (C2 O4 )3 (H2 O)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.1 Magnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2 High Field Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3 Speciﬁc Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5 Na2 Mn2 (C2 O4 )3 (H2 O)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.1 Magnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2 High Field Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6 Na2 Co2 (C2 O4 )3 (H2 O)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1 Magnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 High Field Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7 Na2 Fe2 (C2 O4 )3 (H2 O)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.1 Magnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.2 High Field Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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8 Magnetic Exchange Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Quantum Nanomagnets and Nuclear Spins: An Overview . . . . . . . . . . . . . 125 Andrea Morello 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4 Open Questions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . 135 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Quantum Dimer Models and Exotic Orders . . . . . . . . . . . . . . . . . . . . . . . . . 139 K.S. Raman, E. Fradkin, R. Moessner, S. Papanikolaou, and S.L. Sondhi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 2 Basic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3 Modulated States and the Devil’s Staircase . . . . . . . . . . . . . . . . . . . . 142 4 SU(2)-Invariant Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Imaging Transverse Electron Focusing in Semiconducting Heterostructures with Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 151 Andr´es A. Reynoso, Gonzalo Usaj, and C.A. Balseiro 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2 Transverse Electron Focusing in Presence of Strong Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3 Imaging Techniques in Transverse Focusing with Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Spectroscopic Analysis of Finite Size Effects Around a Kondo Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Pascal Simon and Denis Feinberg 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 2 Model Hamiltonian and Kondo Temperature . . . . . . . . . . . . . . . . . . . 165 3 Spectroscopy of a Kondo Quantum Dot Coupled to an Open Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4 Analysis of Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5 Finite Box Coulomb Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

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Effective Magnus Force on a Magnetic Vortex . . . . . . . . . . . . . . . . . . . . . . . 175 L.R. Thompson and P.C.E. Stamp 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 2 System and Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4 Many Vortex Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Anisotropic Exchange in Spin Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Dmitry Zakharov, Hans-Albrecht Krug von Nidda, Mikhail Eremin, Joachim Deisenhofer, Rushana Eremina, and Alois Loidl 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 2 Microscopic Theory of Exchange Interaction . . . . . . . . . . . . . . . . . . 195 2.1 Exchange Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 2.2 Isotropic Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . 198 2.3 Anisotropic Exchange Interaction . . . . . . . . . . . . . . . . . . . . 205 3 Electron Spin Resonance in Strongly Exchange Coupled Spin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 3.1 Principle of Magnetic Resonance . . . . . . . . . . . . . . . . . . . . 211 3.2 Magnetic Resonance of Interacting Spins . . . . . . . . . . . . . . 212 3.3 One-Dimensional Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . 214 4 Anisotropic Exchange in Low-Dimensional Systems . . . . . . . . . . . . 215 4.1 Universal Relaxation Behavior . . . . . . . . . . . . . . . . . . . . . . 215 4.2 Analysis of the Exchange Paths . . . . . . . . . . . . . . . . . . . . . . 220 4.3 Inﬂuence of the Phase Transition . . . . . . . . . . . . . . . . . . . . . 227 5 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 A Review of Bose-Einstein Condensation in Certain Quantum Magnets Containing Cu and Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Vivien S. Zapf 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 2 Boson Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 3 Experimental Investigation into Bose-Einstein Condensation . . . . . 245 4 Frustration and Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . 246 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

A Gentle Introduction to the Functional Renormalization Group: The Kondo Effect in Quantum Dots Sabine Andergassen, Tilman Enss, Christoph Karrasch, and Volker Meden

Abstract The functional renormalization group provides an efﬁcient description of the interplay and competition of correlations on different energy scales in interacting Fermi systems. An exact hierarchy of ﬂow equations yields the gradual evolution from a microscopic model Hamiltonian to the effective action as a function of a continuously decreasing energy cutoff. Practical implementations rely on suitable truncations of the hierarchy, which capture nonuniversal properties at higher energy scales in addition to the universal low-energy asymptotics. As a speciﬁc example we study transport properties through a single-level quantum dot coupled to Fermi liquid leads. In particular, we focus on the temperature T = 0 gate voltage dependence of the linear conductance. A comparison with exact results shows that the functional renormalization group approach captures the broad resonance plateau as well as the emergence of the Kondo scale. It can be easily extended to more complex setups of quantum dots.

1 Introduction The coupling of a quantum dot with spin degenerate levels and local Coulomb interaction (modeled, e.g., by the single-impurity Anderson model; SIAM) to metallic leads gives rise to Kondo physics [1]. At low temperatures and for sufﬁciently high S. Andergassen Max-Planck-Institut f¨ur Festk¨orperforschung, D-70569 Stuttgart, Germany, and Institut N´eel / CNRS - UJF, 25, rue des Martyrs - BP 166, 38042 Grenoble, France T. Enss INFM–SMC–CNR and Dipt. di Fisica, Universit`a di Roma “La Sapienza”, P.le Aldo Moro 5, I-00185 Roma, Italy C. Karrasch and V. Meden Institut f¨ur Theoretische Physik, Universit¨at G¨ottingen, Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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barriers the local Coulomb repulsion U leads to a broad resonance plateau in the linear conductance G of such a setup as a function of a gate voltage Vg which linearly shifts the level positions [2, 3, 4, 5, 6]. It replaces the Lorentzian found for noninteracting electrons. On resonance the dot is half-ﬁlled implying a local spin1 2 degree of freedom responsible for the Kondo effect [1]. For the SIAM the zero temperature conductance is proportional to the one-particle spectral weight of the dot at the chemical potential [7]. The appearance of the plateau in the conductance is due to the pinning of the Kondo resonance in the spectral function at the chemical potential for −U/2 Vg U/2 (here Vg = 0 corresponds to the half-ﬁlled dot case) [1, 6]. Kondo physics in transport through quantum dots was conﬁrmed experimentally [8, 9], and theoretically using the Bethe ansatz [2, 6] and the numerical renormalization-group (NRG) technique [10, 11]. However, both methods can hardly be used to study more complex setups. In particular, the extension of the NRG to more complex geometries beyond single-level quantum-dot systems [5, 6, 12] is restricted by the computational complexity which increases sharply with the number of interacting degrees of freedom. Alternative methods which allow for a systematic investigation are therefore required. We here propose the functional renormalization group (fRG) approach to study low-temperature transport properties through mesoscopic systems with local Coulomb correlations. A particular challenge in the description of quantum dots is their distinct behavior on different energy scales, and the appearance of collective phenomena at new energy scales not manifest in the underlying microscopic model. An example of this is the Kondo effect where the interplay of the localized electron spin on the dot and the spin of the lead electrons leads to an exponentially (in U) small scale TK . This diversity of scales cannot be captured by straightforward perturbation theory. One tool to cope with such systems is the renormalization group: by treating different energy scales successively, one can often ﬁnd an efﬁcient description for each one. We will here give an introduction to one particular variant, the fermionic fRG [13], which is formulated in terms of an exact hierarchy of coupled ﬂow equations for the vertex functions as the energy scale is lowered. The ﬂow starts directly from the microscopic model, thus including nonuniversal effects on higher energy scales from the outset, in contrast to effective ﬁeld theories capturing only the asymptotic behavior. As the cutoff scale is lowered, ﬂuctuations at lower energy scales are successively included until one ﬁnally arrives at the full effective action (the generating functional of the one-particle irreducible vertex functions) from which all physical properties can be extracted. This allows to control infrared singularities and competing instabilities in an unbiased way. Truncations of the ﬂow equation hierarchy and suitable parametrizations of the frequency and momentum dependence of the vertex functions lead to powerful new approximation schemes, which are justiﬁed for weak renormalized interactions. A comparison with exact results shows that the fRG is remarkably accurate even for sizeable interactions. The outline of this article is as follows. In Section 2 we introduce the fRG formalism and derive the hierarchy of ﬂow equations. The implementation of the fRG technique for a quantum dot modeled by a SIAM is described in Section 3. In the following we present results for the linear conductance including a comparison

A Gentle Introduction to the funRG: The Kondo Effect in Quantum Dots

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with exact solutions and discuss the emergence of the Kondo scale. We conclude in Section 4 with a summary and outlook on further applications and extensions of the present work.

2 The fRG Technique To make this article self-contained, this section aims to give a very short introduction into the fRG formalism by deriving the functional ﬂow equations. Readers not so much interested in the formal beauty might jump ahead to the following Section 3 where these ﬂow equations are applied to a speciﬁc physical model. An introduction to the many-body formalism used here can be found, e.g., in [14]; for the details of the derivation of the functional ﬂow equation see, e.g., [15, 16] and references therein. Exact functional ﬂow equations were derived for bosonic ﬁeld theories in [17, 18, 19] and for fermionic ﬁelds in the one-particle irreducible scheme in [13].

2.1 Green and Vertex Functions We consider a system of interacting fermions described by Grassmann variables ψ , ψ¯ , and an action S[ψ , ψ¯ ] = ψ¯ ,C−1 ψ − V [ψ , ψ¯ ] (1) with bare propagator C(K) =

1 iε − ξk

(2)

where the index K = (ε , k, σ ) collects the Matsubara frequency ε and the quantum numbers of a suitable single-particle basis set, e.g., momentum k and spin projection σ , and ξk = εk − µ denotes the energy relative to the chemical potential. In order to understand the general structure of the ﬂow equation it is useful to include a Nambu (particle/hole) index into K: then each directed fermion line denotes the propagation of either a particle or a hole [13]. The inner productimplies a summation over these indices: for our diagonal propagator, ψ¯ ,C−1 ψ = ∑K ψ¯ K CK−1 ψK . V [ψ , ψ¯ ] is an arbitrary many-body interaction; we will see a speciﬁc example for this below in Eq. (17). Connected Green functions can be obtained from the generating functional [14] ¯ ] −(ψ¯ ,η )−(η¯ ,ψ ) S[ ψ , ψ ¯ e G [η , η¯ ] = −log N Dψψ (3) e by taking derivatives with respect to the source ﬁelds η :

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Gm (K1 , . . . , Km ; K1 , . . . , Km ) = −ψK . . . ψKm ψ¯ Km . . . ψ¯ K1 conn 1

δm δm = G [η , η¯ ] . δ ηK1 . . . δ ηKm δ η¯ Km . . . δ η¯ K1 η =η¯ =0

Equivalently, G [η , η¯ ] can be expanded in powers of the source ﬁelds with expansion coefﬁcients Gm (K1 , . . . , Km ). The normalization factor N = detC cancels the noninteracting vacuum diagrams, such that G [0, 0] = 0 in the absence of interaction. As we explain below, it will be of advantage in our case to describe the system not by connected Green functions but by the one-particle irreducible (1PI) vertex functions. Their generating functional, the effective action, is obtained from G by Legendre transformation, Γ [φ , φ¯ ] = G [η , η¯ ] + φ¯ , η − (η¯ , φ ). (4) This functional can again be expanded in powers of the ﬁelds φ , φ¯ to obtain the vertex functions γm (K1 , . . . , Km ). The usual relations between G and Γ hold, i.e., φ = δ G /δ η¯ , φ¯ = δ G /δ η as well as η¯ = δΓ /δ φ , η = δΓ /δ φ¯ , and δ 2Γ /δ φ δ φ¯ = (δ 2 G /δ ηδ η¯ )−1 .

2.2 Functional Flow Equations We introduce an infrared cutoff into the bare propagator that suppresses all soft modes, which may be a source of divergences in perturbation theory. In bulk systems it is convenient to use a momentum cutoff, which suppresses momenta close to the Fermi surface. On the other hand, if translation invariance is spoiled by impurities or a particular spatial setup as for quantum dots, it is easier to use a frequency cutoff, excluding all Matsubara frequencies below scale Λ using a step function Θ (x): CΛ (K) =

Θ (|ε | − Λ ) . iε − ξk

(5)

This changes the microscopic model to exclude soft modes; in the limit Λ → 0 the original model is recovered. The easiest way to understand how the Green functions change with Λ is to derive the ﬂow equation for the cutoff-dependent generating functional ψ¯ ,QΛ ψ )−V [ψ ,ψ¯ ] −(ψ¯ ,η )−(η¯ ,ψ ) Λ Λ ( ¯ ¯ (6) Dψψ e e G [η , η ] = − log N with (CΛ )−1 ≡ QΛ , and the normalization factor N derivative and denoting ∂Λ QΛ = Q˙ Λ ,

Λ

= (det QΛ )−1 . Taking the Λ

A Gentle Introduction to the funRG: The Kondo Effect in Quantum Dots

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Λ ¯ Dψψ ψ¯ , Q˙ Λ ψ e(ψ¯ ,Q ψ )−V [ψ ,ψ¯ ]−(ψ¯ ,η )−(η¯ ,ψ ) e

Λ δ ˙Λ δ Λ GΛ e−G = −∂Λ log N + e ,Q δη δ η¯

2G Λ δ G Λ δ δGΛ + (7) = Tr(Q˙ Λ CΛ ) − Tr Q˙ Λ , Q˙ Λ δ η δ η¯ δη δ η¯

∂Λ G Λ = −∂Λ log N

Λ

−

1

−G Λ

where the ﬁrst term, −∂Λ log N Λ = Tr(Q˙ Λ CΛ ), comes from the derivative of the normalization factor, and the trace denotes a sum over all K. In the present application to quantum dots, the one-particle potential is strongly renormalized, and it is important to include the feedback of this renormalization nonperturbatively in the ﬂow equations. This is most easily achieved in the one-particle irreducible (1PI) scheme, where the one-particle renormalizations are included in the internal propagators. Hence, we perform a Legendre transform which now also depends on Λ , (8) Γ Λ [φ , φ¯ ] = G Λ [η Λ , η¯ Λ ] + φ¯ , η Λ − η¯ Λ , φ , and we ﬁnd ∂Λ Γ Λ = dG Λ /dΛ + (φ¯ , ∂Λ η Λ ) − (∂Λ η¯ Λ , φ ) = ∂Λ G Λ after taking into account the derivatives also of the η Λ ﬁelds. The fundamental variables of Γ Λ are the φ ﬁelds and the η acquire a Λ dependence via the relation between φ and η [14]. By Legendre transform of the ﬂow Eq. (7), we obtain the 1PI ﬂow equation δ 2Γ Λ [φ , φ¯ ] −1 ¯ , Q˙ Λ φ . + ∂Λ Γ Λ [φ , φ¯ ] = ∂Λ G Λ = TrQ˙ Λ CΛ − φ δ φ δ φ¯

(9)

The inversion of a functional of Grassmann variables with a nonzero complex part is well deﬁned as a geometric series, which involves only products of Grassmann variables. Therefore, we write the Hessian of Γ Λ as

∂ 2Γ Λ ˜Λ ¯ = (GΛ )−1 K,K + ΓK,K [φ , φ ], ∂ φK ∂ φ¯K

(10)

where the inverse full propagator (GΛ )−1 = γ1Λ = QΛ − Σ Λ according to the Dyson equation, and Γ˜ Λ depends at least quadratically on the φ , φ¯ ﬁelds. This decomposition allows us to write the inverse as a geometric series,

∂ 2Γ Λ ∂ φK ∂ φ¯K

−1

= (1 + GΛ Γ˜ Λ )−1 GΛ = 1 − GΛ Γ˜ Λ + [GΛ Γ˜ Λ ]2 − · · · GΛ ,

which we insert into the ﬂow Eq. (9),

∂Λ Γ Λ = Tr(Q˙ Λ [CΛ − GΛ ]) + (φ¯ , Q˙ Λ φ ) + Tr(SΛ [Γ˜ Λ − Γ˜ Λ GΛ Γ˜ Λ + · · · ]).

(11)

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The ﬁrst term describes the ﬂow of the zero-point function (grand canonical potential), while the second term expresses the change of the bare propagator with Λ . The hierarchy of ﬂow equations is encoded in the third term which represents a loop with an arbitrary number of vertices Γ˜ Λ , which contribute at least two external legs each, connected by full propagators GΛ and one single-scale propagator SΛ which has support only at frequency Λ , SΛ = GΛ Q˙ Λ GΛ .

(12)

Given the initial conditions at Λ = ∞, the ﬂow equations determine Γ Λ uniquely for all Λ < ∞. The ﬂow of Γ Λ thus interpolates between the bare action Γ Λ =∞ = S and the full solution of the problem, Γ Λ =0 = Γ .

2.3 Flow Equation Hierarchy Expanding the functional ﬂow equation for Γ Λ in powers of the source ﬁelds φ , φ¯ , we obtain an inﬁnite hierarchy of coupled ﬂow equations for the m-particle vertex functions γmΛ . This hierarchy can be represented diagrammatically and the ﬁrst three levels are shown in Fig. 1, where each line represents the propagation of either a particle or a hole. For instance, the last diagram in the second line includes both a particle-hole and a particle-particle bubble. For our application to the SIAM it is convenient to distinguish these,1 so we denote by γ2 (1 , 2 ; 1, 2) the two-particle vertex with incoming electrons 1, 2 and outgoing electrons 1 , 2 . The ﬁrst line of Fig. 1 represents the ﬂow equation for the self-energy Σ Λ ,

∂ Λ + Σ (1 , 1) = −T ∑ eiε2 0 SΛ (2, 2 ) γ2Λ (1 , 2 ; 1, 2), ∂Λ 2,2

(13)

where the labels 1, 1 , . . . are a shorthand notation for K1 , K1 etc., and the summation includes Matsubara frequencies. The two-particle vertex γ2Λ entering Eq. (13) is determined by the second line in Fig. 1, which in turn depends on the three-particle vertex γ3Λ , etc. This inﬁnite system of coupled differential equations can usually not be integrated analytically; therefore one has to resort to numerical computations which require to truncate the hierarchy by neglecting the ﬂow of the higher vertices. This may be justiﬁed perturbatively for weak interactions since the higher vertices not present in the bare interaction are generated only at higher orders in the effective interaction. As a ﬁrst step, we consider only the ﬂow of the self-energy with all higher vertices remaining at their initial conditions (the bare interaction V ): in our application, this already produces qualitatively the correct result. As a second step, we also include the ﬂow of γ2Λ but neglect γ3Λ , which brings us even quantitatively within a few percent of the known results: 1

A derivation of the ﬂow equation using separate particle and hole propagators from the outset can be found, e.g., in [20].

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SΛ ∂ ∂Λ

= ΣΛ

γ2Λ SΛ SΛ

∂ ∂Λ

= γ2Λ

+

GΛ γ2Λ

γ3Λ

γ2Λ

SΛ SΛ ∂ ∂Λ

= γ3Λ

+

GΛ

+

GΛ γ2Λ

γ4Λ

SΛ γ3Λ

GΛ

Fig. 1 Diagrammatic representation of the ﬂow equations for the self-energy Σ Λ , the two-particle vertex γ2Λ and the three-particle vertex γ3Λ in the 1PI formulation of the fRG

∂ Λ γ (1 , 2 ; 1, 2) = T ∂Λ 2

∑ ∑ GΛ3,3 SΛ4,4

γ2Λ (1 , 2 ; 3, 4) γ2Λ (3 , 4 ; 1, 2)

3,3 4,4

− γ2Λ (1 , 4 ; 1, 3)γ2Λ (3 , 2 ; 4, 2)

+ γ2Λ (2 , 4 ; 1, 3) γ2Λ (3 , 1 ; 4, 2) + (3 ↔ 4, 3 ↔ 4 ) ,

(14)

with the ﬁrst term representing the particle-particle channel and the other two the particle-hole channels, respectively. The last bracket means that the two terms in the second line have to be repeated with the changes of variables as indicated. There is one technical detail which allows to simplify these equations signiﬁcantly: the sharp frequency cutoff in Eq. (5) has the advantage that the frequency integrals on the right-hand side of Eqs. (13) and (14) can be performed analytically. The propagators contain both Θ (|ε | − Λ ) and δ (|ε | − Λ ) = −∂Λ Θ (|ε | − Λ ) functions, which at ﬁrst look ambiguous because the step function has a jump exactly where the δ function has support. However, if one smoothes the step slightly and makes a change of variables from Λ to t = Θ (|ε | − Λ ) and dt = −δ (|ε | − Λ )d Λ in the vicinity of |ε | = Λ , the integration over the step becomes well-deﬁned and is implemented by the substitution [15, 21]

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δ (x − Λ ) f Θ (x − Λ ) → δ (x − Λ )

1 0

f (t)dt.

(15)

Using this substitution, the product of full and single-scale propagators around the loop is replaced by a delta function2 δ (|ε | − Λ ) times smooth propagators of the form Λ −1 G˜ Λ = [G−1 0 −Σ ] .

(16)

Note that the explicit Λ dependence remains only in the self-energy, such that the resulting propagator (and hence the ﬂow equation) is smooth in both Λ and ε and can be easily integrated numerically.

3 Application to Quantum Dots In order to give a pedagogical example how the functional ﬂow equations can be used to solve an interesting physical problem, we choose as a toy model the SIAM [3, 4], which is used to study, e.g., transport through a quantum dot. After integrating out the noninteracting lead degrees of freedom (see below) this model has zero space dimensions, hence spin is the only quantum number, and the resulting ﬂow equations are particularly simple and can even be integrated analytically if the ﬂow of the two-particle vertex is neglected. Nevertheless, the physics is nontrivial, and the exponentially small scale (the Kondo scale) we obtain is seen neither in perturbation theory in U (which in the present example is free of infrared divergencies) nor in self-consistent Hartree-Fock calculations.

3.1 The Single-Impurity Anderson Model Our model consists of a single site with local Coulomb repulsion U ≥ 0, which is connected to two leads l = L, R via tunnel barriers tL,R (see Fig. 2): H = U(n↑ − 12 )(n↓ − 12 ) + ∑ εσ dσ† dσ − ∑ tl (dσ† c0,l + H.c.), σ

(17)

σ ,l

where dσ† is the creation operator for an electron with spin σ on the dot, nσ = dσ† dσ is the spin-σ number operator, and c†l denotes the creation operator at the end of the semi-inﬁnite lead l. The leads may be modeled by tight-binding chains, Hl = † −t ∑∞ m=0 (cm,l cm+1,l + h.c.) with hopping amplitude t. The hybridization of the dot with the leads broadens the levels on the dot by Γl (ε ) = π tl2 ρl (ε ), where ρl (ε ) is the local density of states at the end of lead l which we henceforth assume to be

2

There remain additional frequency constraints if not all propagators are at the same frequency.

A Gentle Introduction to the funRG: The Kondo Effect in Quantum Dots Fig. 2 A quantum dot connected to two leads (reservoirs)

9

U L

tL

tR

R

constant (wide-band limit). A local magnetic ﬁeld h at the dot site splits the spin-up and spin-down energy levels around the gate voltage Vg (σ↑↓ = ±1),

ε↑↓ = Vg + σ h/2.

(18)

Since the leads are noninteracting, we can integrate out the degrees of freedom of the lead electrons in the path integral and thereby obtain a hybridization contribution Γ = ΓL + ΓR to the bare Green function of the dot which depends only on the sign of the Matsubara frequency [1], G0,σ (iε ) =

1 . iε − (Vg + σ h/2) + iΓ sgn(ε )

(19)

By solving the interacting many-body problem we obtain a frequency-dependent self-energy Σσ (iε ) on the dot, and the full dot propagator is given by the Dyson equation, Gσ (iε ) = [G0,σ (iε )−1 − Σσ (iε )]−1 .

(20)

This self-energy could be computed by perturbation theory in the strength of the Coulomb interaction, e.g., in a Hartree-Fock calculation, but it turns out that this does not capture the physical effect that is observed in experiments and that we wish to describe. Instead, we will now show how the fRG can be used to compute with very little effort an approximation for the self-energy that leads to the correct physical properties of the T = 0 conductance. In general the linear response conductance is given by the current-current correlation function (Kubo formula). At zero temperature, zero bias voltage, and for a single interacting site the exact conductance has the simple form [7] G(Vg ) = G↑ (Vg ) + G↓ (Vg ) Gσ (Vg ) =

e2 h

(21)

πΓ ρσ (0)

in terms of the dot spectral function continued analytically to real frequencies, 1 ρσ (ε ) = − ImGσ (ε + i0+). π

(22)

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3.2 Flow Equation for the Self-Energy In order to implement the fRG, following Eq. (5) we introduce a cutoff in Matsubara frequency into the bare propagator on the dot (19), GΛ0,σ (iε ) =

Θ (|ε | − Λ ) . iε − (Vg + σ h/2) + iΓ sgn(ε )

(23)

This is an infrared cutoff which sets the propagator to zero for frequencies smaller than Λ (preventing these modes from being excited), but leaves the high-energy modes unchanged (propagating). As the cutoff scale Λ is gradually lowered, more and more low-energy degrees of freedom are included, until ﬁnally the original model is recovered for Λ → 0. Changing the cutoff scale leads to the inﬁnite hierarchy of ﬂow equations for the vertex functions shown in Fig. 1. We use the ﬂow equation for the self-energy (13) with the single-scale propagator substituted by (16). The two-particle vertex γ2Λ is in general a complicated function of three independent frequencies (justifying the name functional renormalization group) that evolves during the ﬂow by the second line of Fig. 1. As a ﬁrst approximation we neglect the ﬂow of the two-particle vertex and all higher vertex functions. This can be justiﬁed perturbatively if the bare coupling U is small, as all the terms generated during the ﬂow are of higher order in U. As a consequence, the two-particle vertex

γ2Λ (iεσ , iε σ¯ ; iεσ , iε σ¯ ) = U Λ

(24)

remains for all Λ at its initial condition, U Λ ≡ U Λ =∞ = U which is the bare Coulomb interaction in the Hamiltonian (17). We denote σ¯ = −σ . As γ2Λ does not depend on frequency in our approximation, also the self-energy does not acquire a frequency dependence during the ﬂow. The ﬂow equation for the effective level position VσΛ = Vg + σ h/2 + ΣσΛ is

∂Λ VσΛ

UΛ =− 2π =

(Λ

Λ

U d ε δ (|ε | − Λ ) G˜ Λσ¯ (iε ) = − 2π

∑

ε =±Λ

U Λ VσΛ ¯ /π 2 + Γ )2 + (VσΛ ¯ )

1 iε − VσΛ + iΓ sgnε ¯ (25)

with initial condition VσΛ =∞ = Vg + σ h/2 [22]. At the end of the ﬂow, the renormalized potential Vσ = VσΛ =0 determines the dot spectral function (22),

ρσ (ε ) =

1 Γ , π (ε − Vσ )2 + Γ 2

(26)

which is a Lorentzian of full width 2Γ and height 1/πΓ centered around Vσ . Although the true spectral function has a very different form with a very sharp Kondo peak at ε = Vσ , this difference is not seen in the T = 0 conductance (21): it probes only the value of the spectral function at the chemical potential ε = 0,

A Gentle Introduction to the funRG: The Kondo Effect in Quantum Dots

2

G/(e /h)

2

11

U/Γ=1 U/Γ=10 U/Γ=25

1

n

0 2 1 0 -4

-2

0 Vg/U

2

4

Fig. 3 Upper panel: conductance as a function of gate voltage for different values of U/Γ and h = 0. Lower panel: average number of electrons on the dot

Gσ (Vg ) =

e2 Γ 2 . h Vσ2 + Γ 2

(27)

In the noninteracting case, Vσ = Vg + σ h/2 and the conductance is a sum of two Lorentzians of the applied gate voltage. If interaction is switched on, this changes drastically. In Fig. 3 the conductance G as a function of gate voltage Vg for the SIAM is shown for different values of U/Γ in the upper panel, together with the occupation of the dot in the lower, both for the case without magnetic ﬁeld, implying V↑Λ = V↓Λ = V Λ and G↑ = G↓ . For Γ U the resonance exhibits a plateau [6]. In this region the occupation is close to 1 while it sharply rises/drops to 2/0 to the left/right of the plateau. Also for asymmetric barriers we reproduce the exact resonance height 4ΓLΓR /(ΓL + ΓR)2 (2e2 /h) [1, 6]. Here we focus on strong couplings U/Γ 1. The solution of the differential Eq. (25) at Λ = 0 is obtained in implicit form J0 (vg ) vJ1 (v) − γ J0(v) = , vY1 (v) − γ Y0 (v) Y0 (vg )

(28)

with v = V π /U, vg = Vg π /U, γ = Γ π /U, and Bessel functions Jn , Yn . For |Vg | < Vc this equation has a solution with small |V |, where vc = Vc π /U is the ﬁrst zero of J0 corresponding to Vc 0.77U. For U Γ the crossover to a solution with |V | being of order U (for |Vg | > Vc ) is fairly sharp. Expanding both sides of Eq. (28) for small

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2

G/(e /h)

2

1

0 -1

0 Vg/U

1

Fig. 4 Gate voltage dependence of the conductance at U/Γ = 4π and h = 0. Solid line: exact Bethe ansatz solution from [6]. Dashed line: fRG approximation without ﬂow of the interaction. Dotted line: fRG approximation with ﬂow of vertex

|v| and |vg | gives

U . V = Vg exp − πΓ

(29)

The consequent exponential pinning of the spectral weight in Eq. (22) at the chemical potential for small |Vg | and the sharp crossover to a V of order U when |Vg | > Vc leads to the observed resonance line shape represented by the dashed line in Fig. 4. For U Γ the width of the plateau is 2Vc 1.5U, which is larger than the width U found with the Bethe ansatz [2, 6] corresponding to the solid line in Fig. 4. Our approximation furthermore slightly overestimates the sharpness of the box-shaped resonance. The inclusion of the renormalization of the two-particle vertex improves the quantitative accuracy of the results considerably (see below), while the pinning of the spectral function and the subsequent resonance plateau is captured already at the ﬁrst order of the ﬂow-equation hierarchy, even though the spectral function neither shows the narrow Kondo resonance nor the Hubbard bands.

3.3 Flow Equation for the Fermion Interaction With the simple approximation above, we have obtained an exponentially small scale for the local potential from a renormalization group ﬂow equation. This yields a plateau in the conductance that qualitatively agrees already well with the exact Bethe ansatz solution. We will now demonstrate how one can go beyond the simplest approximations in the renormalization group ﬂow equations to improve (systematically for small U) the result and the agreement with the known solutions. Any improvement of the ﬂow equation has to come from a more detailed parametrization

A Gentle Introduction to the funRG: The Kondo Effect in Quantum Dots

13

of the two-particle vertex. The next step is to include the ﬂow of the two-particle vertex γ2Λ from the second line in Fig. 1, but still neglect the three-particle vertex γ3Λ and the frequency dependence of γ2Λ 3 . Using the vertex ﬂow Eq. (14) with a sharp cutoff and the parametrization (24), we obtain [22] ∂ Λ (U Λ )2 G˜ Λ↑ (iε ) G˜ Λ↓ (−iε ) + G˜ Λ↑ (iε ) G˜ Λ↓ (iε ) U = ∑ ∂Λ 2π ε =±Λ 2 2 U Λ V↑Λ V↓Λ /π , = (Λ + Γ )2 + (V↑Λ )2 (Λ + Γ )2 + (V↓Λ )2

(30)

again with initial condition U Λ =∞ = U. A systematic improvement with respect to the previous results can be observed in Fig. 4, with the dotted line resulting from the inclusion of the interaction renormalization. The quantitative agreement with the exact results is excellent and holds for U/Γ = 25, the largest value with available Bethe ansatz data [6]. Also for more complex dot geometries this extension considerably improves the agreement with NRG results [22]. We note that a truncated fRG scheme including the full frequency dependence of the two-particle vertex, and hence a frequency dependent self-energy, reproduces also the Kondo resonance and Hubbard bands in the spectral function [20]. This requires, however, a substantial computational effort.

3.4 Effect of Magnetic Fields We next consider the case of ﬁnite magnetic ﬁelds. For h > 0 the Kondo resonance in the NRG solution of the spectral function splits into two peaks with a dip at ω = 0, resulting in a dip of G(Vg ) at Vg = 0. In Figs. 5 and 6 we compare the total G = G↑ + G↓ and partial G↑ conductance obtained from the above fRG truncation scheme including the ﬂow of the effective interaction with NRG [23] results for different h expressed in units of TKNRG = 0.116Γ , where TKNRG is the width of the Kondo resonance at the particle-hole symmetric point Vg = 0 obtained by NRG [23]. The excellent agreement between NRG and fRG results provides strong evidence of the presence of the Kondo scale within the truncated fRG scheme.

3.5 Determining the Kondo Scale From the above comparison of fRG and NRG data shown in Fig. 6 for different h we infer the appearance of an exponentially small energy scale deﬁning the Kondo 3 Including the frequency dependence of γ Λ is an ambitious project that is beyond the scope of 2 this introductory article [20].

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2

G/(e /h)

2

1

0 -1

0 Vg/U

1

Fig. 5 Gate voltage dependence of the total conductance G of a single dot with U/Γ = 3π and h/Γ = 0, 0.058, 0.116, 0.58 from top to bottom. In units of the Vg = 0 Kondo temperature TKNRG /Γ = 0.116 these ﬁelds correspond to h = 0, 0.5TKNRG , TKNRG , and 5TKNRG . Solid line: NRG data from [23]. Dashed lines: fRG approximation with ﬂow of vertex

2

G /(e /h)

1

0.5

0 -1

0 Vg/U

1

Fig. 6 Gate voltage dependence of the partial conductance G↑ of a single dot for the same parameters as in Fig. 5

scale TK (U,Vg ) by the magnetic ﬁeld required to suppress the total conductance to one half of the unitary limit, G(U,Vg , h = TK ) = e2 /h. For ﬁxed U Γ this deﬁnition applies for gate voltages within the resonance plateau for h = 0. In Fig. 7 we show TK (U,Vg ) for different Vg as a function of U on a linear-log scale. The curves can ﬁtted to a function of the form

U Γ (31) f (U/Γ ) = a exp − b − c Γ U with Vg -dependent coefﬁcients a, b, and c. The above form is consistent with the exact Kondo temperature TK [1] that depends exponentially on a combination of U and the level position

TK/Γ

10 10 10 10

c(Vg)

A Gentle Introduction to the funRG: The Kondo Effect in Quantum Dots

-1

20 0 0

-2

15

10 2 (Vg/ Γ )

-3

-4

10

20 U/Γ

30

Fig. 7 The Kondo scale TK as a function of U for different Vg : Vg = 0 (solid), Vg /Γ = 1 (short dashed), Vg /Γ = 2 (dotted), and Vg /Γ = 4 (long dashed). Inset: The ﬁtting parameter c as a function of (Vg /Γ )2

π 2 U /4 − Vg2 TKexact ∼ exp − Γ 2U π U π V2 Γ g − = exp − . 8 Γ 2 Γ2 U

(32)

The prefactor of the exponential depends on the details of the model considered. To leading order its U and Vg dependence can be neglected. For the ﬁt to Eq. (31) we ﬁnd b(Vg ) ≈ 0.32 for all Vg (see Eq. (29) for comparison), in good agreement with the exact value π /8 ≈ 0.39. The prefactor a depends only weakly on Vg , and c(Vg ) increases approximately quadratically with Vg /Γ as shown in the inset of Fig. 7, according to the behavior of the exact Kondo temperature Eq. (32). We thus conclude that TK (U,Vg ) can be determined from the h dependence of the conductance obtained from the fRG. Similar results are obtained from the local spin susceptibility [1]. For the single dot at T = 0 the exact conductance, transmission phase, and dot occupancies are directly related by a generalized Friedel sum rule [1]: Gσ /(e2 /h) = sin2 (π nσ ), and the transmission phase ασ = π nσ . As 0 ≤ nσ ≤ 1 the argument of sin2 is restricted to a single period and the relation between Gσ , nσ , and ασ is unique. In many approximation schemes the Friedel sum rule does not hold exactly. In contrast, within our method we map the many-body problem onto an effective single-particle one for which the Friedel sum rule is fulﬁlled. For gate voltages within the h = 0 conductance plateau the (spin independent) dot ﬁlling is 1/2 and the (spin independent) phase is π /2. For sufﬁciently large U/Γ the crossover to nσ = 1 and ασ = π to the left of the plateau as well as nσ = 0 and ασ = 0 to the right is fairly sharp.

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4 Summary and Conclusion We presented an fRG scheme developed for the study of electronic transport through quantum-dot systems. The resulting differential ﬂow equations describe the effective renormalized level position and interaction in presence of local Coulomb interactions and magnetic ﬁelds. Analytical estimates capture signatures of the Kondo effect, and a comparison with exact Bethe ansatz and high-precision NRG results shows excellent agreement up to the largest Coulomb interaction for which Bethe ansatz or NRG data are available. The presented fRG scheme constitutes a reliable and promising tool in the investigation of correlation effects and interference phenomena in quantum-dot systems. The application to a single quantum dot can be directly extended to different geometries involving more complicated setups [22] such as parallel double dots [24]. In [25] this method was used to investigate the long-standing phase lapse puzzle of the transmission phase through multi-level quantum dots. Furthermore, the truncated fRG scheme was sucessfully used to study onedimensional fermionic lattice models with two-particle interaction and impurities [26, 27, 28, 29] (inhomogeneous Luttinger liquids). In addition to the universal low-energy asymptotics the fRG captures nonuniversal properties at higher energy scales. Novel low-energy ﬁxed points were found for Y-junctions of onedimensional quantum wires pierced by a magnetic ﬂux [30]. Also a single dot with Luttinger liquid leads [31] was investigated. For the latter the competition between Kondo and Luttinger liquid physics leads to a broad resonance plateau for all experimentally accessible length scales, whereas the low-energy ﬁxed point is described by a sharp resonance characteristic for the Luttinger liquid behavior. Acknowledgements We thank X. Barnab´e-Th´eriault, R. Hedden, W. Metzner, Th. Pruschke, H. Schoeller, U. Schollw¨ock, and K. Sch¨onhammer for useful discussions. We thank T. Costi and J. von Delft for providing their NRG and Bethe ansatz data. This work has been supported by the French ANR (program PNANO) (S.A.), the Deutsche Forschungsgemeinschaft (SFB 602) (V.M.), and a Feoder Lynen fellowship of the Alexander von Humboldt foundation and the Istituto Nazionale di Fisica della Materia–SMC–CNR (T.E.).

References 1. A.C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, UK, 1993) 2. A.M. Tsvelik, P.B. Wiegmann, Adv. Phys. 32, 453 (1983) 3. L. Glazman, M. Raikh, JETP Lett. 47, 452 (1988) 4. T.K. Ng, P.A. Lee, Phys. Rev. Lett. 61, 1768 (1988) 5. T.A. Costi, A.C. Hewson, V. Zlati´c, J. Phys.: Condens. Matter 6, 2519 (1994) 6. U. Gerland, J. von Delft, T.A. Costi, Y. Oreg, Phys. Rev. Lett. 84, 3710 (2000) 7. Y. Meir, N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992) 8. D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, M.A. Kastner, Nature 391, 156 (1998)

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9. W.G. van der Wiel, S.D. Franceschi, T. Fujisawa, J.M. Elzerman, S. Tarucha, L.P. Kouwenhoven, Science 289, 2105 (2000) 10. K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975) 11. H.R. Krishna-murthy, J.W. Wilkins, K.G. Wilson, Phys. Rev. B 21, 1044 (1980) 12. W. Hofstetter, J. K¨onig, H. Schoeller, Phys. Rev. Lett. 87, 156803 (2001) 13. M. Salmhofer, C. Honerkamp, Prog. Theor. Phys. 105, 1 (2001) 14. J.W. Negele, H. Orland, Quantum Many-Particle Systems (Addison-Wesley, Reading, MA, 1987) 15. T. Enss, Renormalization, conservation laws and transport in correlated electron systems, Ph.D. thesis, University of Stuttgart, Germany (2005), arXiv:cond-mat/0504703, URL http://elib.uni-stuttgart.de/opus/volltexte/2005/2258/ 16. W. Metzner, AIP Conf. Proc. 846, 130 (2006) 17. F.J. Wegner, A. Houghton, Phys. Rev. A 8, 401 (1973) 18. J. Polchinski, Nucl. Phys. B 231, 269 (1984) 19. C. Wetterich, Phys. Lett. B 301, 90 (1993) 20. R. Hedden, V. Meden, Th. Pruschke, K. Sch¨onhammer, J. Phys.: Condens. Matter 16, 5279 (2004) 21. T.R. Morris, Int. J. Mod. Phys. A 9, 2411 (1994) 22. C. Karrasch, T. Enss, V. Meden, Phys. Rev. B 73, 235337 (2006) 23. T.A. Costi, Phys. Rev. B 64, 241310(R) (2001) 24. V. Meden, F. Marquardt, Phys. Rev. Lett. 96, 146801 (2006) 25. C. Karrasch, T. Hecht, Y. Oreg, J. von Delft, V. Meden, Phys. Rev. Lett. 98, 186802 (2007) 26. V. Meden, W. Metzner, U. Schollw¨ock, K. Sch¨onhammer, Phys. Rev. B 65, 045318 (2002) 27. S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollw¨ock, K. Sch¨onhammer, Phys. Rev. B 70, 075102 (2004) 28. T. Enss, V. Meden, S. Andergassen, X. Barnab´e-Th´eriault, W. Metzner, K. Sch¨onhammer, Phys. Rev. B 71, 155401 (2005) 29. S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollw¨ock, K. Sch¨onhammer, Phys. Rev. B 73, 045125 (2006) 30. X. Barnab´e-Th´eriault, A. Sedeki, V. Meden, K. Sch¨onhammer, Phys. Rev. Lett. 94, 136405 (2005) 31. S. Andergassen, T. Enss, V. Meden, Phys. Rev. B 73, 153308 (2006)

A Simple View on the Quantum Hall System Emil J. Bergholtz and Anders Karlhede

Abstract The physics of the quantum Hall system becomes very simple when studied on a thin torus. Remarkably, however, the very rich structure still exists in this limit and there is a continuous route to the bulk system. Here we review recent progress in understanding various features of the quantum Hall system in terms of a simple one-dimensional model corresponding to the thin torus.

1 Introduction Even though more than twenty years have passed since the experimental discovery [1] of the fractional quantum Hall effect at ﬁlling factor ν = 1/3 and its basic explanation due to Laughlin [2], the physics of the quantum Hall regime still continues to surprise us with new novel phenomena. Already from the beginning it was clear that the quasiparticles in the Laughlin state have fractional charge and later on it was realized that they obey fractional statistics [3, 4]. Soon after the ﬁrst observations at ν = 1/3 many other gapped quantum Hall states were observed, some of them at fractions that could not be explained by Laughlin’s wave functions. To explain these new fractions, hierarchical schemes were developed by Haldane, Halperin and Laughlin [5, 3, 6] and Jain constructed wave functions for these states and proposed an intriguing interpretation in terms of composite fermions [7], where each of the electrons captures an even number of magnetic ﬂux quanta, mapping the original problem of electrons partially ﬁlling a Landau level onto composite fermions ﬁlling an integer number of Landau levels. This gives a nice picture of how the gap responsible for the quantum Hall effect appears at the fractions ν = p/(2 mp + 1) by mapping the system onto the well understood integer quantum Hall effect. Moreover, the composite fermion theory E.J. Bergholtz and A. Karlhede Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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E.J. Bergholtz and A. Karlhede

offers an appealing explanation for the existence of the gapless states observed at even denominator fractions such as ν = 1/2, where the system is mapped onto free fermions in no magnetic ﬁeld. The mean ﬁeld theory of such states, due to Halperin, Lee and Read [8], has been spectacularly conﬁrmed by surface acoustic wave experiments at ν = 1/2 [9], and by ballistic experiments near this ﬁlling factor [10]. However, in our opinion, a microscopic understanding of composite fermions is still lacking [11]. Gapped quantum Hall states have now been observed that fall outside Jain’s main scheme [12], and the microscopic origin of these states is under debate. Also, in higher Landau levels quantum Hall states exist that might possess even more exotic properties. One such example is the Moore-Read state [13], which is believed to describe the quantum Hall system at ν = 5/2 [14, 15, 16]. This state has attracted great interest recently due to the supposed non-abelian statistics of the quasiholes and its possible application to topologically protected q-bits (decoherence free quantum computational devices) [17]. In a recent line of research it has been shown that studying the quantum Hall system on a thin torus allows for both a simple understanding of already established results and for providing new insights [18, 19, 20, 21, 22, 23, 24]. Here, we give a non-technical review of this work. References [26, 27, 28] contain relevant precursors to the work presented here. We study the quantum Hall system of spin-polarized electrons on a torus as a function of its circumference, L1 , by mapping the problem onto a one-dimensional lattice model. When L1 is small, the range of the electron-electron interaction becomes short (in units of the lattice spacing), and we get a systematic expansion of the quantum Hall system around a simple case—the thin torus. The abelian quantum Hall states are manifested as gapped one-dimensional crystals, ‘TaoThouless states’, and their fractionally charged excitations appear as domain walls between degenerate ground states. At half-ﬁlling, ν = 1/2, the electrons condense into a Fermi sea of neutral dipoles which connects smoothly to the gapless state in the bulk. The non-abelian pfafﬁan (Moore-Read) state believed to describe the ν = 5/2 phase is described by six distinct crystalline states, and the non-trivial quasiparticle and quasihole degeneracies that are crucial for the nonabelian statistics follow simply from the inequivalent ways of creating domain walls between these different vacua. This formulation is manifestly particle-hole symmetric and thus allows for the construction of both quasiparticles and quasiholes. The outline of this paper is the following. In section 2 we set up a onedimensional lattice model of the lowest Landau level. In section 3 we discuss how ground states and excitations have a very simple and appealing manifestation on the thin torus, and in section 4 we discuss the crucial issue of how the thin torus picture is connected to the experimentally realizable bulk system.

A Simple View on the Quantum Hall System

21

2 1D Lattice Model The energy of a charged particle moving in a magnetic ﬁeld is quantized in macroscopically degenerate Landau levels. In the strong magnetic ﬁeld limit, the gap between different Landau levels becomes large and the electrons will populate the lowest available states. Hence the kinetic energy effectively freezes out, leaving a strongly interacting problem in the highest partially populated Landau level (LL). Since a single LL is an effectively one-dimensional system, it is possible to map the two-dimensional quantum Hall system onto a one-dimensional problem. It turns out that this mapping is particularly convenient on the torus. For simplicity we consider the problem of an electron moving in a perpendicular magnetic ﬁeld on the surface of a cylinder (the torus case is obtained by straight forward periodising). In Landau gauge, A = Byx, ˆ the lowest Landau level states are

ψm (r) =

1

π 1/4 L1/2

e2π imx/L1 e−(y+2π m/L1 )

2 /2

,

(1)

where we use units such that = h¯ c/eB = 1, h¯ = 1, and label the states by integers m. The states are centered along the lines ym = −2π m/L1, given by the momentum in the x−direction. This provides an explicit mapping of the two-dimensional electron gas in the lowest Landau level onto a one-dimensional lattice model, where the lines ym can be thought of as the sites, see Figure 1. A general (two-body) interaction Hamiltonian takes the form H =∑

∑ Vkm c†n+m c†n+k cn+m+k cn ,

(2)

n k>m

x 1

y 1

Fig. 1 A cylinder with a magnetic ﬁeld B perpendicular to its surface. The single particle states are centered along the lines ym = −2π m/L1 and can be thought of as sites in a one-dimensional lattice

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E.J. Bergholtz and A. Karlhede

where Vkm are matrix elements that can be calculated for a given real-space interaction. The physics of the interaction can be understood by dividing H into two parts: Vk0 , the electrostatic repulsion (including exchange) between two electrons separated k lattice constants, and Vkm , the amplitude for two particles separated a distance k − m to hop symmetrically to a separation k + m and vice versa. The symmetry of the hopping, which is a consequence of conservation of momentum, implies that the position of the center of mass is conserved. A general Ne −particle state in the lowest Landau level is a linear combination of states characterized by the positions (or, equivalently, the momenta) at which they are centered. We represent these (Slater determinant) states in Fock space as | n1 n2 n3 . . . where ni = 0, 1 according to whether site i is occupied or not. The problem of ﬁnding the ground state and the low lying excitations, at ﬁlling fraction ν = Ne /Ns , is thus a matter of arranging Ne electrons on Ns sites. A very important property of the obtained lattice model is that the lattice constant is 2π /L1. This means that, for a given real space interaction, the interaction in the one-dimensional lattice model becomes short range in units of the lattice spacing when the torus becomes thin and we can hope to be able to solve the problem in this limit. The experimental situation, on the other hand, is obtained as L1 → ∞, where the lattice model becomes inﬁnitely long range measured in units of the lattice constant. When the system is studied as a function of L1 , we ﬁnd that many of the characteristic features of the quantum Hall system is independent of L1 and there is a continuous route between the two extreme cases—we claim that the two cases are adiabatically connected.

3 The Thin Torus Here we consider the quantum Hall system at generic ﬁlling fractions, ν = p/q < 1, in the limit L1 → 0. For reasonable interactions (including Coulomb), the problem becomes a classical electrostatic one-dimensional problem and the ground states are regular lattices of electrons where the particles are as far apart as possible, as shown in Table 1. The reason that the physics is completely determined by electrostatics in the thin limit is actually rather simple. The single particle states are essentially gaussians

Table 1 Examples of ground states in the thin limit, L1 → 0. The underlined unit cells containing p electrons on q sites are periodically repeated in the ν = p/q ground state. The q-fold degeneracy on the torus is reﬂected by q different translations of the unit cell | 100100100100100100 . . . ν = 1/3 | 10100101001010010100 . . . ν = 2/5 | 1010010010010100100100 . . . ν = 4/11

A Simple View on the Quantum Hall System

23

extended roughly one unit length (i.e. one magnetic length) and separated by the lattice constant 2π /L1. Consequently, the overlap between different one-particle wave functions becomes very small and the only non-vanishing matrix elements are those where each electron is created and destroyed at the same site, i.e. the electrostatic matrix elements Vk0 . Thus, L1 is a parameter that controls the strength of the hopping, which can be continuously turned on by increasing L1 . The ground states in the thin limit are regular lattices with unit cells containing p electrons and q sites at ﬁlling ν = p/q. This is true for any repulsive interaction that is monotonic, with positive second derivative—Coulomb falls into this category. The same ground states were obtained by Hubbard when he investigated generalized Wigner lattices in the seemingly very different context of quasi-one-dimensional salts [29]. It is interesting to note that, at ν = 1/3, the thin limit ground state, see Table 1, is the state originally proposed by Tao and Thouless in 1983 to explain the fractional quantum Hall effect [30]. We call these states, at general ﬁlling factor, Tao-Thouless (TT) states, stressing the fact that they are different from ordinary, classical crystals or Wigner crystals. It is important to note that the TT-states have a gap to all excitations—there are no phonons. The reason for this is that once the ﬂuxes through the holes of the torus are ﬁxed, then the positions of the one-particle states along the torus are ﬁxed, and hence no vibrations of the lattice are possible. Note also that the q−fold degeneracy, present for all energy eigenstates on the torus [31], is trivially manifested by the q different translations of the unit cell.

3.1 Gapped Fractions and Fractional Charge At odd denominator fractions in the lowest Landau level, the TT-states describe (but are extreme forms of) the gapped abelian quantum Hall states observed in the laboratory. In section 4 we discuss this connection further, but let us ﬁrst consider the structure of ground states and fractional charge that emerge in the thin limit. At the Jain fractions, ν = p/(2pm + 1), the unit cells are 102m (102m−1 ) p−1 in chemical notation. At ν = 1/3 the unit cell is 100, at ν = 2/9 it becomes 100001000 and so on. These states are gapped and q−fold degenerate. The low energy excitations of the TT-states at arbitrary ﬁlling fractions are domain walls separating sequences of degenerate ground states. These domain walls carry fractional charge and correspond to the quasiparticle and quasihole excitations in the bulk. At ν = 1/q a quasihole (quasiparticle) is constructed by inserting (removing) a zero somewhere in the ground state, see Table 2. This is very similar to Laughlin’s original concept of creating a quasihole by inserting a ﬂux quantum. At ν = p/(2pm + 1) the corresponding quasiparticle (quasihole) excitations are obtained by inserting (removing) 102m−1 somewhere in the TT-state with unit cell 102m (102m−1 ) p−1 .

24

E.J. Bergholtz and A. Karlhede

Table 2 The ν = 1/3 ground state, and the corresponding states with three quasiparticles and three quasiholes respectively. Note that the underlined concentration of electrons (or holes) are domain walls between degenerate ν = 1/3 ground states. The charge (±e/3) of these excitations is determined by Su and Schrieffer’s counting argument | 100100100100100100100100100100100 . . . | 100101001001001010010010010100100 . . . | 100100010010010001001001000100100 . . . Table 3 The ν = 3/7 ground state, and the corresponding states with a quasiparticle and a quasihole respectively. Note that inserting/removing 10 creates domain walls with the correct charge ±e/7. (Inserting/removing 100 would instead create domain walls with charge ±2e/7) | 1010100101010010101001010100 . . . | 101010010101010010101001010 . . . | 1010100101001010100101010010 . . .

The charge of these excitations is determined by Su and Schrieffer’s counting argument [32]. By removing 102m−1 at 2pm + 1 separated position and adding 2m unit cells 102m (102m−1 ) p−1 to keep the number of sites ﬁxed, 2pm + 1 quasiholes, e each with charge e∗ = e (2pm+1)−2pm = 2pm+1 , are created. This readily general2pm+1 izes to generic ﬁllings p/q, where the lowest lying excitations naturally emerge as domain walls carrying charge e (3) e∗ = ± . q

3.2 The Non-Abelian Pfafﬁan State The single particle states differ from (1) in the higher Landau levels, thus the interaction (i.e. Vkm ) is different, and as a consequence, the ground states and their excitations may differ from those in the lowest Landau level. Perhaps most notably, the ground state at half-ﬁlling in the second Landau level appears to be gapped and is believed to be accurately described by the Moore-Read pfafﬁan state [13]. This state, which is motivated by conformal ﬁeld theory, has quasihole excitations with charge e/4 that can only be created in pairs, and obey non-abelian statistics. Here we describe how this state is manifested on the thin torus and give the degeneracies of the quasihole excitations that are crucial for the existence of nonabelian statistics. Moreover, the particle-hole symmetry allows us to construct also quasiparticles, as well as states with general combinations of quasiholes and quasiparticles [22]. The pfafﬁan states on the torus are known to be the exact ground states of a hyper-local three-body potential [14, 16]. In the thin torus limit, this implies that the electrostatic energy (of this three-body potential) is minimized by separating all

A Simple View on the Quantum Hall System

25

Table 4 The six degenerate pfafﬁan ground states on a thin torus | 010101010101 . . . 2 translations | 001100110011 . . . 4 translations

Table 5 Examples of domain walls with fractional charge ±e/4 | 01010100110011001010101 . . . two quasiholes | 01010101100110011010101 . . . two quasiparticles | 0101010110011001010101 . . . a quasiparticle-hole pair

triples of particles as much as possible. At half-ﬁlling this means that there are no sequences of four consecutive sites containing three electrons (or holes). The six states displayed in Table 4 are the unique states at half-ﬁlling that have no such sequences. The extra freedom created by the additional pfafﬁan ground states allows for the creation of domain walls carrying charge e∗ = ±e/4—i.e. half of the fractional charge e∗ = ±e/2 that is implied by the center of mass degeneracy. The domain walls that achieve this are those between the two different kinds of ground states | 10101010 . . . and | 11001100 . . ., as shown in Table 5. Again this charge is readily determined by Su and Schrieffer’s counting argument. Note also that, because of the periodic boundary conditions, these excitations can only be created in pairs. The degeneracy of these excitations is readily determined by considering the various ways of matching the domains. In Ref. [22] we derived that the degeneracy of a state with 2n − k quasiholes and k quasiparticles with ﬁxed positions is 2n−1 . Results similar to ours have also been obtained by Haldane [21], and subsequently also by Seidel and Lee [23] for the closely related bosonic pfafﬁan state at ν = 1.

3.3 The Half-Filled Landau Level The physics of the half-ﬁlled lowest Landau level is known to be very different from the gapped fractions discussed above. There is strong experimental and numerical evidence that the system is gapless. In the composite fermion picture, all magnetic ﬂux is attached to the electrons and the system becomes a free Fermi gas of composite fermions in no magnetic ﬁeld [7, 8, 33]. Furthermore, it has been proposed that the quasiparticles are dipoles [34, 35, 36]. In the thin limit, the ν = 1/2 ground state is |1010101010.... and the (gapped) low lying excitations are the fractionally charged excitations described above. In

26

E.J. Bergholtz and A. Karlhede 1 L1=5.2

0.9

L1=5.4 0.8

0.7

Density

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6 Site

8

10

12

Fig. 2 The evolution of the one-dimensional density c†k ck from the small L1 TT-state (triangles) to the homogeneous state (squares) at ν = 1/2. At L1 ∼5.3 there is a sharp transition from the TT-state to a homogeneous state that is described by our solvable model, and corresponds to a Luttinger liquid of neutral dipoles. At the transition the quantum numbers change

fact, the ν = 1/2 state has a larger energy gap than the ν = 1/3 state on the thin torus. This is clearly different from the observed gapless state in the bulk. In order to explain this discrepancy we consider the situation when L1 increases from zero. Short range hopping terms will now become important and start to compete with the electrostatic terms. However, the shortest range hopping V21 annihilates the TT state |1010101010..... Also, from early numerical investigations it was clear that there is a sharp transition from the TT-state |1010101010.... at L1 ∼ 5.3 to a gapless homogeneous state [28]. It is interesting to contrast ν = 1/2 with ν = 1/3. At ν = 1/2, the ground state is the TT-state | 101010 . . . when L1 → 0. As noted, this state is annihilated by the shortest range hopping term V21 which favours hoppable states of the type | 11001100 . . .. Thus there is a competition between the electrostatic terms and the hopping term and this leads to a phase transition to a gapless state when L1 grows. For ν = 1/3 on the other hand, the TT-state | 100100 . . . favoured by electrostatics is also a maximally hoppable state favoured by the short range hopping term. In this case there is no competition between electrostatics and hopping and there is no phase transition as L1 grows.

A Simple View on the Quantum Hall System

27

We now brieﬂy discuss a solvable model that accurately describes the system at L1 slightly larger than 5.3. The low-energy sector of the model consists of free one-dimensional neutral fermions (dipoles) [18]. The crucial part in the Hamiltonian turns out to be the hopping term V21 —the other terms can be treated as perturbations yielding an interacting Luttinger liquid. We start with the Hamiltonian H ∗ = −V21 ∑ c†n cn+1 cn+2 c†n+3 + H.c.

(4)

n

This provides a good approximation of the interaction on a thin, but not inﬁnitely thin, torus (L1 ∼ 6) as discussed in Ref. [18]. We deﬁne a subspace H of the full Hilbert space by requiring each pair of sites (2p − 1, 2p) to have charge one (the equivalent grouping of the sites (2p, 2p + 1) gives a trivial copy of our solution). In Ref. [18] it is argued that H contains the low-energy sector under fairly general conditions. It agrees with what we ﬁnd in numerical studies, and H contains the maximally hoppable state |100110011001..... Furthermore, H ∗ preserves the subspace H , thus any other ground state candidate may not mix with the states in H . There are two possible states for a pair of sites in H ; | ↓ ≡ |01, | ↑ ≡ |10

(5)

and it is natural to introduce the spin operators † † − s+ p = c2p−1 c2p , s p = c2p c2p−1 .

(6)

On states in H , s+ , s− describe hard core bosons—they commute on different sites but obey anti-commutation relations on the same site. In this subspace, H ∗ is simply the nearest neighbor spin 1/2 XY -chain, − − + H ∗ = V21 ∑(s+ p+1 s p + s p+1 s p ).

(7)

p

The (hard core) bosons can be expressed in terms of fermions d using the JordanWigner transformation, p−1 †

iπ ∑ j=1 d j d j , s− p = Kp d p, Kp = e

(8)

and the Hamiltonian (4) is then that of free fermions. The ground state is obtained by ﬁlling all the negative energy states. The excitations are neutral particle-hole excitations out of this Fermi sea. These excitations have a natural interpretation in terms of dipoles as is seen from (6), and in the limit Ne → ∞, the excitations become gapless. It is also straight forward to show that the state is homogeneous. We would like to stress that this explicitly and exactly maps (the low energy sector of) a system of strongly interacting electrons in a strong

28

E.J. Bergholtz and A. Karlhede

magnetic ﬁeld onto a system of non-interacting particles that are neutral and hence are unaffected by the magnetic ﬁeld. By considering the relation between the real system—where the electrons interact via Coulomb repulsion—and our model, we conclude that the ν = 1/2 system is a Luttinger liquid of these dipoles on a thin torus (L1 slightly larger than 5.3). This conclusion is supported by numerical calculations for both Coulomb [19] and short range interactions [28]. Note also that the obtained solution has striking similarities to the bulk state—both are homogeneous gapless states with quasiparticles (dipoles) that do not couple to the magnetic ﬁeld.

4 Bulk Physics In this section we discuss how the two-dimensional bulk physics is related to the physics in the thin limit. We will argue that the abelian and non-abelian gapped states, as well as the gapless state at ν = 1/2, are adiabatically connected to the states found on the thin torus. The strength of the argument varies with the ﬁlling factor but we believe the over all picture of bulk states at generic ﬁlling factor being adiabatically connected to simple ground states on the thin torus is ﬁrmly established. Before we proceed with a more detailed account for each of the considered cases we make two important remarks: (1) The TT-states and the bulk QH-states do in fact have the the same symmetries and qualitative properties. That the TT-state is not homogeneous is not a result of spontaneous symmetry breaking—in fact the Laughlin/Jain states have periodic density variations on any ﬁnite torus [37]. (2) As indicated in section 3.3, there is actually a simple way of understanding why the TT-state melts at half-ﬁlling while it develops smoothly into the bulk QH-state at e.g. ν = 1/3.

4.1 Abelian States We begin by considering the simplest case, ν = 1/q, q odd. At these ﬁlling factors the Laughlin wave functions describe the bulk physics; moreover, they are the exact and unique ground states to a short range pseudo-potential interaction and there is a gap to all excitations [38, 39]. This holds also on a torus (or cylinder) for arbitrary circumference L1 .1 This is fairly obvious since it depends only on the short distance property of the electron-electron interaction. In our opinion, this establishes that the ground state develops continuously as L1 increases, without a phase transition, from the TT-state to the bulk Laughlin state for this short range interaction. This result is implicit in the work of Haldane and Rezayi [27]. The same is then very likely to be

1

On the torus, the ground state of course has the trivial q-fold center of mass degeneracy.

A Simple View on the Quantum Hall System

29

true for the Coulomb interaction—this is supported by exact diagonalization where no transition is seen as L1 varies. We now show that the Laughlin wave function on a cylinder

Ψ1/q =

∏ (e2π izn/L1 − e2π izm/L1 )q e− 2 ∑n yn , 1

2

(9)

n<m

where z = x + iy, approaches the TT-state as the radius of the cylinder shrinks [27]. Expanding Ψ1/q in powers of e2π iz/L1 and using that the single particle states (1) can be written as ψk =

1 2π iz/L )k e−y2 /2 e−2π 2 k2 /L21 , 1/2 (e π 1/4 L1

Ψ1/q =

one ﬁnds

∑ ∏ c{kn} (e2π izn /L1 )kn e− 2 ∑n yn = 1

2

{kn } n

=

2 2 2 1 c{kn }ψk1 ψk2 · · · ψkNe e2π ∑n kn /L1 , ∑ 1 πNe /4 LNe /2 {kn }

(10)

where c{kn } are coefﬁcients that are independent of L1 . The weight of a particu-

lar electron conﬁguration is multiplied by the factor e2π ∑m km /L1 , thus in the limit 2 will dominate (all terms have the same L1 → 0 the term with the maximal ∑m km ∑m km ). The dominant term is the one that corresponds to the TT-state discussed above, where the electrons are situated as far apart as possible. In this case at every q:th site. This argument can be generalized to the Jain wave functions describing the ground states at ﬁlling factors ν = p/(2mp + 1) showing that they approach the TT-states above as L1 → 0. It can also be generalized to show that the fractionally charged quasiparticles in the TT-state, discussed in section 3.1, are the L1 → 0 limits of the bulk quasiparticles at ﬁlling factor ν = p/(2mp + 1). The TT-state and the bulk Laughlin/Jain state on the torus at ν = p/(2mp + 1) have the same quantum numbers. The symmetry generators that commute with the hamiltonian are T1 and T2 (Tα translates all particles in the α -direction). The Laughlin/Jain state is an eigenstate of T1 and T22mp+1 , with quantum numbers K1 and K2 , whereas T2k , k = 1, 2, . . . , 2mp generate the degenerate states—this is true for any L1 —and the eigenvalues are independent of L1 . The state is inhomogeneous for any ﬁnite L1 , although the inhomogeneity decreases very rapidly as L1 grows. Furthermore, the TT-state and the Laughlin/Jain state both have a gap and have quasiparticles and quasiholes with the same fractional charge. The conclusion is that there is no phase transition separating the TT-states and the bulk Laughlin states. This result has a long history. The very ﬁrst observation was made already in 1983 by Anderson [25] who discussed the TT-state as the ‘parent state’ of the Laughlin state and observed that the fractionally charged quasiparticles could be thought of as domain walls between the degenerate vacua. Rezayi and Haldane noted that the Laughlin state is the exact ground state for the short range interaction on a cylinder of any circumference and showed that the state approaches a crystal as L1 → 0 in 1994 [27]. More recently this was reexamined by the present authors in DMRG calculations [28, 18] and in exact diagonalization [19] 2

2

2

30

E.J. Bergholtz and A. Karlhede 1

0.9

L1=5.0 L1=7.0

0.8

L1=9.0 L1=10.6

0.7

Density

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10

12

14

16

18

Site

Fig. 3 The evolution of the one-dimensional density c†n cn from the small L1 TT-state (triangles) to the nearly homogeneous bulk Laughlin state (circles) at ν = 1/3. This process is smooth and the quantum numbers Kα remain unchanged as L1 changes. Results are obtained from exact diagonalization of an unscreened Coulomb potential

and a careful numerical study of the rapid crossover from the TT- state to a virtually homogeneous state was performed by Seidel et al. using Monte Carlo methods [20]. In the case of the Jain states, there is no known interaction which they are the exact and unique ground states of. However, as we have noted above they have the same qualitative properties as the corresponding TT-states: same quantum numbers, gap and quasiparticles with the same charge. These TT-states, including quasiparticle excitations, are obtained as the L1 → 0 limits of Jain’s wave functions. Furthermore, exact diagonalization of small systems show a smooth development of the ground state from the TT-state to the Jain state as L1 grows. No transition is observed and there is a gap for all L1 [19]. Recent progress strongly suggests that this picture is true also for more general odd denominator fractions in the lowest Landau level, such as the non-Jain state at ν = 4/11; in these cases no phase transition is observed for small systems and a new set of trial wave functions connect the solvable limit to the bulk [24, 40]. We conclude that the adiabatic continuity holds also for the hierarchy states.

A Simple View on the Quantum Hall System

31

4.2 Non-Abelian States Recently, it has been understood that also non-abelian gapped quantum Hall states follow the same pattern as we outlined for the abelian states above [21, 22, 23]. The six Moore-Read pfafﬁan ground states2 are the exact ground states of a hyperlocal three-body interaction on the torus—as in the case of the Laughlin states, this holds for general L1 as it depends on the local properties only. As L1 decreases the states continuously approach the TT-states in Table 4.

4.3 The Gapless State at ν = 1/2 The ν = 1/2 solution on the thin torus, discussed above, has striking similarities to what is expected from theory and experiment for the bulk state. Based on this, we conjectured [18] that this state develops continuously, without a phase transition, to the bulk state as L1 → ∞. This is however a much more delicate issue than it is for the states above since the state at ν = 1/2 is gapless. To investigate this conjecture, we performed exact diagonalization studies of small system for various Ne and L1 using an unscreened Coulomb potential [19]. The obtained ground states were then compared with the Rezayi-Read state [33], that is expected to describe the bulk state, by calculating overlaps. On the torus the Rezayi-Read wave function takes the form

ΨRR = detij [eiki ·R j ]Ψ1 , 2

(11)

where R is the guiding center coordinates and Ψ1 is the bosonic Laughlin state at 2 ν = 1/2. This wave function depends on a set of momenta {ki }, which determine the conserved quantum numbers Kα . For L1 ≤ 5.3 the ground state is the TT-state | 10101010 . . .. At L1 ∼ 5.3 there is a sharp transition into a new state that we identify as our Luttinger liquid solution, discussed above. As L1 is increased further, there is a number of different transitions to new states, but these transitions are all much smoother than the one at L1 ∼ 5.3. As shown in Figure 4 for the case of nine electrons, each of these states corresponds to a given set of momenta {ki } in the Rezayi-Read state. The Fermi seas of momenta develop in a very natural and systematic way. Starting from an elongated sea, which we identiﬁed as the exact solution, a single momentum is moved at each levelcrossing, terminating in a symmetric Fermi sea when L1 ∼ L2 . Since our Luttinger liquid solution corresponds to one of the Fermi seas in the Rezayi-Read state and this state develops smoothly towards the bulk, we conclude

2 There are three distinct pfafﬁan wave functions on the torus. This together with the two-fold center of mass degeneracy gives all the six states on the thin torus.

32

E.J. Bergholtz and A. Karlhede

0.993

0.996

0.995

0.998

10101010.... L1 5.26

8.23 8.43 9.02

10.63

Fig. 4 ‘Phase diagram’ showing the ground states for ν = 1/2 as a function of L1 for nine electrons [19]. The results are obtained in exact diagonalization, using unscreened Coulomb interaction. Overlaps with the Rezayi-Read state with the displayed Fermi seas of momenta are shown above each Fermi sea

that the Luttinger liquid of neutral dipoles is continuously connected to the bulk ground state.

5 Conclusions We conclude that the thin torus provides a simple and accurate picture of both abelian and non-abelian quantum Hall states, and even more surprisingly, also of the gapless state at ν = 1/2. The gapless state is particularly important since it provides an explicit microscopic example of how weakly interacting quasiparticles moving in a reduced (zero) magnetic ﬁeld emerge as the low energy sector of strongly interacting fermions in a strong magnetic ﬁeld. There are strong reasons to believe that the picture presented here is valid also for other quantum Hall states. Indeed, the ground state and quasihole degeneracies of other topological states can be obtained on the thin torus [41, 21]. A one-dimensional picture of the quantum Hall system is very natural, and in some sense almost obvious. After all, a single Landau level is a one-dimensional system. The non-trivial result is, of course, that a model with an interaction that is short range in the one-dimensional sense is relevant. We believe that the evidence reviewed here establishes that this is indeed the case. Acknowledgements We thank Hans Hansson, Janik Kailasvuori, Emma Wikberg and Maria Hermanns for interesting discussions and fruitful collaborations. This work was supported by the Swedish Research Council and by NordForsk.

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Halfvortices in Flat Nanomagnets Gia-Wei Chern, David Clarke, Hyun Youk, and Oleg Tchernyshyov

Abstract We discuss a new type of topological defect in XY systems for which the O(2) symmetry is broken in the presence of a boundary. Of particular interest is the appearance of such defects in nanomagnets with a planar geometry. They are manifested as kinks of magnetization along the edge and can be viewed as halfvortices with winding numbers ±1/2. We argue that the halfvortices play a role in ﬂat nanomagnetics equally important to that of ordinary bulk vortices. We show that domain walls found in experiments and numerical simulations in strips and rings are composite objects containing two or more elementary defects. We also discuss a closely related system: the two-dimensional smectic liquid crystal ﬁlms with planar boundary condition.

1 Introduction It is well known that topological defects play an important role in catalyzing the transitions of physical systems with spontaneously broken symmetries [1, 2]. For instance, in nanorings made of soft ferromagnetic material, the switching process G.-W. Chern Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA, e-mail: [email protected] D. Clarke Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA, e-mail: [email protected] H. Youk Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MD 02139, USA O. Tchernyshyov Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA, e-mail: [email protected]

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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usually involves creation, propagation, and annihilation of domain walls with complex internal structure [3, 4]. We have pointed out in a series of papers [5, 6, 7] that domain walls in nanomagnets of planar geometry are composed of two or more elementary defects including ordinary vortices in the bulk and fractional vortices conﬁned to the edge. As an example, the simplest domain wall in a magnetic strip consists of two edge defects with opposite winding numbers n = ±1/2. In a nanomagnet with the geometry of a disk, the strong shape anisotropy (due to dipolar interaction) forces the magnetization vector M to lie in the disk plane, effectively making the magnet a 2D XY system. At the edge of the ﬁlm, dipolar interaction further tries to align the spins to either of the two tangential directions of ˆ = M/|M| = (cos θ , sin θ ) ± τˆ . The reduction of ground state symmetry the edge: m from O(2) to a discrete Z2 allows for a new type of topological defect conﬁned ˆ along to the edge. These edge defects are manifested as kinks in magnetization m the boundary. In systems with discrete symmetry, such as the Ising ferromagnet, kinks are topological defects connecting different ground states. Their topological properties are rather simple [1]. Nevertheless, as two of us pointed out in Ref. [5], the edge defects can be viewed as halfvortices and have nontrivial topological charge related to the winding number of vortices in the bulk. For a bounded ﬂat nanomagnet, the winding number of vortices in the bulk is not a conserved quantity. This is illustrated by an example shown in Fig. 1, where a bulk vortex with winding number n = +1 is absorbed into the edge. Conservation of topological charges can be restored by assigning winding numbers to edge defects. In this case there are two such kinks at the edge of the ﬁlm. The process shown in Fig. 1 then expresses the annihilation of an +1 bulk vortex with two − 12 edge defects. Numerical simulations exhibiting similar annihilation of bulk vortex with edge defects can be found in Ref. [5]. The winding number of a single edge defect is deﬁned as the line integral along the boundary ∂ Ω [5]: n=−

1 2π

1 ∇(θ − θτ ) · dr = ± . 2 ∂Ω

(1)

Examples of edge defects with winding numbers ± 12 are shown in Fig. 2. For a closed boundary the sum of the winding numbers of edge defects is also given by the above integral, but instead of integrating around one edge defect, the integral is carried out along the entire boundary. It was shown in Ref. [5] that this integral is related to the sum of winding numbers of vortices in the bulk. In general, for a ﬁlm with g holes, we obtained edge

bulk

i

i

∑ ni + ∑ ni = 1 − g.

(2)

Here the winding numbers ni are integers for bulk defects and half-integers for edge defects. This conservation law has important implications for the dynamics of

Halfvortices in Flat Nanomagnets

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Fig. 1 A vortex (n = +1) absorbed by the edge can be viewed as its annihilation with two − 21 edge defects. The annihilation results in a uniform magnetization pointing to the right

−1/2

+1

−1/2

magnetization in nanomagnets [5]. As will be discussed in Section 3, defects with large winding numbers carry signiﬁcant magnetic charge and thus are unfavored energetically in ﬂat nanomagnets. Most of the intricate textures observed involve only bulk vortices with winding number n = ±1 and edge defects with n = ± 12 . Topological considerations also place important constraints on the possible structure of the domain walls in magnetic nanostrips [6]. Such domain walls play an important role in the switching dynamics of magnetic nanorings [16]. Examples of such domain walls are shown in Figs. 3 and 5. Since edge defects are kinks of magnetization along the boundary, a domain wall in a magnetic strip must contain an odd number of kinks at each edge. Furthermore, the angle of magnetization rotation along the two edges must be compensated by the winding number of the bulk. Consequently, the total topological charge including contributions of vortices and edge defects is zero. The edge defects in nanomagnets are analogs of boojums at the surfaces and interfaces of superﬂuid 3 He [8, 9]. In general, “boojum” refers to a topological defect that can live only on the surface of an ordered medium [10]. Boojums were also predicted and observed in liquid crystals [11]. An interesting system which is closely related to our study of ﬂat nanomagnets is the two-dimensional (2D) ˆ is important to the smectic C ﬁlms. The vector nature of the order parameter m conﬁnement of halfvortices at the edge. For example it is well known that vortices with half-integer winding numbers are allowed to exist in the bulk of nematic liquid crystals. On the other hand, in 2D smectic C ﬁlms, the in-plane ordering of molecular orientations is described by a 2D unit vector cˆ parallel to the smectic layers and pointing to the tilt direction [12]. Because rotating a tilted molecule by 180◦ around the normal of layers does not return it to its original conﬁguration, this unit vector ˆ is a true vector. The system thus has edge-conﬁned cˆ , like the magnetization m, halfvortices similar to those in ﬂat nanomagnets. We will discuss their structure in Section 4. In this article we shall review the structure and energetics of halfvortices in nanomagnets. In contrast to the determination of the structure of topological defects in superﬂuids or liquid crystals where the energy is dominated by short range ˆ interactions, ﬁnding solutions of the vector ﬁeld m(r) for topological defects in

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nanomagnets is considerably more difﬁcult due to the nonlocal nature of dipolar interaction. We approached this problem from two opposite limits dominated by the exchange and dipolar interactions, respectively. The results are presented in Sections 2 and 3. Edge defects of smectic C ﬁlms are discussed in Section 4, where we also point out the similarities and differences of the two models. We conclude with a summary of our major results in Section 5.

2 Exchange Limit of Flat Nanomagnets The magnetic energy ofa ferromagnetic nanoparticle has two major contributions: ˆ 2 d 3 r and the dipolar energy (µ0 /2) |H|2 d 3 r. The the exchange energy A |∇m| magnetic ﬁeld H is related to the magnetization through Maxwell’s equations, ∇ × H = 0 and ∇ · (H + M) = 0. Here we disregard the energy of anisotropy, which is negligible for soft ferromagnets such as permalloy. Analytical treatment of topological defects is generally impossible due to the long range nature of dipolar interaction. One usually minimizes the energy numerically to ﬁnd stable structures of the magnetization ﬁeld. Nevertheless exact solutions are possible in a thin-ﬁlm limit [13, 14]: t

w λ 2 /t w log (w/t) deﬁned for a strip of width w and thickness t. Here λ = A/ µ0 M 2 is the length scale of exchange interaction. In this limit the magnetization only depends on the in-plane coordinates x and y, but not on z. Furthermore, the magnetic energy becomes a local functional of magnetization [13, 14]: ˆ E[m(r)]/At =

Ω

ˆ 2 d 2 r + (1/Λ ) |∇m|

∂Ω

ˆ · n) ˆ 2 dr. (m

(3)

Here Ω is the two-dimensional region of the ﬁlm, ∂ Ω is its line boundary, nˆ ⊥ τˆ is unit vector pointing to the outward normal of the boundary, and Λ = 4πλ 2/t log (w/t) is an effective magnetic length in the thin-ﬁlm geometry. This is the familiar XY model [1] with anisotropy at the edge resulting from the dipolar interaction. Minimization of (3) with respect to θ yields the Laplace equation ∇2 θ = 0 in the bulk and boundary condition nˆ · ∇θ = sin 2(θ + θe )/Λ at the edge. Topological defects that are stable in the bulk are ordinary vortices with integer winding numbers, which are well known in the XY model [1]. The boundary term of model (3) introduces yet another class of topological defects that have a singular core outside the edge of the system. To be explicit, consider an inﬁnite semiplane y > 0. Solutions satisfying the Laplace equation in the bulk and the boundary condition ∂y θ = sin 2θ /Λ at the edge y = 0 are [5, 13] tan θ (x, y) = ±

y+Λ . x−X

(4)

The singular core is at (X , −Λ ), distance Λ outside of the edge. Figure 2 shows the magnetization ﬁelds of Eq. (4). As can be easily checked using Eq. (1) the winding

Halfvortices in Flat Nanomagnets

39

−1/2

+1/2

Fig. 2 Edge defects with winding numbers n = + 12 (left) and − 12 (right) in the exchange limit

numbers of these solutions are ± 12 , respectively. The halfvortex cannot live in the bulk: as its singular core is moved inside the boundary, a string of misaligned spins occurs which extends from the core of halfvortex to the boundary [5]. The edge thus provides a linear conﬁning potential for halfvortices. In the limit Λ /w → 0, magnetization at the edge is forced to be parallel to the ˆ = ±τˆ . By exploiting the analogy between XY model and 2D electroboundary, m statics, one can use the method of images to deal with the effects introduced by the boundary [1, 5]. In this analogy the vortex is mapped to a point charge whose strength is given by the corresponding winding number. However, unlike in electrostatics, the “image” charge induced by the boundary has the same sign as the original. The above solution (4) with Λ = 0 looks just like a n = ±1 vortex with its core sitting at the edge. The assignment of half-integer winding number n = ± 12 to the edge defect is thus also consistent with the electrostatics analogy in the sense that the winding number is doubled by the reﬂection at the edge [5]. An exact solution for a domain wall was also obtained in this limit [5]. Consider a strip |y| < w/2. It has two ground states with uniform magnetization: θ = 0 or π . Domain walls interpolating between the two ground states are given by tan θ (x, y) = ±

cos ky , sinh k(x − X )

(5)

where the wavenumber k ≈ π /(w + 2Λ ). The magnetization ﬁeld of Eq. (5) (shown in Fig. 3) is reminiscent of ‘transverse’ domain walls (Bottom panel of Fig. 3) observed in micromagnetic simulations [15]. Unlike domain walls (kinks) in Ising magnet, a domain wall in a nanomagnet, e.g. Eq. (5), is a composite object containing two edge defects with opposite winding numbers ± 12 . The singular cores of the two halfvortices reside outside the ﬁlm, a distance Λ away from the edges. One can understand the stability of the domain wall using the electrostatics analogy: the attractive ‘Coulomb’ force pulling together the two halfvortices is balanced by the conﬁning force from the edges.

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Fig. 3 Top: Magnetization of the head-to-head domain wall solution (5). It is composed of two edge defects with opposite winding numbers ± 21 . Bottom: A transverse domain wall observed in a micromagnetic simulation using OOMMF [20] in a permalloy strip of width w = 80 nm and thickness t = 20 nm

The total energy of the domain wall solution Eq. (5) evaluates to E ≈ 2π At(1 + log(w/πΛ )). As expected for the XY model, the exchange energy depends logarithmically on the system size, the width of the strip w in our case. It also depends logarithmically on a short distance cutoff which is provided by Λ here. After restoring the energy units and expressing Λ in terms of the relevant parameters, we obtain the following domain wall energy in the exchange limit ewt log(w/t) . EDW ≈ 2π At log πλ 2

(6)

The energy depends linearly on the thickness of the ﬁlm t and only weakly (logarithmically) on the width. These relations are important to the understanding of the hysteresis curves of asymmetric magnetic nanorings [16].

3 Dipolar Limit of Flat Nanomagnets The thin-ﬁlm limit discussed in the previous section is inaccessible to most experimental realizations of nanomagnets, in which the dipolar interaction is the primary driving force. In this section we discuss the structure and energetics of topological defects and domain walls in the opposite limit where the energy is dominated by the dipolar interaction. Our strategy here is ﬁrst to ﬁnd structures which min imize the magnetostatic energy (µ0 /2) |H|2 d 3 r and then to include exchange interaction as a perturbation. However, energy minimization in the dipolar limit

Halfvortices in Flat Nanomagnets

41

is relatively difﬁcult due to following reasons. Firstly, as opposed to the local exchange interaction, the dipolar interaction is long-ranged. Secondly, in many cases the magnetostatic energy has a large number of absolute minima. One thus has to search among these minima for one with the lowest exchange energy, making it a degenerate perturbation problem. ˆ The magnetostatic energy of a given magnetization ﬁeld m(r) can also be expressed as the Coulomb interaction of magnetic charges with density ρm (r) = ˆ where M0 is the saturation magnetization. Being positive deﬁnite, the −M0 ∇ · m, magnetostatic energy has an absolute minimum of zero, which corresponds to a complete absence of magnetic charges. A general method to obtain the absolute minima of magnetostatic energy was provided by van den Berg in 1986 [17]. For magnetic ﬁlms with arbitrary shapes, his method yields domains of slowly varying magnetization separated by discontinuous N´eel walls. In the following we look for structures that have the desired winding number and are free of magnetic charges, ˆ = 0 in the bulk and nˆ · m ˆ = 0 on the boundary. i.e. ∇ · m We start by examining the vortex solutions of XY model. In polar coordinate, a vortex with winding number n is described by θ (x, y) = nφ + θ0 , where θ0 is a constant and φ = arctan(y/x) is the azimuthal angle. Among these solutions, only the n = 1 vortex with θ0 = π /2 has zero charge density and survives in the dipolar limit. Its energy then comes entirely from the exchange interaction and diverges logrithmically with system size R: E ≈ 2π At log(R/λ ). Here the short distance cutoff is given by the exchange length λ . The antivortex solutions of the XY model always carry a ﬁnite density of magnetic charge and thus are not a good starting point to obtain the n = −1 defect in the dipolar limit. Fortunately, a magnetization ﬁeld with winding number −1 and free of bulk charges is realized by a conﬁguration known as the cross tie (top panel of Fig. 4) [6, 18]. It consists of two 90◦ N´eel walls normal to each other and intersecting at the singular core. The magnetization ﬁeld of an antihalfvortex (winding

Fig. 4 An antivortex (left), and a − 12 edge defect (right) in the dipolar limit

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number − 12 ) is obtained by placing the core of a cross tie at the edge of the ﬁlm (bottom panel of Fig. 4). Since the magnetization along the edge is parallel to the boundary, the structure is also free of surface charge. As one moves from left to right along the edge the magnetization rotates counterclockwise through π . This is in agreement with the deﬁnition (1) for an antihalfvortex. The energy of an antivortex or an antihalfvortex grows linearly with the length of the N´eel walls L emanating from it: E ∼ σ tL + Ecore

(7)

The surface tension of the wall σ has contributions from both exchange and dipolar interactions. In magnetic ﬁlms with thickness exceeding the N´eel-wall width (of order λ ), it is given by [6] √ σ = 2 2(sin θ0 − θ0 cos θ0 ) A/λ , (8) where 2θ0 is the angle of magnetization rotation across the wall. In thinner ﬁlms (t ≤ λ ) the magnetostatic term becomes substantially nonlocal and the N´eel walls acquire long tails [19]. There is no charge-free conﬁguration for the + 12 edge defect. In addition, one could also observe from micromagnetic simulations that most of the magnetic charges of a transverse domain wall in a strip is accumulated around the + 12 defect. Thus, in the dipolar limit, the magnetically charged + 12 defect is prone to decay into a − 12 edge defect and +1 vortex in the bulk. We next turn to the discussion of the structure of domain walls in this limit.

Fig. 5 Top: A magnetization conﬁguration free of bulk magnetic charges, −∇ · M = 0, and containing two −1/2 edge defects and a +1 vortex in the middle. Parabolic segments of Neel walls are shown by dashed lines. Bottom: A head-to-head vortex wall obtained in a micromagnetic simulation using OOMMF [20] in a permalloy strip of width w = 500 nm and thickness t = 20 nm

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An intrinsic problem arises when one tries to apply van den Berg’s method to ﬁnd the structure of domain walls. That is because a head-to-head domain wall carries a ﬁxed nonzero amount of magnetic charge: Qm = 2M0tw. However, these magnetic charges tend to repel each other and spread over the surface of the sample, much the same as the electric charges in a metal. Based on this principle, we provided in Ref. [7] a construction of the head-to-head domain wall that is free of bulk magnetic charges. All of the charge Qm is expelled to the edges. The resulting structure is shown in the top panel of Fig. 5. It resembles the structure known as the ‘vortex’ domain wall (bottom panel of Fig. 5) predicted to be stable in regimes dominated by dipolar interaction [15]. Both structures contain two − 12 edge defects sharing one of their N´eel walls and a +1 vortex residing at the midpoint of the common wall. The variational construction contains charge-free domains with uniform and curling magnetization separated by straight and parabolic N´eel walls. In a strip |y| < w/2, the two − 12 edge defects share a N´eel wall x = y with the vortex core residing at (v, v) is located. The two curling domains in the regions ±v < ±y < w/2 are separated by parabolic N´eel walls (x − v)2 = (2y ± w)(2v ± w) from domains with horizontal magnetization; they also merge seamlessly with other uniform domains along the lines x = v and y = v. The location (v, v) of the +1 vortex on the shared N´eel wall is a free parameter of our variational construction. The structure remains free of bulk charge as the vortex core moves along the diagonal x = y. When it reaches one of the edge, its annihilation with the − 12 edge defect creates a widely extended + 12 edge defect. The resulting structure (top panel of Fig. 6) is topologically equivalent to the transverse domain wall (bottom panel of Figs. 3 and 6) discussed in the previous section.

Fig. 6 Top: A model vortex wall when the vortex is absorbed by the edge forming an extended + 12 edge defect along the upper boundary. Bottom: Transverse wall observed in micromagnetic simulation

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The equilibrium structure for given strip width w and thickness t is determined by minimizing the total energy of the composite domain wall with respect to vortex coordinate v. The total energy contains the following terms. (a) The exchange energy of the two curling domains Ω around the vortex core. It is given by At Ω (∇θ )2 d 2 r λ ). (b) The energy of N´eel walls, which can be computed and of the order At log(w/ as a line integral t σ () d, where d is a line element of the wall. The surface tension σ , which depends on the angle of spin rotation across the wall, is given by Eq. (8). This term is of order Atw/λ . (c) The magnetostatic energy coming from the Coulomb interaction of magnetic charges spreading along the two edges. It is of the order Aw(t 2 /λ 2 ) log(w/t). By combining the above three contributions, the total energy curve E(v) for a ﬁxed width w and varying thickness t is shown in Fig. 7. For substantially wide and thick strips, the curve attains its absolute minimum as the vortex is in the middle of the strip, in agreement with numerical simulations [15]. A local minimum develops with the vortex core at the edge of the strip as the thickness decreases corresponding to the transverse wall shown in the bottom panel of Fig. 6. The transverse wall becomes the absolute minimum as the thickness is further reduced and the vortex wall (v = 0) is locally unstable. It should be noted that the above calculation for thin ﬁlms, e.g. t = 1 nm, is only an extrapolation. For ﬁlms with small cross section (but not in the exchange limit), our variational approach can not be trusted. Nonetheless, the method is illustrative and indeed shows that the three-defects wall structure is unstable when approaching the exchange limit.

1.3 1.2

E(v)/E(0)

1.1 1 0.9 0.8 t=20 nm t=10 nm t=5 nm t=1 nm

0.7 0.6 -0.5 -0.4 -0.3 -0.2 -0.1

0 v

0.1

0.2

0.3

0.4

0.5

Fig. 7 Energy of the vortex domain wall as a function of the vortex position v at a ﬁxed strip width w = 50 nm for several thicknesses t

Halfvortices in Flat Nanomagnets

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4 Halfvortices in Smectic Films The XY model discussed in the thin-ﬁlm limit preserves a symmetry between topological defects with opposite winding numbers, namely, ±1 vortices have exactly the same energy in model (3) (so do ± 12 edge defects). Since vortices of opposite winding numbers carry different magnetic charges, the degeneracy is lifted in thicker and wider strips where the dipolar interaction becomes more important. The conﬁguration of the topological defects in the extreme dipolar limit discussed previously clearly shows this asymmetry. One can also break this symmetry by assigning ˆ = 0) and bending (∇ × m ˆ = 0) deformations: different penalties to splay (∇ · m ˆ E[m(x)] =

Ω

ˆ 2 + K2 (∇ × m) ˆ 2] d2r [K1 (∇ · m)

+ (1/Λ )

∂Ω

ˆ 2 dr, (nˆ · m)

(9)

In the units where the energy is dimensionless, the elastic constants Ki are also dimensionless, whereas the edge anisotropy 1/Λ scales as the inverse length. With ˆ identiﬁed with the cˆ -director ﬁeld, this energy functional describes the unit vector m the elastic energy of a chiral smectic ﬁlm [21] or a Langmuir monolayer [22] with planar boundary conditions. The case K1 = K2 corresponds to the XY model, which represents the exchange limit. By choosing K1 > K2 we discourage splay, which is similar to a penalty for magnetic charges in the bulk. The dipolar limit is similar to the regime where the bend energy is small compared to those of splay and edge anisotropy. In what follows we discuss the structure and energetics of topological defects in the extreme dipolar limit, K2 = 0. The +1 vortex solution θ (r) = φ + π /2 remains an energy minimum of model (9) for arbitrary K1 and K2 . The +1/2 edge defect in the XY limit, Eq. (4) with the ‘+’ sign, also remains a stable conﬁguration for arbitrary Ki and Λ except that the singular core is pushed further outside the boundary, a distance (1 + ε )Λ away from the edge. Here ε = (K1 − K2 )/(K1 + K2 ). Since the bulk term of the energy functional (9) does not have an intrinsic length scale, an exact scale-invariant solution for an antivortex has been obtained in the ‘dipolar’ limit K2 = 0: √ θ (x, y) = φ − arcsin( 2 sin φ ), (10) where φ = arctan(y/x) is the azimuthal angle. The solution is singular at φ = ±π /4, where the ﬁrst derivative d θ /d φ diverges. A complete solution of the antivortex nevertheless can be obtained by continuing the above solution outside of the interval |φ | < π /4 periodically. The result is shown in Fig. 8(a). Analytical solutions for antihalfvortex for arbitrary Λ is yet to be found. In the limit Λ → 0 achieved in boundaries with very strong anchoring force, the unit vecˆ is forced to be parallel to the edge. In this limit, the − 12 edge defect can be tor m constructed following the same trick for antihalfvortex in the dipolar limit of nanomagnets. The resulting conﬁguration is shown in Fig. 8(b). Compared with their

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(a)

(b)

Fig. 8 An antivortex (a), and a − 21 edge defect (b) in the elastic model with K1 = 1, K2 = 0, and Λ =0

counterparts in the XY model, the antivortex and antihalfvortex in Fig. 8 are closer to the cross tie conﬁguration (or half of it) shown in Fig. 4. Although topological defects of the generalized elastic model show some similarities with those of the magnetic problem in the dipolar limit, there is an important difference regarding the scaling of their energy with system size. Since solution (10) is scale-invariant, the energy of the defects also diverges logrithmically with system size R: E ∼ const × K1 log(R/a). Here a is a short distance cutoff. In the case of − 12 edge defect, a is of the order of Λ . However, as discussed in the previous section, the energy of both antivortex and antihalfvortex scales linearly with the length of N eel wall. The nonlocal dipolar interaction in the magnetic problem results in a natural length scale λ = A/ µ0 M 2 . By contrast, there is no such length scale in the elastic model (9), so the dependence of the energy on the system size is logarithmic.

5 Conclusion We have discussed the topological properties of edge defects in XY systems with a broken O(2) symmetry at the boundary. In particular, we discussed two physical systems containing such edge defects: the 2D smectic C ﬁlms and nanomagnets with a planar geometry. Since spins at the boundary have two degenerate preferred directions (parallel or antiparallel to the boundary), the edge defects are manifested as kinks of magnetization along the edge. Moreover, they carry half-integer winding numbers and thus can be viewed as half-vortices conﬁned at the edge. Conservation of the winding number can only be established by including contributions from the edge defects. As we have pointed out before [5, 6, 7], edge defects should be included along with ordinary vortices as the elementary topological defects in ﬂat

Halfvortices in Flat Nanomagnets

47

nanomagnets. Indeed, domain walls in ﬂat nanomagnets are composite objects consisting of two or more of the elementary defects. These walls play an important role in the dynamics of magnetization, especially in magnetic switching in strip or ring geometries. We have also reviewed structures and energetics of the edge defects in two opposite limits dominated by the exchange and dipolar interactions, respectively. Analytical solutions of halfvortices and transverse domain walls were obtained in a thin-ﬁlm limit where the exchange interaction is the dominant force determining the shape of topological defects. In this limit, the magnetic problem is reduced to the familiar XY model with an anisotropy at the edge. Domain walls stable in this regime are composed of two edge defects with winding number ± 12 . By analogy with 2D electrostatics, the stability of transverse domain wall can be understood as resulting from a balance of the attractive Coulomb force between the oppositely charged halfvortices and the conﬁning force from the edges. Energy minimization is relatively difﬁcult in the opposite limit dominated by the nonlocal dipolar interaction. Nevertheless, by treating the exchange interaction as a perturbation, we are able to ﬁnd structures of topological defects stable in this regime. The +1 vortex of XY model with circulating magnetization remains a stable defect in the dipolar limit. The −1 vortex survives in this limit but is severely deformed; it has the cross tie structure consisting of two 90◦ N´eel walls intersecting at the singular core. The conﬁguration of the − 12 edge defect is constructed by placing the core of a cross tie at the boundary. The + 12 defect carries a ﬁnite amount of magnetic charge and is unstable in this limit. A vortex domain wall composed of two − 12 edge defects and a +1 vortex in the bulk is stable in the dipolar limit. We have presented a variational construction of the vortex domain wall which is free of bulk magnetic charge. By varying the location of the center +1 vortex, the construction interpolates between the vortex wall and the transverse wall. Variational calculation of the domain wall energy reveals that the vortex wall is indeed stable in the dipolar limit whereas it becomes an energy maximum in thin and narrow strips. Finally, we have discussed structures of topological defects in an elastic model which generalizes the XY model of the thin-ﬁlm limit. Calculations in this model are simpliﬁed by the replacement of non-local interactions between magnetic charges by a term that penalizes the existence of magnetic charge in a local fashion. This model is strictly applicable to smectic C ﬁlms, but may provide insight into magnetic conﬁgurations. In particular, the allowed topological defects are the same in both systems, due to the identical group structure of the two models both in the bulk and on the boundary. Acknowledgements We thank C.-L. Chien, P. Fendley, D. Huse. R. L. Leheny, P. Mellado, O. Tretiakov, and F. Q. Zhu for helpful discussions. The work was supported in part by the NSF Grant No. DMR05-20491.

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References 1. Chaikin, P.M., Lubensky, T.C.: Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, (2000). 2. Mermin, N.D.: Rev. Mod. Phys. 51, 591 (1979). 3. Zhu, J.-G., Zheng, Y., Prinz, G. A.: J. Appl. Phys. 87, 6668 (2000). 4. Kl¨aui, M., Vaz, C.A.F., Lopez-Diaz, L., Bland, J.A.C.: J. Phys.: Condens. Matter 15, R985 (2003). 5. Tchernyshyov, O., Chern, G.-W.: Phys. Rev. Lett. 95, 197204 (2005). 6. Chern, G.-W., Youk, H., Tchernyshyov, O.: J. Appl. Phys. 99, 08Q505 (2006). 7. Youk, H., Chern, G.-W., Merit, K., Oppenheimer, B., Tchernyshyov, O.: J. Appl. Phys. 99, 08B101 (2006). 8. Mermin, N.D.: pp. 3–22 In: Quantum Fluids and Solids, eds. S.B. Trickey, E.D. Adams and J.W. Dufty, Plenum, New York (1977). 9. Misirpashaev, T.Sh.: Sov. Phys. JETP 72, 973 (1991). 10. Volovik, G.E.: The Universe in a Helium Droplet, Clarendon Press, Oxford (2003). 11. Kleman, M., Lavrentovich, O.D.: Soft Matter Physics, Springer, New York (2003). 12. deGennes, P.G., Prost, J.: The Physics of Liquid Crystals, Clarendon Press, Oxford (1993). 13. Kurzke, M.: Calc. Var. PDE 26, 1 (2006). 14. Kohn, R.V., Slastikov, V.V.: Proc. Roy. Soc. (London) Ser. A 461, 143 (2005). 15. McMichael, R.D., Donahue, M.J.: IEEE Trans. Magn. 33, 4167 (1997). 16. Zhu, F.Q., Chern, G.-W., Tchernyshyov, O., Zhu, X.C., Zhu, J.G., Chien, C.L.: Phys. Rev. Lett. 96, 027205 (2006). 17. van den Berg, H.A.M.: J. Appl. Phys. 60, 1104 (1986). 18. Londorf, M., Wadas, A., van den Berg, H.A.M., Wiesendanger, R.: Appl. Phys. Lett. 68, 3635 (1996). 19. Hubert, A., Schaefer, R.: Magnetic Domains, Springer, Berlin (1998). 20. Donahue, M.J., Porter, D.G.: OOMMF User’s Guide, Version 1.0, In: Interagency Report NISTIR 6376, NIST, Gaithersburg (1999). http://math.nist.gov/oommf/ 21. Langer, S.A., Sethna, J.P.: Phys. Rev. A 34, 5035 (1986). 22. Fischer, T.M., Bruinsma, R.F., Knobler, C.M.: Phys. Rev. E 50, 413 (1994).

Spin Structure and Dynamical Magnetic Response of Spin-Orbital Polarons in Lightly Doped Cobaltates M. Daghofer, P. Horsch, and G. Khaliullin

Abstract We present numerical results on a spin-orbital polaron for lightly doped cobaltates with x 0.7 < 1. We compare magnetic susceptibility of the polaron with realistic parameters to experiment and ﬁnd perfect agreement. Further, we analyze magnetic excitations and argue that they are responsible for observed spin-wave scattering. The cobaltates Nax CoO2 have recently enjoyed much interest for a variety of reasons. One of their intriguing features is an unconventional superconductivity, which can be observed in the hydrated compound Nax CoO2 :yH2 O for x ≈ 0.3, y ≈ 1.4 [1, 2]. The half-doped Na0.5 CoO2 is a charge ordered insulator and separates two different metallic phases: Low sodium content leads to paramagnetism (this doping range contains the superconducting compounds) while high sodium induces a Curie-Weiss-like susceptibility [3]. The latter part of the phase diagram is of interest because of its high thermo-electric power [4, 5], i.e., the capability to transform temperature differences into electricity, combined with relatively low resistivity. The thermoelectric properties are strongly affected by a magnetic ﬁeld, which seems to suggest a strong role of the spin degree of freedom and to indicate strong electronelectron correlation effects [6]. The magnetic properties for 0.7 x < 1 – where thermopower is particularly enhanced [2] – are rather peculiar: While neither of the endpoints x = 1 and x 0.7 shows magnetic structure, A-type antiferromagnetism below TN ≈ 20K [7, 8] is found in between, where ferromagnetic planes are stacked antiferromagnetically. This regime is rather unconventional, because its high-temperature magnetic behavior shows a negative Curie-Weiss temperature [5, 9, 6, 10, 11], while a positive one can be inferred from its low-temperature spin-wave data, as pointed out in Ref. [7]. This contradiction can be resolved by assuming that the charge carriers are spin-orbital polarons [12, 13]. Spin-orbital polarons can likewise explain why the ferromagnetic in-plane coupling is rather small – only as large as the antiferromagnetic inter-layer coupling, despite the much short in-plane bond lengths. M. Daghofer, P. Horsch, and G. Khaliullin Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany, e-mail: [email protected] B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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1 Spin-Orbital Polarons in Nax CoO Nax CoO2 has a layered structure with a triangular Co lattice within the planes and a considerably larger lattice spacing between the planes than within them. In this structure, crystal ﬁeld splits Co-d orbitals into three energetically lower t2g orbitals and two more energetic eg orbitals, see Fig. 1(a). The splitting is large enough to 6 conﬁguration in NaCoO , i.e., the t orbitals are fully occupied by six lead to a t2g 2 2g electrons of the Co3+ ion. The Co ions consequently show no magnetic moment and the compound has low susceptibility [14]. However, the splitting is just large enough to achieve the low-spin state: Hund’s-rule coupling, which would instead promote a higher-spin state, is only slightly smaller than the crystal ﬁeld splitting [12]. When 5 sodium content is reduced, holes are doped into the Co ions. Each t2g hole carries a spin 1/2, and one would thus expect to observe a higher magnetic susceptibility in the doped compounds. However, the observed susceptibility far exceeds what one could explain by the small number of spin-1/2 sites [5]. The large susceptibility can be explained by taking into account the orbital degree of freedom [15, 12]. Each hole (Co4+ ion) destroys the cubic symmetry for the six surrounding Co3+ ions, which affects the crystal ﬁeld splitting: Two of the t2g orbitals are slightly raised, one of the eg orbitals is lowered, so that the difference between the highest t2g and the lower eg orbital becomes smaller than Hund’s rule splitting. It is then energetically favorable to form an intermediate-spin state with spin one by transferring one electron in the lower eg level, see Fig. 1(a). As this happens on all six sites surrounding the hole, one ﬁnds the spin-orbital polaron depicted in Fig. 2.

a

e Jh t 0

1

c

b

Jij

Co3+

Jij

Fig. 1 Orbital occupation of ions and their magnetic coupling. (a): Left: Non-magnetic ions with fully occupied t2g orbitals as found in NaCoO2 . Right: One of the six Co3+ ions next to a Co4+ hole in doped Nax CoO2 with x < 1. jH is the Hund’s rule coupling, which stabilizes the intermediate-spin state with S = 1. (b) Antiferromagnetic coupling Ji j between two S = 1 sites promoted by t2g –eg hoppings. (c) Ferromagnetic coupling J0 j between the hole and an S = 1 site via eg –eg hopping is suppressed because of the 90-degree Co-O-Co bond angle. t2g –eg charge transfer results in competing antiferromagnetic and ferromagnetic exchange contributions

Spin-Orbital Polarons in Cobaltates

51

J Jd J’

Fig. 2 Spin-orbital polaron. The black arrow in the center denotes the hole (Co4+ ion) with spin s = 1/2, the six surrounding arrows indicate the Co3+ sites with induced spin S = 1, see Fig. 1(a). J ∼ 10 − 20 meV is their antiferromagnetic coupling, J is the coupling between the hole and the S = 1 sites, Jdiag ∼ J couples the S = 1 along the diagonal

While this situation seems similar to the orbital polaron obtained for perovskite manganites [16], there is an important difference: In the manganites, the couplings within the polaron are ferromagnetic and the polaron therefore has a large total spin. In the present case, however, the antiferromagnetic couplings within the polaron result in a total low-spin state [12]. The most important antiferromagnetic coupling is caused by virtual hopping processes between two spin-one sites. Figure 1(b) shows one of the relevant processes involving t2g and eg orbitals. In addition, there is another, small, antiferromagnetic contribution and some ferromagnetic coupling via Hund’s rule, which is likewise small enough to be neglected. In total, the antiferromagnetic coupling has been estimated to be J = 10 − 20 meV [12]. A similar process mediated by longer-range hopping leads to an antiferromagnetic coupling Jdiag along the diagonals of the spin-orbital polaron Fig. 2. Figure 1(c) shows another virtual hopping process that involves two eg orbitals and could connect the central hole to S = 1 sites. It is a double exchange process, similar to the one responsible for the large total spin of the polarons in perovskite manganites, and would induce a large ferromagnetic coupling. In Nax CoO, however, the relevant Co-O-Co bonds have a 90-degree angle, which strongly suppresses this process, in contrast to the 180-degree bond angle found in the cubic perovskites. One of the two remaining virtual hopping processes is ferromagnetic, the other is antiferromagnetic and both are of a similar strength [12]. We do therefore not know the strength or the sign of this interaction J , apart from the fact that it is expected to be weaker than the coupling J between two Co3+ ions. It has, however, been found that a ferromagnetic coupling ﬁts experimental data better than an antiferromagnetic one [13]. These considerations lead to an effective spin Hamiltonian: 6

3

6

i=1

i=1

i=1

H = J ∑ Si Si+1 + Jdiag ∑ Si Si+3 + J ∑ s0 Si ,

(1)

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where Si , i = 1, . . . , 7, S7 ≡ S1 denote the S = 1 spins in the outside ring and s0 the s = 1/2 in the center. J, Jdiag , and J are the coupling constants introduced above, see also Fig. 2. The relevant Hilbert space contains 1,458 states, and we can easily diagonalize it numerically and calculate thermodynamic quantities of interest.

2 Results and Discussion If the central spin s0 = 1/2 does not couple to the outside ring (J = 0), the S = 1 sites obviously order antiferromagnetically for the physically relevant J, Jdiag > 0. Since a large ferromagnetic coupling of the central hole to the surrounding sites is responsible for the large spin of orbital polarons in manganites [16], it is interesting to examine how large a coupling J is necessary to overcome J and to induce a larger spin. Figure 3 shows the ground state energy of the polaron depending on J for J = 1. The energy does not depend on J as long as the S = 1 ring remains in a singlet state. Both antiferromagnetic and ferromagnetic J can polarize the ring, if enough energy can be gained from activating J . However, quite large values of J would be necessary even for Jdiag = 0 – in all probability larger than what is plausible for Nax CoO2 . In the following, we will use J = Jdiag = 1 and J = −0.5. Next, we want to examine the magnetic susceptibility of such a polaron: z 2 χpol (T ) = (Stot ) /T = ∑

e

E − k lT

b

ZT

l

El |

2

6

∑ Siz

|El ,

(2)

i=0

−8 S

ring

E0 / J

−8.5

=0

−9 Sring = 1 −9.5

−10 −3

Sring = 1

−2

−1

0

1

2

J’ / J

Fig. 3 Ground state energy E0 vs. J for J = 1, Jdiag = 0. ×: The outside ring has Sring = 1, which couples to the central spin to give total spin Stot = 3/2. : Sring = 0, Stot = 1/2. : Sring = 1, Stot = 1/2

Spin-Orbital Polarons in Cobaltates

53

z with Stot the z component of the total spin and temperature T . El gives the eigen energy of state |El , Siz the z-component of the spin at site i, kb denotes the E − l

Boltzmann constant and Z = ∑l e kb T the partition function. From the susceptibility of a polaron (2), we obtain the theoretical susceptibility

χtheory =

g2 µB2 NA (1 − x) χpol, J

(3)

with Land´e g-factor g = 2, Bohr magneton µB , Avogadro’s constant NA and doping x. In order to compare our results to those of Ref. [11], we choose x = 0.82, J = 20 meV, Jdiag = J = 20 meV and J = −J/2 = −10 meV. The polaron susceptibility depending on temperature T is shown in Fig. 4 together with experimental results taken from Ref. [11]. In addition to the total susceptibility (), Fig. 4 also shows the susceptibilities of the central spin-1/2 site (◦) and of the outside spin-one ring (×) separately. The former is proportional to 1/(4T ) and is the result one expects if hole doping does not induce spins at the surrounding sites. At high temperatures, it is clearly much smaller than the experimental values, which means that the spins of the holes alone can not explain the observed susceptibility. The susceptibility of the ring, on contrast, increases with T at low temperatures and is quite high at intermediate and large ones. The total susceptibility (due to J < 0, this is not exactly the sum of the two contributions, but slightly larger) can explain the experimental values quite well: The

0.0020 J

diag

= J, J’ = − 0.5 J J = 20 meV

−1

χ (emu mol )

0.0015

total polaron ring spin S = 1/2 exp., c exp., ab

0.0010

0.0005

0.0000

0

100

200

300

400

T/K

Fig. 4 Susceptibility for x = 0.82 depending on temperature. : Susceptibility of the polaron Fig. 2; ◦: susceptibility of a single s = 1/2; ×: susceptibility of the S = 1 ring. Solid (dashed) line without symbols: Experimental data from Ref. [11] for Na0.82 CoO2 with magnetic ﬁeld parallel (perpendicular) to the ab plane. For the theoretical data, x = 0.82, J = 20 meV, Jdiag = J = 20 meV and J = −J/2 = −10 meV were used

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antiferromagnetic correlations in the ring lead to a ﬂat temperature dependence, and hence to the large high-temperature susceptibilities seen in experiment. At very low temperatures, ≈20 K, the measured susceptibility shows an antiferromagnetic ordering; the low temperature phase is A-type antiferromagnetic order [7, 8]. In this structure, the correlations within the layers containing the polarons are ferromagnetic, while the correlations between the layers are antiferromagnetic. At such low temperatures, antiferromagnetism within the polarons is strong – see the small susceptibility of the S = 1 ring in Fig. 4 – but these act only on very short distances. The long-range coupling is determined by the interaction between individual polarons. While their effective mass is large, they can still move and when two polarons come near each other, they may form bipolarons for a short time, which leads to a weak ferromagnetic interaction [13]. The interlayer coupling is antiferromagnetic and of similar strength; these couplings result in an A-type antiferromagnetic order. Further insight into the internal degrees of freedom of the polaron can be gained from the dynamic spin-structure factor 6

S(k, ω ) = ∑ δ (ω − Em + E0 )|Em | ∑ eikrl Slz |E0 |2 m

(4)

l=0

depending on momentum k and frequency ω . Em (E0 ) is the eigen (ground-state) energy for eigen (ground) state |Em (|E0 ). Slz is the z-component of spin l at located at rl in real space. Figure 5 shows S(k, ω ) of the polaron in addition to a spin wave dispersion inferred from neutron scattering data [7]. The internal degrees of freedom of the polaron lead to dispersionless signals at energies of ∼15 meV, which are strongest at the zone boundary. Once the spin waves reach this frequency, they start

20 18 16

ω / meV

14 12 10 8 6 4 2 0 0

0.05

0.1

0.15 0.2 (h, h, 0)

0.25

0.3

Fig. 5 Dynamic spin structure factor Eq. (4) of a polaron J = 20 meV, Jdiag = J = 20 meV and J = −J/2 = −10 meV (gray shades, peaks have been broadened with Gaussians). The signal at the K-point (1/3, 1/3, 0) shows the antiferromagnetism within the polaron. Solid line: Ferromagnetic (within the layers) spin-wave with parameters from Ref. [7]

Spin-Orbital Polarons in Cobaltates

55

to scatter on these internal degrees of freedom, which might explain why spin waves broaden and become hard to measure [7].

3 Conclusions We have discussed the properties of spin-orbital polarons in Nax CoO2 at small hole doping, i.e., large x. Apart from their large thermopower, these compounds also have very peculiar magnetic properties: The Curie-Weiss temperature obtained from high-temperature magnetic susceptibility has opposite sign to the one inferred from low-temperature spin-wave data [7]. This means that magnetic correlations within the Co-planes appear to be antiferromagnetic at high temperatures and small distance, while they seem ferromagnetic at low temperature and large distance. Not only do the spin-orbital polarons offer a solution to this seeming contradiction and give a qualitative explanation of the observed magnetic peculiarities [12, 13], but we can even obtain a quantitative description of experimental data for the magnetic susceptibility [11] and can suggest a mechanism for the observed spin-wave broadening [7].

References 1. K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R. A. Dilanian, T. Sasaki, Nature 422, 53 (2003) 2. M. Lee, L. Viciu, L. Li, Y. Wang, M. L. Foo, S. Watauchi, R. A. Pascal Jr, R. J. Cava, N. P. Ong, Nature Materials 5, 537 (2006) 3. M. L. Foo, Y. Wang, S. Watauchi, H. W. Zandbergen, T. He, R. J. Cava, N. P. Ong, Phys. Rev. Lett. 92, 247001 (2004) 4. K. Fujita, T. Mochida, K. Nakamura, Jpn. J. Appl. Phys. 40, 4644 (2001) 5. M. Mikami, M. Yoshimura, Y. Mori, T. Sasaki, R. Funahashi, M. Shikano, Jpn. J. Appl. Phys. 42, 7383 (2003) 6. Y. Wang, N. S. Rogado, R. J. Cava, N. P. Ong, Nature 423, 425 (2003) 7. S. P. Bayrakci, I. Mirebeau, P. Bourges, Y. Sidis, M. Enderle, J. Mesot, D. P. Chen, C. T. Lin, B. Keimer, Phys. Rev. Lett. 94, 157205 (2005) 8. L. M. Helme, A. T. Boothroyd, R. Coldea, D. Prabhakaran, D. A. Tennant, A. Hiess, J. Kulda, Phys. Rev. Lett. 94, 157206 (2005) 9. J. L. Gavilano, D. Rau, B. Pedrini, J. Hinderer, H. R. Ott, S. M. Kazakov, J. Karpinski, Phys. Rev. B 69, 100404(R) (2004) 10. T. Motohashi, R. Ueda, E. Naujalis, T. Tojo, I. Terasaki, T. Atake, M. Karppinen, H. Yamauchi, Phys. Rev. B 67, 064406 (2003) 11. S. P. Bayrakci, C. Bernhard, D. P. Chen, B. Keimer, R. K. Kremer, P. Lemmens, C. T. Lin, C. Niedermayer, J. Strempfer, Phys. Rev. B 69, 100410(R) (2004) 12. G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005) 13. M. Daghofer, P. Horsch, G. Khaliullin, Phys. Rev. Lett. 96(21), 216404 (2006). URL http://link.aps.org/abstract/PRL/v96/e216404 14. G. Lang, J. Bobroff, H. Alloul, P. Mendels, N. Blanchard, G. Collin, Phys. Rev. B 72, 094404 (2005) 15. C. Bernhard, A. V. Boris, N. N. Kovaleva, G. Khaliullin, A. V. Pimenov, Li Yu, D. P. Chen, C. T. Lin, B. Keimer, Phys. Rev. Lett. 93, 167003 (2004) 16. R. Kilian, G. Khaliullin, Phys. Rev. B 60, 13458 (1999)

Quantum Corrections to the Ising Interactions in LiY1−xHox F4 A. Chin and P.R. Eastham

Abstract We systematically derive an effective Hamiltonian for the dipolar magnet LiY1−x Hox F4 , including quantum corrections which arise from the transverse dipolar and hyperﬁne interactions. These corrections are derived using a generalised Schrieffer-Wolff transformation to leading order in the small parameters given by the ratio of the interaction energies to the energy of the ﬁrst excited electronic state of the Holmium ions. The resulting low-energy Hamiltonian involves two-level systems, corresponding to the low-lying electronic states of the Holmiums, which are coupled to one another and to the Holmium nuclei. It differs from that obtained by treating the electronic states of the Holmium as a spin-1/2 with an anisotropic g-factor. It includes effective on-site transverse ﬁelds, and both pairwise and threebody interactions among the dipoles and nuclei. We explain the origins of the terms, and give numerical values for their strengths.

1 Introduction The rare-earth compound LiY1−x Hox F4 has often been described as a model quantum magnet, and as such has been studied for over three decades [1, 2, 3, 4, 5, 6]. The basic model for this material is a diluted Ising model with dipole interactions, and depending on the dilution the low-temperature phase is expected to be either a ferromagnet or a spin-glass [7]. Applying a strong transverse ﬁeld to the material introduces quantum ﬂuctuations which can lead to domain wall tunnelling in the ferromagnet [8], and to quantum melting of the ferromagnet [3] and the spin-glass A. Chin T.C.M., Cavendish Laboratory, J J Thomson Avenue, Cambridge, CB3 0HE, U.K. e-mail: [email protected] P.R. Eastham T.C.M., Cavendish Laboratory, J J Thomson Avenue, Cambridge, CB3 0HE, U.K. e-mail: [email protected]

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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[9, 10, 11, 12, 13]. Even in the absence of an applied transverse ﬁeld, however, quantum tunnelling of magnetic dipoles can be observed in LiY1−x Hox F4 [4, 14]. Furthermore, samples with a Holmium concentration of 4.5% do not appear to behave as a spin-glass on cooling, as expected for a classical Ising model. Instead they show unexplained “antiglass” properties which have been attributed to quantum mechanics [5]. This attribution is supported by theoretical predictions of the static susceptibility, which agree with experiments only once quantum corrections are included [6]. Although quantum effects are implicated in several phenomena in LiY1−x Hox F4 in the absence of an applied transverse ﬁeld, there is no derivation in the literature of an effective low-energy Hamiltonian which contains them. Here we derive such a Hamiltonian, projecting out the high-energy electronic states to obtain a theory describing the low-lying electronic doublet and the nuclei. We shall ﬁnd a Hamiltonian which has a different form from that appropriate in a strong applied ﬁeld [15], and whose electronic part differs from the two-level model previously proposed for the zero-ﬁeld case [6].

2 Model The magnetic degrees of freedom in LiY1−x Hox F4 are the f electrons on the Ho3+ ions. The strong spin-orbit coupling of the ions leads to a well-deﬁned J = 8 for the ions, and an associated dipole moment µ = gL µB , with the Land´e g-factor gL = 5/4. The 2J + 1 = 17-fold degeneracy of the free ion is broken by the crystal-ﬁeld Hamiltonian, leaving a degenerate ground-state doublet. All matrix elements of Jx and Jy are zero within this ground-state subspace, but there are non-zero matrix elements of Jz . This is the source of the strong Ising anisotropy in the interactions and response to an applied ﬁeld. The interactions between Holmium ions are dipolar, giving an interaction Hamiltonian Hint =

1 µ0 µB2 g2L (Ji · J j − 3(ˆri j · Ji )(ˆri j · J j )). ∑ 2 i, j 4π r3i j

(1)

The crudest way to obtain a low-energy effective Hamiltonian from (1) is to truncate the electronic state-space to the ground-state doublet on each Holmium ion. It is possible to choose a basis for the doublet in which Jz has no off-diagonal matrix elements, while the diagonal matrix elements are α and −α . With this choice of basis, truncating leads to the dipolar Ising model Hint =

1 α 2 µ0 µB2 g2L ˆ 2 )σ z σ z , (1 − 3(ˆri j · k) i j 3 2∑ 4 π r i, j ij

(2)

Quantum Corrections to the Ising Interactions in LiY1−x Hox F4

59

where σ x , σ y and σ z are the usual Pauli matrices. As discussed in previous studies of this system [10, 15, 3, 4] the contact hyperﬁne interaction between the Holmium nuclei and the electronic degrees of freedom also plays an important role, as it is typically of similar strength to the dipolar interaction. Under the simple two-state truncation scheme described above the hyperﬁne interaction also takes a simple Ising form, Hhyp =

∑ AJ Ii · Ji

(3)

−→ α ∑ AJ Iiz σiz ,

(4)

i

i

as do other couplings to the Holmium moments. The procedure of projecting into the low-energy subspace of the single-ion Hamiltonian has been used in the presence of an applied transverse ﬁeld [15]. The ﬁeld changes the low-energy states, leading to ﬁnite matrix elements for Jx and Jy within the low-energy doublet. This leads to corrections to the Ising forms (2,4) that are pair interactions involving at least one of σ x , σ y , I x , I y , and to effective ﬁeld terms. To obtain quantum terms in the absence of an applied transverse ﬁeld we must go beyond a simple projection onto the low-lying single-ion states. We thus now consider the lowest three levels of the crystal-ﬁeld Hamiltonian, denoting the two states of the doublet as | ↑, | ↓ and the ﬁrst excited state, with energy ∆ , as |Γ . The parameters in such a three-level model are ∆ and the matrix elements of the angular momentum operators, and can be obtained from a numerical diagonalisation of the full crystal-ﬁeld Hamiltonian. With an appropriate choice of basis this gives [15] ↑ |Jz | ↑ = −↓ |Jz | ↓ = 5.52 = α , ↓ |Jx |Γ = ↑ |Jx |Γ = 2.4 = ρ ,

(5) (6)

↓ |Jy |Γ = −↑ |Jy |Γ = iρ ,

(7)

∆ = 10.8 K.

(8)

All other matrix elements of angular momentum among the three states vanish.

3 Derivation of the Effective Hamiltonian Quantum corrections to the Ising Hamiltonian (2, 4) appear once one includes the state at |Γ because the interaction terms (1) and (3) couple |Γ to the electronic ground-state doublet. The typical nearest-neighbour dipolar interaction energy scale is ≈300 mK and AJ ≈ 38 mK [4, 15]. Since these interaction scales are small compared with ∆ the couplings to the |Γ states can be treated in perturbation theory, and an effective Ising model obtained using the standard Schrieffer-Wolff procedure

60

A. Chin and P.R. Eastham

[16]. The resulting model will include quantum terms arising from virtual transitions between the ground-state doublet and the |Γ state. To derive the effective Hamiltonian we begin by writing the three-level model as H = H0 + HT , where H0 contains terms which do not couple the doublet to |Γ , and HT contains those which do. Measuring energy from the electronic ground-state doublet, H0 contains the crystal ﬁeld term Vc = ∆ ∑i |Γi Γi | and the Ising parts of the interactions, while HT contains the parts of the interactions which involve Jx and Jy . Including the hyperﬁne interaction the Hamiltonian can be split as H0 = Vc + HT =

1 Jizzj Jiz J zj + ∑ AJ Iiz Jiz , 2 i∑ i =j

(9)

1 νµ ν µ Ji j Ji J j + AJ ∑(Iix Jix + Iiy Jiy ), 2 i∑ i =j

(10)

ν ,µ

where ν , µ = x, y, z, but the prime on the sum indicates that we must exclude all terms that have ν = µ = z. We then seek a unitary transformation H → eS He−S which decouples the electronic doublet from the |Γ state to ﬁrst order in HT . Such a transformation obeys (11) [S, H0 ] = −HT . An S which approximately satisﬁes (11) can be constructed from HT using projection operators, S=

νµ ν µ ν µ ν µ ν µ Ji j (Γi Γj Ji J j Pi Pj − Pi Pj Ji J j Γi Γj )

∑

2εν µ ∆

i= j ν ,µ

−

AJ ∆

+

AJ ∆

∑ Pi (Iix Jix + Iiy Jiy )Γi ,

∑ Γi (Iix Jix + Iiy Jiy )Pi i

(12)

i

Pi = | ↑i ↑i | + | ↓i ↓i |, |Γ Γ | if ν = x, y ν Γi = , if ν = z Pi 2 if ν = µ εν µ = . 1 if ν = µ

(13) (14) (15)

The form (12) for S actually obeys the relation [S,Vc ] = −HT , and although we have successfully eliminated the linear term in HT the remainder of the commutator [S, H0 − Vc ] generates new couplings between the doublet and |Γ . However these tunnelling terms are of order (interaction)2 /∆ , and as they can only couple between the ground states and |Γ , they can only contribute to the effective low-energy Hamiltonian in higher order perturbation theory. Therefore they do not contribute to the effective low-energy Hamiltonian to leading order in (interaction)2 /∆ , and

Quantum Corrections to the Ising Interactions in LiY1−x Hox F4

61

vanish when we project onto the Ising basis at the end of the Schrieffer-Wolff procedure. Having eliminated HT , the effective two-state Hamiltonian is given to lowest order in (interactions)2 /∆ by [16] 1 Heff = ∏ Pi (H0 + [S, HT ]) ∏ Pi . 2 i i

(16)

This form extends the Ising interaction Hamiltonian by including second-order processes in which HT causes virtual transitions from the electronic doublet to |Γ and then back again.

4 Effective Hamiltonian Substituting the form (12) for S into (16) and discarding irrelevant energy shifts we ﬁnd that the effective Hamiltonian takes the form Heff = H0 + HD + HDTB + HN + HND .

(17)

This Hamiltonian operates in a space with sixteen states for each ion: two low-lying electronic states, each with the eight nuclear states. The various contributions to Heff are labelled according to the interactions that give rise to them.

4.1 Dipolar Processes The ﬁrst term in (17) arises from the second-order dipolar processes that involve only single pairs of spins, and is νµ

µ

HD = ∑(hxi σix + hij σiy ) + ∑ ∆i j σiν σ j . i

(18)

i= j

We see that the pairwise dipolar interaction generates both transverse interaction terms and effective magnetic ﬁelds. In the appendix we give expressions for the strengths of the parameters hi and ∆ ν µ in terms of the parameters of the underlying three-level model. Using these relations we estimate the characteristic magnitude of these corrections by calculating the contribution from a single nearest-neighbour. This yields hx ≈ 3 mK, hy ≈ 0 and ∆ ν µ ≈ 0.06 mK, which may zz ≈ 300 mK, and the Ising hyperﬁne be compared to the dipolar Ising energy, JNN splitting, α AJ ≈ 210 mK. Note that terms such as (18) describe tunnelling between the low-lying electronic states. It is important to stress that this is not sufﬁcient to generate tunnelling between the two electro-nuclear ground states of H0 [10].

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In addition to the dipolar processes involving only a single pair of Holmium ions there are processes involving three ions. In such a process the interaction between one pair of spins, say i, j, virtually excites the jth spin into |Γ , and the dipolar interaction of this spin with the kth spin brings the jth spin back into the Ising doublet. This generates two- and three-body interaction terms in the effective theory, HDTB = −

α 2ρ 2 (Jikxz Jixzj + Jikyz Jiyzj )σkz σ zj ∆ i=∑ j=k

−

α 2ρ 2 (Jikxz Jixzj − Jikyz Jiyzj )σkz σ zj σix ∆ i=∑ j=k

−

α 2ρ 2 (Jikxz Jiyzj + Jikyz Jixzj )σkz σ zj σiy . ∆ i=∑ j=k

(19)

For an isolated group of three spins in which {i, j} and {i, k} are nearest neighbours we obtain 3 mK for the magnitude of the interaction terms σkz σ zj , σkz σ zj σix and σkz σ zj σiy .

4.2 Hyperﬁne Processes The contributions to the effective Hamiltonian from the hyperﬁne interactions are simpler to deal with, as they are conﬁned to each site. They are HN =

ρ 2 A2J 2 (Ii − (Iiz )2 ) + ((Ii+ )2 + (Ii− )2 )σix ∑ ∆ i −

ρ 2 A2J + 2 i((Ii ) − (Ii− )2 )σiy − 2Iiz σiz . ∆ ∑ i

(20)

We see that in second-order perturbation theory the hyperﬁne interaction leads to spin ﬂipping terms which require the z component of the nuclear spin to change by two units, gives a slight correction to the longitudinal hyperﬁne interaction, and introduces a weak nuclear anisotropy. The factors outside the sum give 0.86 mK for the interaction energy.

4.3 Mixed Hyperﬁne-Dipolar Processes Finally, the term in the effective Hamiltonian labelled HND arises due to secondorder processes whereby the electrons are virtually excited into the state |Γ by the dipolar interaction and then de-excited by the hyperﬁne interaction, or vice versa:

Quantum Corrections to the Ising Interactions in LiY1−x Hox F4

HND = − −

αρ 2 AJ ∆

i= j

αρ 2 AJ ∆

i= j

63

∑ Jizxj (Iix (1 + σix) + Iiyσiy )σ zj ∑ Jizyj (Iiy (1 − σix) + Iixσiy )σ zj .

(21)

Note that the nuclei mediate a coupling between electronic states of the form σ x,y σ z which does not occur in HD .

5 Discussion In recent papers [11, 12, 10, 13] it was shown that the non-Ising parts of the dipolar coupling provide a route by which an applied transverse ﬁeld can destroy spin-glass order in LiY1−x Hox F4 . Although our model contains spontaneous transverse ﬁelds we do not expect them to destroy the spin-glass phase in this way, because the model retains time-reversal symmetry. An interesting feature of the effective Hamiltonian is that the spontaneous transverse ﬁelds and the two-and three-body interactions are all correlated. Theoretical work on spin models with several forms of random couplings, such as the random transverse-ﬁeld Ising model [17], generally considers different interaction terms as independent. The approach given here could be straightforwardly extended to derive an effective Hamiltonian for LiY1−x Hox F4 in an applied transverse ﬁeld, so long as the ﬁeld strength is sufﬁciently small for perturbation theory to apply. This generates a variety of new processes at second order. The straightforward process, in which the applied ﬁeld excites to the |Γ state and back again, leads to an electronic tunnelling term ∝ B2 /∆ . But note that there will also be mixed processes, for example where the applied ﬁeld causes the transition to |Γ and the dipole interaction causes the transition back into the electronic doublet, which will generate electronic tunnelling terms ∝ B/∆ . We stress that although our Hamiltonian contains terms which generate transitions between the states of the electronic Ising doublet, and further such terms will be introduced by an applied ﬁeld, these terms alone do not couple the doublydegenerate electro-nuclear ground states of the Ising single-ion Hamiltonian. To generate quantum ﬂuctuations between the electro-nuclear ground states requires processes which ﬂip the nuclear spin, and hence the splittings of the electro-nuclear ground states will be much smaller than the tunnelling terms connecting the bare electronic states. Although we do not expect the quantum corrections to formally destroy the ordered phase unless there is an applied ﬁeld, they can still affect the thermodynamics, changing the susceptibilities and moving the phase boundary. However, we only expect this to occur at very low temperatures. In this context we note that the experimental susceptibility of LiY1−x Hox F4 in the 10–100 mK temperature regime of the antiglass experiment has been reproduced by a quantum theory of a two-level model

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[6]. That model is obtained by neglecting the nuclei and treating the Holmium ions µ µ as spin-1/2 ions with an anisotropic g-factor, i.e. writing Ji = g µ σi , with µ = x, y, z. We note that the resulting Hamiltonian is very different from that which would be obtained by neglecting the nuclei in our model: there are no spontaneous ﬁeld terms and the non-Ising interactions decay as 1/r3 (here 1/r6 ), while the quoted values for g give the energy scale for the largest non-Ising coupling of ≈30 mK, whereas here we have 3 mK. While we do not expect the small corrections derived here to affect thermodynamics except at very low temperatures, they may be relevant to understanding dynamics at much higher temperatures. Quantum tunnelling of the magnetisation has been observed in the magnetisation relaxation and susceptibility of the dilute compound LiY0.998 Ho0.002 F4 , due to both single-ion [4] and two-ion processes [14], and it would be interesting to compare the details of these results with the processes given here. An understanding of the single or few-ion electro-nuclear dynamics, based on the Hamiltonian (17), may also help to explain the antiglass experiments [5]. In the high-temperature regime these experiments show a characteristic relaxation time for the magnetisation which is activated, with a barrier similar to the width of the hyperﬁne multiplet and the nearest-neighbour interaction strength. Extrapolating this behaviour indicates that this particular activated dynamics freezes out on the experimental frequency scale as the temperature is lowered into the antiglass regime. The quantum terms derived here, although small, may perhaps then provide a route to the observed dynamics. The potential signiﬁcance of quantum electro-nuclear dynamics for the antiglass experiment has been noted by Atsarkin [18], who proposes a speciﬁc relaxation mechanism due to interactions of the form (18).

6 Conclusions In this work we have motivated the need to go beyond the simple dipolar electronuclear Ising Hamiltonian commonly used to describe LiY1−x Hox F4 , and have derived an effective low-energy Hamiltonian which includes the quantum corrections caused by the transverse elements of the dipolar and hyperﬁne interactions. We have given estimates for the typical magnitudes of these correction terms, and have shown that they are typically about one percent as strong as the energy scale associated with the Ising interactions. As a result, we do not expect any qualitative changes to the Ising phase diagram, but as we have highlighted in this work, these quantum corrections can describe a large variety of single and many-body processes which might play signiﬁcant roles in the observed dynamics. Thus, for low temperatures, our effective Hamiltonian should serve as a good starting point for a microscopic investigation of the dynamical physics of LiY1−x Hox F4 , and as it can be used across the whole dilution series, should contain the rich low-energy physics which characterises the spin glass, free ion, and presumably, the “antiglass” phases of this material.

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65

Acknowledgements We are grateful to Misha Turlakov, Peter Littlewood, and to the participants of the summer school, for helpful and interesting discussions of this problem.

Appendix The components of the effective magnetic ﬁeld hi , deﬁned in (18), are related to the original microscopic Hamiltonian by hxi =

ρ 2α 2 ρ4 zy 2 zx 2 ((J ) − (J ) ) + ij ij ∆ ∑ 2∆ j

hyi = −2

ρ 2α 2 ρ4 Jizyj Jizxj − ∑ ∆ 2∆ j

hzi = 0.

∑((Jiyyj )2 − (Jixxj )2 ),

(22)

j

∑ Jixyj (Jixxj + Jiyyj ),

(23)

j

(24)

The components of the effective magnetic ﬁeld vanish in the undiluted crystal, as expected from the crystal symmetry. The transverse dipolar interactions between spins have coupling strengths ρ 4 2(Jixyj )2 − (Jixxj )2 − (Jiyyj )2 ∆ixxj = , (25) 4∆ −ρ 4 Jixxj Jiyyj + Jixyj Jixyj ∆iyyj = , (26) 2∆ ρ 4 Jixxj Jiyyj − Jixyj Jixyj ∆izzj = , (27) 2∆ ρ 4 Jiyyj Jixyj − Jixxj Jixyj ∆ixyj = , (28) 2∆

∆iyxj = ∆ixyj .

(29)

These are the only couplings generated; there are no terms such as σ x σ z , since these would break time-reversal symmetry.

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References 1. A.H. Cooke, D.A. Jones, J.F.A. Silva, M.R. Wells, J. Phys. C 8, 4083 (1975) 2. D.H. Reich, B. Ellman, J. Yang, T.F. Rosenbaum, G. Aeppli, D.P. Belanger, Phys. Rev. B 42, 4631 (1990) 3. D. Bitko, T.F. Rosenbaum, G. Aeppli, Phys. Rev. Lett. 77, 940 (1996) 4. R. Giraud, W. Wernsdorfer, A.M. Tkachuk, D. Mailly, B. Barbara, Phys. Rev. Lett. 87, 057203 (2001) 5. S. Ghosh, R. Parthasarathy, T.F. Rosenbaum, G. Aeppli, Science 296, 2195 (2002) 6. S. Ghosh, T.F. Rosenbaum, G. Aeppli, S.N. Coppersmith, Nature 425, 48 (2003) 7. M.J. Stephen, A. Aharony, J. Phys. C 14, 1665 (1981) 8. J. Brooke, T.F. Rosenbaum, G. Aeppli, Nature 423, 610 (2001) 9. W. Wu, B. Ellman, T.F. Rosenbaum, G. Aeppli, D.H. Reich, Phys. Rev. Lett. 67, 2076 (1991) 10. M. Schechter, P.C.E. Stamp, Phys. Rev. Lett. 95, 267208 (2005) 11. M. Schechter, N. Laﬂorencie, Phys. Rev. Lett. 97, 137204 (2006) 12. S.M.A. Tabei, M.J.P. Gingras, Y.J. Kao, P. Stasiak, J.Y. Fortin, Phys. Rev. Lett. 97, 237203 (2006) 13. M. Schechter, P.C.E. Stamp, N. Laﬂorencie, J. Phys.: Condens. Matter 19, 145218 (2007) 14. R. Giraud, A.M. Tkachuk, B. Barbara, Phys. Rev. Lett. 91, 257204 (2003) 15. P.B. Chakraborty, P. Henelius, H. Kjønsberg, A.W. Sandvik, S.M. Girvin, Phys. Rev. B 70, 144411 (2004) 16. J.R. Schrieffer, P.A. Wolff, Phys. Rev. 149, 491 (1966) 17. Z.Q. Liu, X.M. Kong, X.S. Chen, Phys. Rev. B 73, 224412 (2006) 18. V.A. Atsarkin, Phys. Lett. A 130, 492 (1988)

Spin-Orbital-Lattice Physics in Ca-Based Ruthenates Mario Cuoco, Filomena Forte, and Canio Noce

Abstract In this paper, we review some recent results, obtained by means of exact diagonalization technique, about the competition between the octahedral distortions and the Coulomb interactions for the t2g electrons in layered Ca-based ruthenates. We provide a scenario where the ﬂattening of the octahedra is the driving mechanism for yielding an antiferromagnetic state with C- or G-type structure and different orbital conﬁgurations. On the other hand, we show that the elongation of the octahedra gives rise to a ground state with incomplete ferromagnetism. To further account for unconventional spin-orbital conﬁgurations in the Ca2 RuO4 system, where all the t2g degrees of freedom contribute, the role of the spin-orbit coupling and its competition with compressed octahedral deformations are investigated. One of the main ﬁndings is the occurrence of anisotropic spin patterns with partially ﬁlled orbital occupation and coexisting ferro- and antiferro-type correlations in the spin/orbital channel.

1 Introduction The role of orbital degree of freedom has been investigated as one of the most important issues for the comprehension of the physical properties of t2g transition metal oxides (TMO) such as for instance tithanates, vanadates and ruthenates, through strong couplings with charge, spin, and lattice degrees of freedom (see [1] for a M. Cuoco, Laboratorio Regionale SuperMat, INFM-CNR, and Dipartimento di Fisica “E. R. Caianiello”, Universit`a di Salerno, I-84081 Baronissi (Salerno), Italy, F. Forte Laboratorio Regionale SuperMat, INFM-CNR, and Dipartimento di Fisica “E. R. Caianiello”, Universit`a di Salerno, I-84081 Baronissi (Salerno), Italy, e-mail: [email protected] C. Noce Laboratorio Regionale SuperMat, INFM-CNR, and Dipartimento di Fisica “E. R. Caianiello”, Universit`a di Salerno, I-84081 Baronissi (Salerno), Italy

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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review). Among the t2g TMO materials, considerable interest has been focused on the origin and nature of the complex and rich phase diagram shown in single-layered ruthenate Ca2−x Srx RuO4 [2]. Indeed, this compound exhibits different ground state properties including a Mott transition, accompanied with orbital ordering for x = 0, heavy mass Fermi liquid behavior near the critical point for x = 0.5, and the orbital degenerated triplet superconductivity for x = 2 [3, 4, 5, 6, 7, 8]. It has been shown that the evolution of physical properties with the doping x is non-monotonous, and a close relationship among electrical, magnetic, and structural properties does exist [7, 5]. The isovalent substitution of Sr for Ca does not change the number of electrons, but, due to their different ionic radii, modiﬁes the crystal structure systematically [6, 7]. Since Ca2−x Srx RuO4 has four 4d electrons in the t2g orbitals, the relevance of the orbital degree of freedom is invoked. However, the orbital states have not been established: even the propagation vector of orbital ordering for x = 0 has not been determined yet. Referring to this speciﬁc doping, we note that Ca2 RuO4 undergoes a ﬁrst order transition from a high temperature metallic to a low temperature insulating phase at TMI = 357 K [4, 8, 7] triggered by a structural transition from orthorhombic to tetragonal lattice symmetry, which leads to a fattening of the RuO6 octahedra perpendicular to the RuO2 layers. A further cooling down below TN = 110 K stabilizes a G-type antiferromagnetic order between the Ru spins. To understand the G-type antiferromagnetism, it would be naturally expected that the dxy orbital should be dominantly occupied due to the tetragonal crystal ﬁeld effect. On the other hand, the X-ray experiments estimated 0.5 holes/site in the dxy orbital and 1.5 holes in the dyz , suggesting that the ground state may favor the occupation of complex orbitals due to the strong spin-orbit coupling [9, 10]. Such a scenario has been confuted and further puzzled by the observation, via resonant X-ray diffraction, of an orbital ordering transition at a wave vector characteristic of the antiferromagnetic ordering [11]. Thus, the clear understanding of electronic structure of Ca2 RuO4 is still left to be established. In this paper, we review some recent ﬁndings and theoretical results about the orbital ordered states that were proposed as possible candidates of the insulating ground state of Ca2 RuO4 . The calculations usually start from a 3-orbital extended Hubbard model, where at each site the Coulomb repulsions among electrons in the same orbital, between electrons in different orbitals with opposite spins, and between electrons in different orbitals with the same spin, are fully taken into account. This model, which is believed to provide a good starting point to study the electronic properties of ruthenates, is usually combined with some ab-initio descriptions and investigated within different schemes of approximation. Indeed, there have been systematic Hartree-Fock studies to determine the most favorable phases with respect to the multiplet interaction parameters and the electron-lattice coupling [12, 13, 14]. Many works are based on the so-called LDA+U method, where an effective Coulomb repulsion Ue f f is introduced to reproduce the band gap of Ca2 RuO4 . Controversial scenarios have been proposed. Anisimov et al. [25] suggested a ferro-orbital dxy orbital ordered state (FO), where the strong superexchange interaction between the occupied majority-spin and unoccupied minority-spin dyz ,

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dzx orbitals stabilizes the AF ground state. Combining the LDA+U scheme with lattice distortion (ﬂattening, rotation, and tilting of RuO6 ), Fang et al. [14, 15] conclude the cooperative occupation of dxy orbital, favored by the two-dimensional crystal structure, even without the Jahn-Teller distortion of RuO6 . By combining the spectroscopic results with the transport and structural features as well as with the LDA+U calculation, Jung et al. [16]. have also demonstrated the crucial role of dxy orbital ordering. More recently Lee et al. [17] suggested the coexistence of antiferro- and ferro-orbital ordering in their optical study, the former favored by the kinetic energy gain of allowing hoppings between the nearest neighbors, the latter by the orthorhombic lattice distortion. Many attempts have been performed for going beyond the Hartree-Fock approximation, to take into account the strong correlation effects associated with the Mott metal-insulator transition. Routinely, the dynamical mean-ﬁeld method (DMFT) has been applied [25]. In this frame, due to the presence of the orbital degree of freedom, the Mott physics contains extra elements of unconventional character. Studies in this direction have indicated the possibility of orbital selective Mott transitions in reduced multi-orbital models, showing that separate Mott transitions occur at different Coulomb strengths, eventually merging into a single critical point only for special conditions [18] even though the debate on the phenomenology of the Mott physics in this type of systems is still under study [19, 20]. On the numerical side, by using exact diagonalization method based on Lanczos algorithm, Hotta and Dagotto [21] proposed a novel orbital ordered state, which results as a consequence of the combination of Coulomb interactions and lattice effects. In this antiferroorbital ordered state (AFO), the dxy orbital is fully occupied in a bipartite way in the plane, while two electrons ﬁll the dyz and dzx orbitals alternately. The related lattice distortions should occur in pairs, resulting in a cooperative distortion in the total in-plane lattice. Finally, the detection of a substantial orbital angular momentum, suggested the existence of strong spin-orbit coupling in ruthenates. Based on the experimental evaluation, by X-ray absorption spectroscopy, of 0.5 holes in the dxy orbital and 1.5 holes in the dyz and dzx orbitals, Mizokawa et al. [9] argued that the ground state may favor the occupation of complex orbitals due to the strong spin-orbit coupling. With the aim to shed new light on the physics of the single layered Ca-based ruthenates, we review here our recent results on the study of the spin, orbital and charge (SOC) patterns that develop into an extended 3-orbital Hubbard model [10, 22]. This analysis has been performed within an unbiased scheme of computation, as the exact diagonalization method applied on ﬁnite size clusters, in such a way to get more insight on the nature of the ground state conﬁgurations, by treating the competing microscopic mechanisms on equal footing. Speciﬁcally, in this paper we will address the following questions: (i) the competition between the charge ﬂuctuations induced by the Coulomb repulsion and those due to the crystal ﬁeld energy associated with ﬂat or elongated octahedra. Particularly, we will underline the role played by the dynamics of the double occupied conﬁgurations in selecting quantum states with different spin, orbital and charge structure. We will show that the compressive c-axis octahedral

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deformation generally leads to antiferromagnetic states of C- (G-type), depending on the orbital distribution of the double occupied states, with main antiferro(ferro-) orbital correlations, respectively. The inversion of the c-axis distortions, in the regime of expansive octahedra, modiﬁes the character of the magnetic exchange giving rise to ferromagnetic states with partial spin alignment, reﬂecting the subtle competition between the ferro and antiferro magnetic exchange. The nature of the ferromagnetism is strongly related to a mechanism of orbital frustration due to the peculiar connectivity of the t2g orbitals and to the two-dimensionality of the problem under examination; (ii) the role played by the spin-orbit interaction in modifying the conﬁgurations stabilized by the Coulomb repulsion and the tetragonal distortions. In particular, we will analyze how the spin-orbit interaction in presence of ﬂat distortions is able to stabilize an antiferromagnetic ground state with predominantly ferro-like orbital correlations and a partial ﬁlled charge distribution in the xy orbital sector. The outline of the paper is the following. In Section 2 we introduce the adopted microscopic model, while in Section 3 we present the results for the phase diagram in presence of Coulomb correlations and c-axis octahedral distortions. In Section 4, the coupling between spin and orbital dynamics is introduced and analyzed in the case of an environment with compressed octahedra. Section 5 is devoted to the temperature phase diagram exhibited by the systems when the spin-orbit coupling is neglected, and ﬁnally, the last section contains a discussion on the relevance of the results presented, when referred to the speciﬁc case of ruthenates, together with the concluding remarks.

2 Hamiltonian The microscopic ingredients we have to take into account are the following: (1) in single layered ruthenates, the Ru4+ ion contains four electrons in the 4d orbitals. Since the crystal ﬁeld splitting between eg and t2g orbitals is larger than the Hund coupling, the Ru4+ ion is in the low-spin state (S = 1); (2) the constraint to accommodate four electrons in the t2g manifold imposes a double occupation in one of the three orbitals; (3) the t2g orbitals are expected to have different energies because the RuO6 octahedra are deformed. A basic model Hamiltonian capturing the above features is built up by different contributions that reproduce the correlated local dynamics of electrons and electrons coupled to the structural distortions in the t2g manifold: H = Hkin + Hel−el + Hc f + Hso. (1) The ﬁrst term in Eq. 1 is the kinetic operator deﬁning the connectivity between the Ru t2g orbitals via the oxygen ions, Hkin = −t ∑ (di†ασ djασ + h.c.), i j,σ

(2)

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di†ασ being the creation operator for an electron with spin σ at the i site in the α orbital. The hopping amplitude is assumed to be t for all the orbitals in the t2g manifold, due to the symmetry relations of the connections via oxygen π ligands. The second term stands for the local Coulomb interaction between t2g electrons: Hel−el = U ∑ niα ↑ niα ↓ − 2JH ∑ Siα · Siβ iα

iαβ

JH + (U − ) ∑ niα niβ + J ∑ di†α ↑ di†α ↓ diβ ↑diβ ↓ , 2 iαβ iαβ

(3)

where niασ , Siα are the on site charge for spin σ and the spin operators for the α orbital, respectively. U (U ) is the intra (inter)- orbital Coulomb repulsion, JH is the Hund coupling, and J is the pair hopping term. Due to the invariance for rotations in the orbital space, the following relations hold: U = U + 2JH , J = JH . The Hc f part of the Hamiltonian H is the crystal ﬁeld term, directly related to the expansive or compressive mode of the RuO6 octahedra: Hc f = ∆ ∑(nixy − nixz − niyz ).

(4)

i

Zero amplitude for the parameter ∆ indicates no distortions and full local orbital degeneracy. Positive (negative) values of ∆ are related to elongated (ﬂat) RuO6 octahedron along the c-axis, and thus the occupation in the dγ z (dxy ) sector is favored, respectively. Within this description, the microscopic parameter ∆ contains both the contribution for symmetry lowering due to the static Coulomb potential of the surrounding oxygens and the energy shift due to the formation of antibonding molecular orbitals between the transition metal atom and the neighbors oxygens. Finally, since the spin-orbit interaction in the 4d shell is relevant, the orbital angular momentum L can strongly couple to the spin. By introducing the local orbital operator for the total angular momentum Li = (Lix , Liy , Liz ), the Hamiltonian for the spin-orbit interaction is then written as: Hso = −λ ∑ Li · Si ,

(5)

i

where λ is the coupling constant. We point out that, the present model may be considered as an effective one since the electron variables of the oxygens have been projected out implying that we are conﬁning ourselves to the pure dynamics of the 4d bands.

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3 Coulomb Interaction Versus c-axis Octahedral Distortions By means of the Lanczos technique, we performed the numerical simulation on a 2 × 2 plaquette, the smallest size that contains the symmetry features of a planar structure. One important aspect of the single particle connectivity is that for each direction there are only two active hoppings within the t2g sector. Hence, for the two-dimensional system, the xy orbital has a link both along the x and y direction, while the xz (yz) are connected only on the x(y) axis, respectively. The analysis of the ground state (GS) has been performed by evaluating the relevant correlation functions linked to the SOC pattern and by evaluating the spin gap between the GS and the lowest energy conﬁgurations in the other subspaces with different total spin projection. For this purpose, it is useful to introduce the following correlation functions in the available momentum space: S(q) = ∑ ei q(Ri −R j ) Siz Sjz

(6)

i, j

Pγγ (q) = ∑ ei q(Ri −R j ) piγ pjγ

(7)

i, j

where S(q) and Pγγ (q) are the spin and doublon structure factor, q being the characteristic wave vector associated with the spin and charge pattern. Here, Siz = ∑α Siα z is the total z projection on the site i, while piα = niα ,↑ niα ,↓ is the local double occupancy operator counting the double occupied conﬁgurations in the orbital α . We notice that we assume that α = z indicates the double occupancy operator in the γ z sector, i.e. piz = (pixz + piyz ). Moreover, when the phase diagrams are concerned, we adopt the notation Z(l) to indicate a phase with a magnetic character Z, whose local distribution of double occupied states is such that pixy = l and pixz = piyz = 1 − l. In particular, the Z character can be antiferromagnetic AF or ferromagnetic, like Fk with k being the average total projection of the spin momentum per plaquette (see Fig. 1). The F state stands for a GS conﬁguration where all the spins are polarized.

3.1 Fully Degenerate Case We start from the fully degenerate case (∆ = 0), where, due to the constraint of the two-dimensionality, there occurs a dynamical symmetry breaking related to the unequal kinetic energy for the xy and γ z bands. The results are summarized in Fig. 2, where we have reported the phase diagram as a function of the scaled Coulomb repulsion (U − JH ) versus the Hund coupling JH . The main part of the diagram is characterized by a C-AF(1/2) (See Fig. 1(f)) with antiferromagnetic correlations of C-type. Indeed, it turns out that in this region S(q) has a maximum for a wave vector amplitude of q = (0, π ) [q = (π , 0)], indicating the coexistence of antiferromagnetic exchange along the x (y) direction, and ferromagnetic spin coupling in the y (x),

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Fig. 1 A schematic view of the representative spin/charge conﬁgurations that contribute to the ground state on the 2 × 2 plaquette. The on-site orbital conﬁguration indicates the state occupied by a doublon. The arrow indicates the spin momentum of the remaining two electrons on the Ru site. (a) shows the orbital conﬁguration of the states which are doubly occupied. From (b) to (e) are reported the different ferromagnetic conﬁgurations. Here the circle on one bond indicates the tendency to form a singlet-like state that quenches the magnetic moments on the related sites. From (f) to (i) are plotted the possible antiferromagnetic-like states with different charge and orbital occupation. The orbital correlations between the DO states are mainly antiferro-orbital in the C-AF, ferro-orbital in the G-AF, and without any preferential pattern in the ferromagnetic conﬁgurations, respectively

respectively. Considering the orbital and doublon channel, the value of P in the xy sector indicates that the double occupied conﬁgurations distribute almost homogeneously without any preferential arrangement, though the value of Pγ ,γ (π , π ) has an amplitude larger than the longitudinal Pxy,xy (0, π ) [Pxy,xy (π , 0)]. This result indicates the tendency of the system towards an antiferro-type orbital pattern for the double occupied states (DO). Still, the average distribution of the DO conﬁgurations is such that one has half occupation in the xy band and half in the γ z sector. The occurrence of a C-AF state is a consequence of the one-dimensional character of the hopping connectivity for the γ z bands and of the orbital exchange on a given bond when the double occupation is not sitting on homologue orbitals.

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Fig. 2 Ground state diagram as a function of the scaled Coulomb repulsion (U − JH ) vs the Hund coupling energy JH at zero crystal ﬁeld amplitude for a 2 × 2 sites problem (from Ref. [22])

As expected, the large Hund limit is characterized by a region with full polarized spin distribution. The ferromagnetic F(1/4) (See Fig. 1(e)) region is marked by a low density per orbital of DO conﬁgurations in the xy band compared to the γ z sector. There occurs a local inter-orbital charge redistribution in the transition from the C-AF state to the F one, underlining the correlation between the DO density and the magnetic character of the ground state. This spin/charge coupling reﬂects the tendency of the system to exhibit a jump of the magnetization accompanied by a structural change under the application of an external ﬁeld, which is usually observed in the (Ca,Sr)-based family of ruthenates [23, 24].

3.2 Evolution from Flatten to Elongated Octahedra Now, we look at the evolution of the different GS conﬁgurations by simulating an inversion of the c-axis distortions, tuning the amplitude of the crystal ﬁeld potential from negative to positive values. To get insight into the competition between the charge ﬂuctuations induced by the c-axis distortions, and those due to the Coulomb repulsion, we ﬁx the ratio JH /U and we study the phase diagrams in the (JH /t, ∆ /t) plane. This parameterization allows for scanning the parameter phase space of the JH ,U along a speciﬁc direction in various coupling regimes. In particular, we consider two representative cases, (a) JH /U = 4/5 and (b) JH /U = 1/4 corresponding to the comparable (a) and small (b) Hund coupling with respect to the Coulomb repulsion.

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The phase diagram for the case (a) is reported in Fig. 3. For negative values of ∆ /t the system exhibits an antiferromagnetic behavior. Speciﬁcally, at large values of the crystal ﬁeld potential, as a consequence of the gain in the crystal ﬁeld energy, the C-AF state undergoes a transition into a G-AF state, characterized by a complete occupied xy orbital (dixy = 1) and singly occupied states for the γ z sector (diγ z = 0). This orbital conﬁguration is usually indicated as ferro-orbitally ordered (see Fig. 1(g)). Concerning the antiferromagnetic pattern, the spin structure factor gives a dominant peak at a wave vector q = (π , π ), indicating a G-type state, namely the effective exchanges between the local Ru spins are antiferromagnetic along both the planar directions. The magnetic character is easily understood in terms of the G-AF(1) conﬁguration, since now the degree of freedom of the xy orbital is completely frozen, and in the half-ﬁlled γ z sector the Coulomb correlations lead to dominant AF superexchange. The critical value of ∆ for the transition from the C-AF state to the G-AF state is of the order of the bare hopping amplitude in the range of [t, 2t], even for large Hund coupling and correspondent Coulomb repulsions. This aspect indicates a weak renormalization of the effective bandwidth via the Coulomb interaction, as the energy required to quench the DO conﬁgurations within the xy orbital is given by the direct competition between the gain in the crystal ﬁeld potential and the loss of kinetic energy. Another interesting feature is represented by the fact that the transition between the C-AF(1/2) conﬁguration and the G-AF(1) state is separated by an intermediate phase characterized by a non integer distribution of DO between the γ z and the xy sector. This state, indicated as G-AF(3/4), exhibits G-type antiferromagnetic correlations with weak ferromagnetic correlations (Fig. 1(i)). Then, we consider the other distortive mode (∆ /t > 0). As one can observe, the region with a C-AF ground state is replaced by different conﬁgurations with ferromagnetic character. In the regime of c-axis elongation, the γ z orbitals are the two lowest degenerate levels separated by an energy |∆ | from the single xy orbital. Hence, the gain of the crystal ﬁeld potential would lead to the largest number of DO conﬁgurations in the γ z orbital sector. This behavior is counteracted by the requirement of optimizing the kinetic energy between the correlated electrons in the γ z and those in the xy orbitals, leading to competing ferro- or antiferro-patterns between the DO conﬁgurations. Furthermore, the constraint of the hopping connectivity within the γ z manifold (only one of the two z orbitals is active for each planar direction) acts as a frustration in the dynamics of the DO conﬁgurations. Looking at the phase diagram, the interface of the C-AF(1/2) region separates the antiferromagnetic GS from different ferromagnetic conﬁgurations, characterized by a non zero value of the total spin momentum, depending on the amplitude of the Hund coupling. Particularly, the regions labelled with Fk indicate conﬁgurations with an incomplete ferromagnetism (See Fig. 1(b–e)). The change, due to c-axis expansive distortions, tends to weaken the antiferromagnetic correlations within the C-AF state enhancing the tendency to have an incomplete ferromagnetism. This behavior is the evidence of some residual antiferromagnetic correlations within the GS. The transition from the C-AF to a F-like state is always accompanied by a modiﬁcation of the charge distribution due to the different density of DO per orbital. From a general point of

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Fig. 3 Evolution of the phase boundaries vs the CF amplitude from ﬂat to elongated conﬁguration for a ratio between the Hund and the Coulomb repulsion given by JH /U = 4/5 (from Ref. [22])

view, the ferromagnetic states are characterized by a small density of DO in the xy band. Concerning the orbital correlations, both the ferromagnetic and antiferromagnetic conﬁgurations with full or partially full DO in the γ z sector, do not manifest any speciﬁc tendency in the orbital arrangement of the DO within the manifold, namely there is an almost equal probability of ferro- or antiferro-orbital correlations between the xz and yz orbitals. Let us now discuss the case (b). The phase diagram presents some features that are different if compared to the previous case. Looking at the Fig. 4 we deduce that the scenario for the compressive octahedral distortions is qualitative similar to the case (a). We do notice that the transition from the C-AF conﬁguration to the G-AF one does not go through the G-AF(3/4) ground state, thus revealing a characteristic that manifests when the Hund and Coulomb coupling are comparable. Furthermore, when compared to the previous situation, the critical crystal ﬁeld amplitude to drive the transition from the C-AF to the G-AF state has an opposite trend. In the case (a) the upturn of the critical line was moving towards larger amplitude of ∆ , as the Hund coupling is varied. In the present condition, the G-AF region is more easily stabilized as in the limit of JH /t ∼ 2 the value for crossing from one to another AF state is renormalized down to ∆ /t ∼ 0.25. The place where the boundary starts to bend indicates the crossover between the weak and the intermediate/strong coupling, followed by a renormalization in the relevant scale that controls the orbital exchange. Concerning the region corresponding to elongated octahedra, it emerges a new portion that is characterized by a G-AF state with all the DO sitting in the γ z band, G-AF(0). This conﬁguration is somehow symmetric with respect to the

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Fig. 4 As in Fig. 3 assuming a ratio between the Hund and the Coulomb repulsion JH /U = 1/4 (from Ref. [22])

G-AF(1) obtained in the extreme ﬂat situation. Now, at the interface of the CAF(1/2) state there are two conﬁgurations with incomplete ferromagnetism. It is not possible in this regime to stabilize the full ferromagnetic state in the c-axis elongated distortions. This aspect reﬂects the subtle competition between the magnetic and orbital exchange in different limits of Coulomb and Hund coupling. Therefore, we can conclude that, irrespective of the ratio between the Coulomb repulsion and the electron bandwidth, upon a given c-axis compressive distortion, the system can be driven from a state with competing ferromagnetic and antiferromagnetic exchange to an isotropic AF conﬁguration.

4 Interplay Between Spin-Orbit Coupling and Octahedral Compressive Distortions In this section we analyze the interplay between the spin-orbit interaction and the octahedron distortions and their effects on the correlated ground state. The main effect of the spin-orbit coupling is to allow for a rearrangement of the charge and spin giving rise to a local non-zero angular momentum linked to the direction of the local spin polarization. This unquenching of the orbital degree of freedom is peculiarly linked to the orbital occupation. Indeed, as a consequence of the crystal ﬁeld potential, the relevant component of the angular momentum depends on the way the charge distributes on the different orbitals. Hence, it turns out that the spin degrees of freedom follow to the orbital part.

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Fig. 5 Ground state diagram as a function of the scaled Coulomb repulsion (U − JH ) versus the spin-orbit coupling λ /t for a crystal ﬁeld amplitude ∆ /t = −0.15 (from Ref. [10])

Particularly, we will concentrate on the problem of two effective t2g sites in a octahedral environment, in the relevant case of ﬂatten octahedral distorstions, and we will assume that the three dimensional connectivity for the t2g orbitals is projected on one effective bond. This projection procedure allows to simulate a real system with one t2g site connected with three locally equivalent sites along the crystallographic directions. The analysis of one effective bond, with all the homologue orbitals connected via a single particle hopping, can averagely simulate one t2g site embedded in a three dimensional environment whose feedback of the charge ﬂuctuations along all the directions is included. In this way, the symmetry is not broken explicitly and the information of the directional electron transfer is contained in the correspondent connectivity of each orbital in the t2g sector. Therefore, the occurrence of a speciﬁc spin/orbital pattern on the bond in exam would refer to a situation where there is an isotropic exchange along the three crystallographic directions. If the octahedra are compressed (∆ < 0), we infer that the spin-orbit coupling would favor a conﬁguration with one hole in the |ψx ∼ |xy + i|xz (|ψy ∼ |xy + i|yz) and one in the yz(xz) orbital state, respectively. Hence, the orbital and spin angular momentum will be aligned along the x(y) direction. Moreover, the results shown in the previous section allow to conclude that the crystal ﬁeld potential, depending on the strength of the local Coulomb potential, is able to modify the orbital character of antiferromagnetic state, and to separate regions with most favorable ferromagnetic correlations from others where the antiferromagnetism is dominant. Now, we are interested in investigating how the Coulomb correlations interfere with the formation of locked spin and orbital momentum states. To explore the possible ground states in the space of the microscopic parameters, we have chosen a ratio of JH /U and a representative value of the crystal ﬁeld equal

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to ∆ /t = −0.15. The phase diagram is reported in Fig. 5. As one can see, the phase diagram is dominated by a large region where the orbital correlations are ferro-type with a not-integer occupation of the different sectors of the t2g manifold. We can distinguish between two different behaviors (in the following indicated as (i) and (ii)), depending on the amplitude of the Coulomb correlations. (i) In the weak coupling regime, the switch of the spin-orbit interaction produces a sudden removal of the rotational spin invariance, giving rise to a canted- antiferromagnetic region Cxy (3/4), where the correlation are antiferro-like but stronger in the xy plane, where a non-zero ferromagnetic moment mainly manifests, while the orbital correlations have large off-diagonal amplitude in the channel of double occupied correlators. At a critical value of λ , whose amplitude decreases as the electron correlation (U − JH ) increases, the system exhibits two transitions. Firstly, it changes into a C1xy state and then it smoothly crosses over in a C2xy . The main modiﬁcations do not occur in the spin channel, because both the ending regions are characterized by off-site spin correlations that are more antiferromagnetic-like with respect to the Cxy (3/4), with a reduction of the total spin momentum (weak ferromagnetism). On the contrary, in the orbital part one can observe a redistribution of charge that strongly renormalizes the off-diagonal orbital channel leading ﬁrstly to a disordered conﬁguration and then, at larger spin-orbit coupling, to more pronounced diagonal amplitude giving a substantial FO character to the ground state. In this evolution (boundary delimited by the stars), the xy-double occupation density changes as a function of λ in a continuous way from 1 towards a value of ∼1/2 reaching the strong coupling limit. Such feature is related to the tendency of the spin-orbit coupling towards the mixing of xy with the γ z orbitals. The quantum superposition which emerges, on the other hand, cannot allow for an equal charge population in the two sectors due to the crystal ﬁeld potential. Hence, there occurs a competition which manifests as a gradual charge transfer between the two parts of the t2g manifold as the spin-orbit coupling gets comparable with the crystal ﬁeld energy. (ii) In the limit of strong coupling and compressed octahedron distortion, all the main transformations occur in the spin channel. A smooth crossover in the correlation functions separates the region at zero spin-orbit coupling from that at ﬁnite λ . Until the crossover points, the system stays in a AF(1) region, having zero spin angular momentum and isotropic antiferromagnetic correlations, that realize in a fully frozen xy orbital conﬁguration. After that, it changes by evolving into the C2xy , where the antiferro-type correlations keep almost the maximum value in the xy plane (still there is a small ferromagnetic net momentum), while tend to quench along the z direction. The freezing of superexchange mechanisms along the z direction is accompanied by a formation of weak ferromagnetism. In conclusion, as far as the case of compressed octahedron in presence of spinorbit coupling is concerned, we have seen that the correlated ground state is quite sensitive to this interaction. In particular, the FO conﬁguration, where the xy orbital is quenched, smoothly evolves to a state with not integer orbital population, whose xy occupation is not completely quenched.

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5 Thermal Evolution of Correlation Functions In this section, we present and discuss the phase diagram (T /t vs ∆ /t), when the spin-orbit coupling is neglected. To this end, we have evaluated the correlation functions deﬁned in Eqs. (6, 7) considering that the expectation values in these quantities are now statistical expectation values. In other words, for a generic operator O, the quantity we need to calculate is N

O = Z −1 ∑n|e−β H O|n

(8)

n

where β = 1/kB T , kB being the Boltzmann constant, and Z is the partition function. The sum runs over the chosen complete basis set of normalized wave vectors |n, with n = 1, . . . , N. In order to study the temperature behavior of these quantities we make use of thermal Lanczos method [26]. Within this method, the sampling over all states is reduced to a random partial sampling, while only approximate ground state and excited states, produced by Lanczos method, are used for the evaluation of matrix elements. More speciﬁcally, for each basis state one starts a Lanczos procedure generating a set of M orthonormalized states; then the tridiagonal matrix of coefﬁcients produced is diagonalized, and the eigenvectors and the corresponding eigenvalues determined. Assuming these eigenvectors as basis, within this restricted basis one can approximate the thermal averages used to calculate the expectation values of a generic operator at general temperature T . Performing a sampling over the full basis, W being the number of samplings, it has been shown that even when the averaging is performed on M N, where N is the full dimension of the Fock space of the Hamiltonian model, and for a partial random sampling of basis states W N, very accurate results have been obtained. We notice that these calculations require quite modest computational efforts when compared to the zero temperature case, and also the size limitations are comparable to those concerning the Lanczos diagonalization procedure applied to the ground state calculations. The results obtained applying this procedure to evaluate the thermal correlation functions are reported in Fig. 6, where the (normalized) temperature phase diagram is shown as function of the normalized amplitude of the crystal ﬁeld coupling. Moreover, the curves have been obtained by ﬁxing JH /U = 1/4 and JH /t = 1.5. In this respect, the results of Fig. 6 represent the temperature evolution of the ground state conﬁgurations reported in Fig. 4, when the normalized Hund coupling is ﬁxed to JH /t = 1.5. Looking at this ﬁgure, we notice immediately that the energy scale of spin excitations is always lower than the corresponding orbital one. Indeed, at low temperatures the system exhibits a magnetic/orbital conﬁguration that becomes orbital only by increasing the temperature, thus suggesting that the increase of the temperature tends to disorder ﬁrstly the magnetic ordering. In meantime, the orbital correlations still remain effective disappearing at highest temperatures. More precisely, when compressed octahedra are considered, at low temperatures the system is in an antiferromagnetic-like conﬁguration, of G or C type, while a ferro-like orbital

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Fig. 6 Normalized temperature behavior versus normalized crystal ﬁeld amplitude ∆ /t of orbital correlation function (full triangles and diamonds) and spin correlation function (full circles)

state is present at large |∆ /t|, and an AFO at lower values of |∆ /t|. At higher temperatures and large values of |∆ /t|, the system still retains robust orbital correlations exhibiting a ferro-orbital conﬁguration. When |∆ /t| is reduced the orbital correlations reduce too, almost overlapping with the spin ones, while in a region around ∆ /t −0.4 a new FO conﬁguration comes in at higher temperatures. When elongated octahedra are concerned, the situation is similar to the previous case even though a larger number of regimes may be individuated. Considering the low temperatures regime, and by increasing the crystal ﬁeld amplitude, the system passes through the following conﬁgurations: (1) C-AF/AFO; (2) F2 ; (3) G-AFO/FO, reﬂecting the tendency previously reported in Figs. 4 and 5. As before, the energy scale of magnetic excitations is lower than the orbital one, these two scales merging at around ∆ /t = 0.2. In closing this section, we make a summary the main effect played by the temperature on the thermal evolution of the spin/orbital conﬁgurations assumed by the system: the increase of the temperature tunes the system towards a disordered magnetic phase that is generally characterized by enhanced FO correlations that still remains active at highest temperatures. This trend is recovered in the full ∆ /t space, independently on the sign of this quantity.

6 Relevance for Ruthenate Oxides Physics and Conclusions In this paper we have summarized recent achievements in the description of some relevant properties exhibited by Ca-based ruthenates. The results of our study have been obtained applying the Lanczos technique to a three-band Hubbard model

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which also includes the crystal ﬁeld term and the spin-orbit interaction. The subtle interplay between spin, orbital and lattice degrees of freedom gives rise to a large variety of ground states that may account for some regions of the phase diagram of the above mentioned ruthenates. Speciﬁcally, looking at he spin/orbital correlations 4 system upon the competition of the Coulomb that emerge in the dynamics of a t2g correlations and the c-axis distortions, we have shown that the control of the double occupation distribution among the different orbitals is the key aspect in characterizing the spin and orbital correlations of the ground state. When the system is in the extreme ﬂatten conﬁguration, the DO is quenched in the xy sector and the spin coupling turns out be isotropically antiferromagnetic. Our study has shown that the onset of this state, as a function of the crystal ﬁeld potential, has a different behavior depending on the relative ratio between the Hund rule interaction and the Coulomb repulsion. By increasing the Hund coupling and more generally any mechanisms that would reinforce the ferromagnetic correlations, the system stabilizes the CAF ground state with respect to the G-AF one. Indeed, the threshold of the crystal ﬁeld potential for the crossing between the two conﬁgurations shifts at values of the order of the bare xy half-bandwidth. The G-AF conﬁguration with ferro-orbital order has been already proposed as the candidate for the low temperature ground state in the Ca2 RuO4 compound [25, 14]. Nevertheless, as pointed out in the Introduction, there have been experiments showing that the orbital order conﬁguration is more complex and may contain contributions of antiferro-orbital character. Our analysis suggests that in the region (see Fig. 4) of intermediate Coulomb coupling, small variations in the structural parameters may lead to signiﬁcative changes in the magnetic and orbital correlations. Assuming that the compressive octahedral mode stabilizes the xy ferro-orbtial pattern, the closest conﬁguration which can be activated has a weaker antiferromagnetism and enhanced antiferro-orbital correlations, as it occurs in the C-AF(1/2) state. It is worth reminding that, due to the degeneracy of the C-AF state with respect to the x and y axis, there is no a priori symmetry breaking within this analysis. Nevertheless, a small perturbation due to the orthorhombic distortions would symmetrize such a conﬁguration in a form where the antiferromagnetic correlations are more isotropic along the x and y directions. To account for the possibility of concomitant occurrence of ferro- and antiferroorbital patterns, we investigated the role of the spin-orbit coupling and its interplay with octahedral ﬂat distortions. The analysis shows that, in presence of the spinorbit coupling, the FO type of correlation persists but without a full occupation of the xy orbital, thus indicating the formation of an hybrid state with dominant FO conﬁgurations but with a still active charge dynamics in the xy sector. This interaction would tend to give a non vanishing overlap between the G-AF and the C-AF conﬁguration, by removing the constraint of local orbital quenching in the angular momentum, thus leading to a complex ground state that, to the lowest order, can be seen as a quantum superposition of the two antiferromagnetic conﬁgurations [10]. These results may be considered when one wants to address the question of the occurrence of an antiferromagnetic ground state in presence of ﬂattening along the c-axis with about 0.5 holes in the xy sector as it may occur in the Ca2 RuO4 compound. The subtle interplay between the spin-orbit and the charge ﬂuctuations

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controlled via Coulomb and crystal ﬁeld potential turns out to be a key aspect in determining the spin and orbital character of the ground state. Looking at the elongated case, we have seen that the evolution from the ferromagnetic to the antiferromagnetic state, as due to the competition of Coulomb correlation and c-axis distortions, can be smooth so that the systems passes gradually all the intermediate spin polarized conﬁgurations, from the maximum to the unpolarized AF state (see Fig. 3). This feature, that occurs in a speciﬁc range of Coulomb, Hund and elongated octahedra, reveals the subtle competition between the ferromagnetic and antiferromagnetic exchange, as tuned by the different orbital correlations within the ground state. The ferromagnetism that emerges in this region has a novel character being related to the occurrence of competing antiferromagnetic/ferromagnetic coupling and the presence of isotropic in plane orbital resonance. This may be interesting in connection with the debate on the nature of the ferromagnetic correlations induced in the Ca2−x Srx RuO4 , by doping or pressure,in proximity of x = 0.5, where the the octahedra are in the regime of expansive distortions and the experiments [5] suggest the existence of large ferromagnetic correlations, at least at short range. Finally, we have presented preliminary results on the thermal evolution of spin and orbital correlation functions. Speciﬁcally, we have shown that the temperature is able to ﬁrstly disorder the magnetic ground state conﬁgurations and then to drive the system into FO conﬁgurations, this result being not inﬂuenced by the values of ∆ .

References 1. M. Imada, A. Fujimori, Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). 2. S. Nakatsuji, Y. Maeno, Phys. Rev. Lett. 84, 2666 (2000). 3. Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J.G. Bednorz, F. Lichtenberg, Nature (London) 372, 532 (1994); Y. Maeno, T.M. Rice, M. Sigrist, Phys. Today 54, 42 (2001). 4. S. Nakatsuji, S. Ikeda, Y. Maeno, J. Phys. Soc. Jpn. 66, 1868 (1997). 5. S. Nakatsuji, Y. Maeno, Phys. Rev. B 62, 6458 (2000); S. Nakatsuji, Y. Maeno, Phys. Rev. Lett. 84, 2666 (2000). 6. M. Braden, G. Andr´e, S. Nakatsuji, Y. Maeno, Phys. Rev. B 58, 847 (1998). 7. O. Friedt, M. Braden, G. Andr´e, P. Adelmann, S. Nakatsuji, Y. Maeno, Phys. Rev. B 63,174432 (2001). 8. C.S. Alexander, G. Cao, V. Dobrosavljevic, S. McCall, J.E. Crow, E. Lochner, R.P. Guertin, Phys. Rev. B R8422R8425 (1999). 9. T. Mizokawa, L.H. Tjeng, G.A. Sawatzky, G. Ghiringhelli, O. Tjernberg, N.B. Brookes, H. Fukazawa, S. Nakatsuji, Y. Maeno, Phys. Rev. Lett. 87, 077202 (2001). 10. M. Cuoco, F. Forte, C. Noce, Phys. Rev. B 73, 094428 (2006). 11. I. Zegkinoglou, J. Strempfer, C.S. Nelson, J.P. Hill, J. Chakhalian, C. Bernhard, J.C. Lang, G. Srajer, H. Fukazawa, S. Nakatsuji, Y. Maeno, B. Keimer, Phys. Rev. Lett. 95, 136401 (2005). 12. T. Nomura, K. Yamada, J. Phys. Soc. Jpn. 69, 1856 (2000). 13. S. Okamoto, A.J. Millis, Phys. Rev. B 70, 195120 (2004). 14. Z. Fang, K. Terakura, Phys. Rev. B 64, 020509(R) (2001).

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15. Z. Fang, N. Nagaosa, K. Terakura, Phys. Rev. B 69, 045116 (2004). 16. J.H. Jung, Z. Fang, J.P. He, Y. Kaneko, Y. Okimoto, Y. Tokura, Phys. Rev. Lett. 91, 056403 (2002). 17. J.S. Lee, Y.S. Lee, T.W. Noh, S.-J. Oh, Jaejun Yu, S. Nakatsuji, H. Fukazawa, Y. Maeno, Phys. Rev. Lett. 89, 257402 (2002). 18. A. Koga, N. Kawakami, T.M. Rice, M. Sigrist, Phys. Rev. Lett. 92, 216402 (2004). 19. A. Liebsch, Phys. Rev. B 70, 165103 (2004). 20. S. Biermann, L. de Medici, A. Georges, Phys. Rev. Lett. 95, 206401 (2005). 21. T. Hotta, Dagotto, Phys. Rev. Lett. 88, 017201 (2002). 22. M. Cuoco, F. Forte, C. Noce, Phys. Rev. B 74, 195124 (2006). 23. M. Kriener, P. Steffens, J. Baier, O. Schumann, T. Zabel, T. Lorenz, O. Friedt, R. M¨uller, A. Gukasov, P.G. Radaelli, P. Reutler, A. Revcolevschi, S. Nakatsuji, Y. Maeno, M. Braden, Phys. Rev. Lett. 95, 267403 (2005). 24. J.F. Karpus, R. Gupta, H. Barath, S.L. Cooper, Phys. Rev. Lett. 93, 167205 (2004). 25. V.I. Anisimov, I.A. Nekrasov, D.E. Kondakov, T.M. Rice, M. Sigrist, Eur. Phys. J. B 25, 191 (2002). 26. J. Jakliˇc, P. Prelovˇsek, Phys. Rev. B 49, 5065(R) (1994); J. Jakliˇc, P. Prelovˇsek, Adv. Phys. 49, 1 (2000).

Local Moment Approach to Multi-Orbital Anderson and Hubbard Models Anna Kauch and Krzysztof Byczuk

Abstract The variational local moment approach (V-LMA), being a modiﬁcation of the method due to Logan et al., is presented here. The existence of local moments is taken from the outset and their values are determined through variational principle by minimizing the corresponding ground state energy. Our variational procedure allows us to treat both fermi- and non-fermi liquid systems as well as insulators without any additional assumptions. It is proved by an explicit construction of the corresponding Ward functional that the V-LMA belongs to the class of conserving approximations. As an illustration, the V-LMA is used to solve the multi-orbital single impurity Anderson model. The method is also applied to solve the dynamical mean-ﬁeld equations for the multi-orbital Hubbard model. In particular, the MottHubbard metal–insulator transition is addressed within this approach.

1 Introduction The single impurity Anderson model (SIAM) is one of the most investigated models in condensed matter physics [1]. This model is regarded as a prototype to understand and describe: (i) properties of metals with magnetic atoms [16], (ii) charge transport through quantum dots [24], (iii) Mott-Hubbard metal-insulator transitions (MIT) within the dynamical mean-ﬁeld theory (DMFT) [23, 6, 14, 28, 30, 27], and (iv) a crossover between weak and strong coupling limits and conﬁnement phenomena. The SIAM consists of a term describing band electrons coupled by hybridization to a term corresponding to a single impurity where the local Coulomb interaction A. Kauch Institute of Theoretical Physics, University of Warsaw, ul. Ho˙za 69, PL-00-681 Warszawa, Poland, e-mail: [email protected] K. Byczuk Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute for Physics, University of Augsburg, D-86135 Augsburg, Germany, e-mail: [email protected]

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is taken into account [1]. In the featureless hybridization limit the SIAM is solved exactly within the Bethe ansatz or conformal ﬁeld theory techniques so the ground state and the whole excitation spectrum as well as thermodynamics are exactly known [16]. Unfortunately, these methods cannot in practice provide dynamical quantities, for example one-particle spectral functions or dynamical susceptibilities, for all interesting energies. Also the (asymptotic) exact solvability is not possible for a general hybridization term. For practical applications of the SIAM one has to rely on either a numerically exact or an analytical but approximate solution. Numerically exact methods, like the numerical renormalization group (NRG) [5] or the determinant quantum Monte Carlo (QMC) [14] are very time (CPU) consuming. In particular, the CPU is very long when the number of orbitals is large in the NRG case and when the temperature is low in the QMC case. Also to extract dynamical quantities is a rather tricky task [17]. Reliable analytical methods are therefore needed. One of such methods, which recovers properly both weak and strong coupling limits, is a local moment approach (LMA) invented recently by Logan et al. [26]. The LMA is a perturbative method around an unrestricted Hartree-Fock solution with broken symmetry, i.e. with a non-zero local magnetic moment. The broken symmetry is restored at the end by taking the average of the solutions corresponding to different directions of the local magnetic moment [26]. In the present contribution we describe the LMA method and our implementation of it, which is different from the original one [26] by the way of how the value of the local moment is determined. Namely, we use the variational principle demanding that the ground state energy is minimized by the physical value of the local moment. Therefore we use the name variational local moment approach (V-LMA) for this method. Such a procedure allows us to easily generalize the V-LMA for multi-orbital models as well as for ﬁnite temperatures and systems with disorder [8, 7, 9, 10]. We also discuss the Luttinger-Ward generating functional for the V-LMA and claim that this method belongs to the class of conserving approximations. The application of LMA for studying the electron ﬂow through quantum dots and the Mott-Hubbard MIT is addressed at the end of the contribution.

2 Local Moment Method in One Orbital SIAM The single impurity Anderson model is given by the Hamiltonian HSIAM = Hc + Himp + Hhyb ,

(1)

where the conduction electrons are described by Hc = ∑ εk c†kσ ckσ , k,σ

(2)

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where εk is an energy (a dispersion relation) for an electron in a state k and spin σ = ±1/2, the impurity electrons with the local Coulomb interaction U are represented by Himp = ∑(εd + Und−σ )nd σ , (3) σ

with is

nd σ = dσ† dσ ,

and the hybridization between conduction and impurity electrons (4) Hhyb = ∑ Vk dσ† ckσ + h.c. . k,σ

All local (on impurity site) properties are expressed by the hybridization function

∆ (ω ) = ∑ k

|Vk |2 , ω − εk

(5)

and not by εk and Vk separately. This can be proved by tracing out the noninteracting conducting electrons.

2.1 Mean Field Solution of the Single Impurity Anderson Model The Hartree-Fock mean-ﬁeld solution of the SIAM is obtained by factorizing the interacting term nd↑ nd↓ ≈ nd↑ nd↓ + nd↑ nd↓ − nd↑ nd↓ [1]. For the interaction U above Uc and corresponding impurity electron densities n¯ d the mean-ﬁeld solution is unstable toward the local moment formation with non-zero moment µ ≡ nd↑ − nd↓ . The solution is doubly degenerate because of two equivalent directions of the local moment µ = ±|µ |, which give the same energy of the system. The local (impurity) Green function within the Hartree-Fock solution is GHF σ (ω ) =

1 ω − εd − ∆ (ω ) − ΣσHF + iδ sgnω

(6)

where the static Hartree-Fock self-energy ΣσHF = Unσ¯ and δ → 0+ . Since there are in principle two possible signs of the local moment, there are two different possible Hartree – Fock Green functions denoted by GAσ (ω )HF and GBσ (ω )HF that differ only by the sign of the local moment and depend parametrically on its value |µ |. The fundamental deﬁciency of the Hartree-Fock approximation is that it leads to a broken symmetry solution which cannot persist in the thermodynamic limit, i.e. a single impurity cannot lead to the magnetic solution in the inﬁnite system. Also this solution does not recover the singlet ground state known from the exact Bethe ansatz solution. Nevertheless it turns out to be useful as a starting point in the further perturbative calculation combined with the symmetry restoration.

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Fig. 1 The frequency dependent part of self-energy expressed as the RPA series around the broken symmetry Hartree-Fock solution. The transverse spin polarization bubbles constitute a geometric series which can be summed up to inﬁnity

2.2 Two Self-Energy Description HF are used in the time-dependent The two Hartree-Fock Green functions GA,B σ (ω ) many-body perturbation expansion. Within the random phase approximation (RPA) the polarization diagrams are

ΠσAAσ¯ (ω ) =

0 Π AA (ω ) σ σ¯ 1 − U 0ΠσAAσ¯ (ω )

(7)

and correspond to spin ﬂip processes as represented by the Feynmann diagrams in Fig. 1. HF we have the corresponding For each type of the mean-ﬁeld solution GA,B σ (ω ) self-energy

ΣσA (ω )

= ΣσHF

+U

2

d ω AA A Π (ω )Gσ¯ (ω − ω )HF 2π i σ¯ σ

(8)

depending on frequency and parametrically on |µ | as well. The full RPA-Green functions GA,B σ (ω ) are constructed by using the Dyson equation separately for A and B solutions. Note that GA,B σ (ω ) depends parametrically on still unknown | µ |.

2.3 Symmetry Restoration Ansatz To restore the spin-rotational symmetry Logan et al. [26] proposed the following ansatz for the full symmetrized Green function 1 Gσ (ω ) = (GAσ (ω ) + GBσ (ω )). 2

(9)

Within the LMA the physical Green function is an average of the two solutions with equal probabilities. Although each GA,B σ (ω ) is determined within the renormalized perturbation scheme the ﬁnal Green function turns out to capture nontrivial

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non-perturbative physics as was shown by Logan et al. [26] and is also reproduced below. In particular, the LMA is able to recover the Kondo peak in the spectral function correctly with the exponential width.

2.4 Determining the Value of Local Moment The value of the local moment is a free parameter and must still be determined. In the original approach, Logan et al. [26] imposed the Fermi liquid condition to determine |µ | at zero temperature. This condition might be too restrictive at ﬁnite temperatures or in the multi-orbital cases. Therefore we decided to ﬁnd the physical solution to the problem by minimizing the relevant thermodynamical potential with respect to |µ | [20]. At zero temperature the relevant potential is just the ground state energy of the system, i.e. Ephysical = min EG (µ , n), { µ ,n}

(10)

where in the case away of half-ﬁlling the particle density n must also be determined. The variational method reproduces the Fermi liquid properties where they are expected.

2.5 Ground State Energy in the Anderson Impurity Model The ground state energy of the SIAM is given by EG = 0|H|0. This quantummechanical average consists of two parts: the bulk, which is proportional to the system volume and is independent of the local moments, and the impurity part, which depends explicitly on |µ |. The impurity part of the ground state energy, expressed by the local Green function Gσ (ω ) and the hybridization function ∆ (ω ), is equal to [21]

1 ∂ ∆ (ω ) σ ω + εd + ∆ (ω ) − Gimp (ω ), d ω ω (11) Eimp = 2π i ∑ 2 ∂ω σ C where the contour integral is over the half circle in the upper complex plane.

2.6 LMA as a Conserving Approximation According to Kadanoff and Baym [3] any approximate theory is conserving if there exists a Luttinger-Ward functional Φ[G] for this theory. It is necessary that this functional: (i) is universal, i.e. it dependents only on the full propagator Gσ (ω ) and

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not on the atomic properties of the system and (ii) has a functional derivative with respect to Gσ (ω ) which is by deﬁnition equal to the self-energy of the system. It can be shown [20] that the LMA is a conserving approximation and we can construct explicitly the Luttinger-Ward functional 1 1 A Φ[G] = Φ[GA , GB ] = ΦRPA + ΦBRPA + Tr log GAσ GBσ + 2 2

1 A Gσ + GBσ , (12) − Tr log 2 +

where Tr = T ∑σ ∑iωn ei0 and the functionals ΦA,B RPA are represented diagrammatically by the RPA diagrams with GA,B ( ω ) respectively. The constraint that G = σ 1 (G + G ) must be satisﬁed. Finally, the free energy functional is given by B A 2

Ω [G] = Φ [G] + Tr log G − TrΣ G

(13)

and the stationarity condition δ Ω [G]/δ G gives the Dyson equation and the physical solution for G. The fact that the LMA is a conserving approximation, as we proved above, makes this theory reliable in describing correlated electron systems, in particular in the intermediate regimes of parameters.

3 Local Moment Approach for the Multi-Orbital SIAM In reality the magnetic impurities in metals are atoms with partially ﬁeld d- or forbitals. Such orbitals have degenerate levels. Even when a particular environment which decrease the symmetry and leads to eg and t2g split levels, partial degeneracy between orbitals remains. The appropriate model to describe such situations is the multi-orbital single impurity Anderson model. It describes a single impurity with many orbital levels α , which can be degenerate or split depending on the single-body matrix element εα . In this case the electrons can interact via direct (density-density) type of the interaction and via the exchange (Hund) interaction. Microscopically, the single impurity Anderson model with many orbital levels is given by the Hamiltonian: HSIAM = ∑ (εα + Uα nα ,σ¯ ) nα ,σ + ∑ ∑ Uαβ − J δσ σ nασ nβ σ + α ,σ

+

∑

k,σ ,α

σ ,σ α =β

† Vkα dασ ckσ + c†kσ dασ + ∑ εk c†kσ ckσ ,

(14)

k,σ

where the direct U and U as well as exchange J interactions between the electrons of spin σ and on orbitals α or β are taken into account.

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This multi-orbital version of the SIAM is also of interest in quantum dot physics, where dots with a few orbitals can be prepared and investigated experimentally. One of the interesting aspect of such system is the possibility to observe the orbital Kondo effect [18].

3.1 LMA Generalization In the mean ﬁeld approximation of the multi-orbital SIAM we also encounter a doubly degenerate solution, where the two possible Green functions differ only by the sign of the impurity magnetic moment. Within the LMA, we introduce for each pair αβ ,A αβ ,B of orbital indices α and β the two Green functions Gσ (ω )HF and Gσ (ω )HF that correspond to the two possible directions of the total magnetic moment on the impurity. These Hartree-Fock Green functions depend now parametrically on values of local moments on each of the orbitals µα . Next we use the RPA approximation αβ ,A αβ ,B to obtain two Green functions Gσ (ω ) and Gσ (ω ), which are parametrically dependent on the local moments on each orbitals µα .

3.2 Symmetry Restoration and Determining the Local Moment Values The symmetry restoration in the multi-orbital case is a straightforward generalization of the previous ansatz, i.e. αβ

Gσ (ω ) =

1 αβ ,A αβ ,B Gσ (ω ) + Gσ (ω ) , 2

(15)

αβ

except that now the symmetrized Green functions Gσ (ω ) depend explicitly on local moments on all of the orbitals, i.e. |µα |. The parameters |µα | have to be determined independently. They are found by the minimization of the ground state energy of the impurity with respect to both local moment values on orbitals µα and particle number on each of the orbitals nα Ephysical = min EG (µα , nα ). { µα ,nα }

(16)

As mentioned above, the variational procedure allows us to extend the LMA on the multi-orbital cases, where the Luttinger (Fermi liquid) condition for each orbital is absent. Also the possibility of non-Fermi liquid solution is naturally included within present generalization of the LMA [20], i.e. the variational local moment approach.

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4 Application to Multilevel Quantum Dots A single quantum dot with many atomic-like levels coupled to leads are described by a multi-orbital single impurity Anderson model: H = Hdot + Hleads + Hdot−leads , where Hdot is the local impurity part of the SIAM Hamiltonian, Hleads corresponds to the conduction electron part of the SIAM Hamiltonian, and Hdot−leads is equal to the hybridization term in SIAM [24].

4.1 V-LMA in Quantum Dots The properties of transport in a quantum dot in equilibrium, i.e. with inﬁnitesimally small bias voltage between the leads, are determined by the spectral functions on each of the orbitals. Examples of the spectral functions are presented in Fig. 2 for the one-orbital case (left panel) and for the two orbital case (right panel). In the two orbital case the atomic levels are shifted such that one of the orbitals is at half ﬁlling (dashed line) and the other is away of half ﬁlling (solid line). The Kondo peak in the symmetric case is suppressed by the exchange (Hund) interaction (J = 0), which favors parallel spin orientations. In the asymmetric case the Kondo peak survives due to the presence of uncompensated magnetic moment and is shifted toward the lower Hubbard band. Further investigation of multilevel quantum dots including transport properties will be presented elsewhere [20].

EG

Spectral function

0,4

|µ|0 0,85

0,9

|µ|

0,3

0,95

0,2

1

0,2

0,1

0,1

0

-10

-5

0

ω

5

10-10

-5

0

ω

5

10

0

Fig. 2 Spectral functions for one level (left panel) and two level (right panel) quantum dots. Left panel: spectral function at half-ﬁlling and U = 6 (inset: the ground state energy as a function of the absolute value of local moment | µ |; the axis starts at the Hartree-Fock value | µ | = | µ0 |). Right panel: orbitally resolved spectral functions in the dot for U = 3, J = 0.25U, |ε1 − ε2 | = 0.1U, and the total ﬁlling nd = 1.95. All curves are for semi-elliptic hybridization function with the width W = 20. The Fermi level is at zero

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5 Application to the Multi-Orbital Hubbard Model The generalized variational LMA is also applied to solve the multi-orbital Hubbard model HHubb = ∑ ∑ tiαj di†ασ d jασ + Hlocal , i j α ,σ

where the local part is a lattice sum of the terms which are of the same form as the atomic part in the SIAM. This model is solved within the DMFT where the selfconsistency condition relates the local matrix Green functions with the matrix of the self–energies [14]. In this way the lattice problem is mapped onto the Anderson impurity problem which has to be solved for different hybridization functions until self-consistency is achieved. In order to solve the Hubbard model within DMFT we need to solve the SIAM for arbitrary hybridization functions. The self–consistency condition simpliﬁes greatly for the Bethe lattice which is used in this contribution.

5.1 V-LMA Method in DMFT

Spectral function

In the recent few years the orbital-selective Mott-Hubbard metal-insulator transition has been the subject of extensive studies [22, 25, 13, 4, 2]. Using the V-LMA to obtain the solution of the SIAM in each of the DMFT loops the spectral functions for two–orbital Hubbard model at zero temperature were found. As an example, Fig. 3 shows the results for the case with different bandwidths and non-zero Hund coupling J. Since one of the spectral function is metallic-like (ﬁnite at ω = 0) and the other is insulating-like (vanishes at ω = 0) we conclude that the orbital selective MIT occurs in this model system. At the end we discuss the V-LMA in perspective to other methods used to solve the impurity problem and DMFT equations. The V-LMA belongs to the class of approximate, analytical methods like for example the iteration perturbation theory (IPT) [15], the non-crossing approximation (NCA) [12], or slave-boson theory (SB) [29], and various extenstions of these methods. As we showed here, the V-LMA is a conserving approximation, contrary for example to the IPT, and correctly describes

0,6

0,3

0

-4

-2

0

ω

2

4

-4

-2

0

ω

2

4

Fig. 3 Spectral functions for two-orbital Hubbard model with different band widths W = 2 (solid line) and W = 4 (dashed line) on Bethe lattice with inﬁnite coordination number. Left panel: U = 1.2, J = 0.1U. Right panel: U = 2, J = 0.1U. The inter-band interaction U = U − 2J is ﬁxed preserving SU(4) symmetry. In both cases the Fermi level is at zero energy and the bands are half ﬁlled

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high- and low-energy parts of the spectra, recovering the Kondo peak and Luttinger pinning. We tested this theory at zero temperature but there is no conceptual obstacle why the V-LMA should not work at ﬁnite temperatures as well. The V-LMA is not numerically exact like the quantum Monte Carlo method [14], the numerical renormalization group (NRG) [5], dynamical matrix renormalization group (DMRG) [31], or exact diagonalization (ED) [11]. However, each of the numerically exact methods suffers from principal obstacles in practical applications, in particular when the temperature is too low (QMC) or too high (NRG), or number of orbitals increases (NRG, DMRG, ED). Therefore we conclude that the V-LMA is a method of choice for solving the DMFT equations and can be used as a relatively fast and accurate impurity solver. The only technical difﬁculty in the variational LMA is to compute with high accuracy the system energy and to ﬁnd its minimum. This should be performed with a great care.

Summary The generalized variational LMA to the multi-orbital SIAM allows us to efﬁciently solve the problems of correlated electron systems such as multilevel quantum dots and the Hubbard model within the DMFT. In particular it is relatively easy to address the problems of different band widths and also the removing of the orbital degeneracy [19]. We experienced that the local moment approach is an efﬁcient method in studying these problems, in particular, when the number of the orbitals is larger than two. Acknowledgements We thank Dr. R. Bulla and Prof. D. Vollhardt for the discussions. The hospitality at Augsburg University is also acknowledged. This work is supported by Sonderforschungsbereich 484 of the Deutsche Forschungsgemeinschaft (DFG) and by the European Commission under the contract No MRTN-CT-2003-504574.

References 1. 2. 3. 4. 5. 6. 7. 8.

P.W. Anderson, Phys. Rev. 124, 11 (1961). R. Arita, K. Held, Phys. Rev. B 72, 201102(R) (2005). G. Baym, L.P. Kadanoff, Phys. Rev. 124, 287–299 (1961). S. Biermann, L. de’ Medici, A. Georges, Phys. Rev. Lett. 95, 206401 (2005). R. Bulla, Phys. Rev. Lett. 83, 136 (1999). R. Bulla, Phil. Mag. 86, 1877 (2006). K. Byczuk, W. Hofstetter, D. Vollhardt, Phys. Rev. B 69, 045112 (2004). K. Byczuk, W. Hofstetter, D. Vollhardt, Phys. Rev. Lett. 94, 056404 (2005).

Local Moment Approach to Multi-Orbital Andersonand Hubbard Models 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

27. 28. 29. 30. 31.

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K. Byczuk, M. Ulmke, D. Vollhardt, Phys. Rev. Lett. 90, 196403 (2003). K. Byczuk, M. Ulmke, Eur. Phys. J. B 45, 449 (2005). M. Caffarel, W. Krauth, Phys. Rev. Lett. 72, 1545. T. A. Costi, J. Kroha, P. W¨olﬂe, Phys. Rev. B 53, 1850 (1996). P.G.J. van Dongen, C. Knecht, N. Blmer, Phys. Stat. Sol. (b) 243, 116 (2006). A. Georges, et al., Rev. Mod. Phys. 68, 13 (1996). A. Georges, G. Kotliar, Phys. Rev. B 45 6479, (1992). A. Hewson, The Kondo problem to heavy fermions (Cambridge University Press, 1993). M. Jarell, J.E. Gubernatis, Phys. Rep. 269, 1333 (1996). P. Jarillo-Herrero, J. Kong, H.S.J. van der Zant, C. Dekker, L.P. Kouwenhoven, S. De Franceschi, Nature 434, 484 (2005). A. Kauch, K. Byczuk, Physica B, 378–380, 297–298 (2006). A. Kauch, K. Byczuk, in preparation. B. Kj¨ollerstr¨om, D.J. Scalapino, J.R. Schrieffer, Phys. Rev. 148, 665–671 (1966). A. Koga, et al., Phys. Rev. Lett. 92, 216402 (2004). G. Kotliar, D. Vollhardt, Physics Today 57, No. 3 (March), 53 (2004). L. Kouwenhoven, L. Glazman, Physics World 14, 33 (2001). A. Liebsch, Phys. Rev. B 70, 165103 (2004). D. Logan, M.P. Eastwood, M.A. Tusch, J. Phys. Condens. Matter 10, 2673–2700 (1998), D. Logan, M.T. Glossop, J. Phys. Condens. Matter 12, 985 (2000); M.T. Glossop, D. Logan, ibid. 15, 7519 (2003); V.E. Smith, D.E. Logan, H.R. Krishnamurthy, Eur. Phys. J. B 32, 49–63 (2003). W. Metzner, D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989). Th. Pruschke, M. Jarrell, J.K. Freericks, Adv. in Phys. 44, 187 (1995). N. Read, D. Newns, J. Phys. C 16, 3273 (1983). D. Vollhardt, Correlated Electron Systems, vol. 9, ed. V.J. Emery, (World-Scientiﬁc, Singapore, 1993), p. 57. S. White, Phys. Rev. Lett. 69, 2863 (1992).

High Field Level Crossing Studies on Spin Dimers in the Low Dimensional Quantum Spin System Na2T2(C2 O4)3 (H2 O)2 with T = Ni, Co, Fe, Mn C. Mennerich, H.-H. Klauss, A.U.B. Wolter, S. S¨ullow, F.J. Litterst, C. Golze, R. Klingeler, V. Kataev, B. B¨uchner, M. Goiran, H. Rakoto, J.-M. Broto, O. Kataeva, and D.J. Price

Abstract In this paper we demonstrate the application of high magnetic ﬁelds to study the magnetic properties of low dimensional spin systems. We present a case study on the series of 2-leg spin-ladder compounds Na2 T2 (C2 O4 )3 (H2 O)2 with T = Ni, Co, Fe and Mn. In all compounds the transition metal is in the T2+ high spin conﬁguation. The localized spin varies from S = 1 to 3/2, 2 and 5/2 within this series. The magnetic properties were examined experimentally by magnetic susceptibility, pulsed high ﬁeld magnetization, speciﬁc heat measurements and high ﬁeld ESR. The data are analysed using a spin hamiltonian description. Although the transition metal ions form structurally a 2-leg ladder, an isolated dimer model consistently describes the observations very well. This behaviour can be understood in terms of the different coordination and superexchange angles of the oxalate ligands along the rungs and legs of the 2-leg spin ladder. All compounds exhibit magnetic ﬁeld driven ground state changes which at very low temperatures lead to a multistep behaviour in the magnetization curves. In the Co and Fe compounds a strong axial anisotropy induced by the orbital magnetism leads to a nearly degenerate ground state and a strongly reduced critical ﬁeld. We ﬁnd a monotonous decrease of the intradimer magnetic exchange if the spin quantum number is increased.

C. Mennerich, H.-H. Klauss, A.U.B. Wolter, S. S¨ullow, and F.J. Litterst Institut f¨ur Physik der Kondensierten Materie, TU Braunschweig, Mendelssohnstr.3, D-38106 Braunschweig, Germany, corresponding author, e-mail: [email protected] C. Golze, R. Klingeler, V. Kataev, and B. B¨uchner Leibniz-Institute for Solid State and Materials Research IFW Dresden, P.O. Box 270116, D-01171 Dresden, Germany M. Goiran, H. Rakoto, and J.-M. Broto Laboratoire National des Champs Magn´etiques Puls´es, 31432 Toulouse Cedex 04, France O. Kataeva Arbuzov Institute of Organic and Physical Chemistry, RAS, 420088 Kazan, Russia D.J. Price WestCHEM,Department of Chemistry, University of Glasgow, Glasgow, G12 8QQ, UK

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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1 Introduction In recent years, the physical properties of low dimensional spin systems have attracted a lot of attention. For isotropic magnetic interactions described in the Heisenberg model and low spin quantum numbers the ground state properties are strongly inﬂuenced by quantum ﬂuctuations and often pure quantum ground states are found in a macroscopic system [53, 34, 12, 13, 14]. Therefore these systems are ideal model systems for quantum mechanics. Low dimensional spin systems can be realized by linking transition metal ions via organic molecules. Using the rich variety of organic ligands on the transition metal complex the dimensionality and strength of the magnetic interaction can be controlled. Within this Springer series of lecture notes in physics a recent volume is devoted to high magnetic ﬁeld studies in physics [4]. In that book several contributions describe the physics of one dimensional magnets in high magnetic ﬁelds [9, 48] and electron spin resonance (ESR) on molecular magnets [19]. In this article we demonstrate the application of high magnetic ﬁelds to study the magnetic properties of spin dimers and determine the parameters describing the system. The magnetic properties are examined experimentally by magnetic susceptibility, pulsed high ﬁeld magnetization, high ﬁeld ESR and speciﬁc heat measurements. We present a case study on a series of structural 2-leg spin ladders with different transition metal ions, namely Na2 T2 (C2 O4 )3 (H2 O)2 . This series consists of four isostructural compounds with magnetic ions T = Ni(II), Co(II), Fe(II) and Mn(II) with spin 1, 3/2, 2 and 5/2 respectively. One aim of this work is to elucidate the effect of a gradual increase of the spin multiplicity towards classical magnetism. In particular, the characteristic signatures of a strong single ion anisotropy in high ﬁeld magnetization measurements are shown. We also discuss the dependence of the magnetic superexchange interaction strength on the number of 3d electrons, the inﬂuence of orbital moments and the validity of a description in the spin Hamiltonian model using isotropic Heisenberg exchange. In Sections 2 and 3 we describe the chemical synthesis and crystal structure of the samples and give a short overview of the spin hamiltonian used in the analysis. In the following four parts, we discuss the different compounds starting with the Ni(II) and Mn(II) compounds which have an orbital singlet ground state, followed by the Co(II) and Fe(II) compounds with an orbitally degenerate ground state. Finally, we compare the results obtained on the four samples.

2 Synthesis and Crystal Structure The compounds Na2 T2 (C2 O4 )3 (H2 O)2 with T = Ni, Co, Fe and Mn (we abbreviate the chemical structure with STOX for the general structure and SNOX, SCOX, SIOX and SMOX for the individual compounds) are synthesized in a hydrothermal reaction from solutions containing a very high sodium halide concentration [47, 40]. They occur as green crystals of SNOX, purple crystals of SCOX, yellow crystals of

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SIOX and white crystals of SMOX. Phase homogeneity and purity of all compounds were established by a combination of optical microscopy, powder X-ray diffraction and elemental analysis. The crystal size of the different compounds is microcrystalline for the samples SCOX, SIOX and SMOX. Only for the SNOX compound single crystals up to 2 mg were grown in lower yielding reactions by slowing down the cooling rate, and reducing the concentration of Ni2+ and (C2 O4 )2− . Single crystal X-ray structure determination reveals STOX to crystallize in the monoclinic space group P21 /c (#14) with slightly different crystallographic axes lengths below 2%. The key features of the structure are the following: The T(II) ion experiences a pseudo-octahedral coordination environment. It is coordinated in a cis geometry by two chelating and crystallographically independent oxalate dianions. The remaining coordination sites are ﬁlled by a monodentate oxalate oxygen ˚ We atom and a water molecule. T-O bond lengths lie within the range 2.0 to 2.1 A. note that opposite pairs of O have similar T-O lengths, and that the O· · · O separations between opposite pairs, for which the bond vectors are nearly directed ˚ with a slightly bigger along the crystallographic axes, are in the range of 4 A separation along the a-axes (Fig. 1). The overall structure can be regarded as an anionic [T2 (C2 O4 )3 ]2n− network with a 1-D ladder like topology (Fig. 1), where n the metal ions form the vertices and the bridging oxalate ions form the bonds. The two oxalate ions bridge 3d ions in quite different ways. One oxalate forms a symmetric bis-chelating bridging mode (with a crystallographic inversion located at ˚ forming the the C-C centroid) and links pairs of nickel ions (dT ···T (rung) ≈ 5.3 A) rungs of the ladder. The second oxalate is unsymmetrically coordinated, chelating to one T(II) ion and forming a monodentate coordination to a second T ion, with 1,3-syn anti geometry through the bridging carboxylate, this mode forms the ladder ˚ These structural linkages are likely to provide the only legs (dT ···T (leg) ≈ 5.8 A). signiﬁcant pathways for magnetic superexchange. The low dimensionality of the dominant magnetic exchange interaction in all compounds of this series is already evident from the magnetic susceptibility measured on powder samples between 2 K and 300 K (Fig. 2). All compounds exhibit a Curie-Weiss behaviour at high temperatures, a maximum between ≈10 K and 50 K and a strong decrease towards lower temperatures. This behaviour is typical for a system with a dominant antiferromagnetic interaction in less than three dimensions since no indications of static long range order are found down to 4 K. A Curie-Weiss analysis of the high temperature behaviour above 100 K yields effective magnetic moments close to the free ion high spin state for the Mn system only. While the data indicate moderate orbital contributions for the Ni and Fe systems, these contributions are strong for the Co system.

3 Theoretical Model To describe the magnetic properties we use the spin Hamiltonian approach. This approach neglects an explicit description of the orbital degrees of freedom using pure spin coordinates. The orbital contributions treated as a perturbation lead to

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Fig. 1 Crystal structure of Na2 T2 (C2 O4 )3 (H2 O)2 . For clarity, Na and H2 are not shown

an anisotropic g-factor and an anisotropy of the spin orientation described by an anisotropy tensor D. The spin Hamiltonian approach is often valid for transition metal ions because their orbital moments are known to be quenched [1, 32, 20, 5]. It is usually working well for ions with orbitally non-degenerate ground states. Taking into account the two different exchange pathway topologies, we assume a stronger magnetic interaction J along the rungs than the magnetic interaction K along the legs. This leads to an isolated dimer approximation where each dimer consists of two S = 1, 3/2, 2 or 5/2 spins for Ni(II), Co(II), Fe(II) or Mn(II), respectively, on the rungs of the ladder. Since all T ions on a ladder are crystallographically equivalent they share the same strength and orientation of the single ion anisotropy tensor D. Including the Zeeman energy in the external magnetic ﬁeld, the spin Hamiltonian of the system in the dimer approximation is given by H = J S1 S2 + g µB B(S1 + S2 ) +

∑

Si D Si .

(1)

i=1,2

Here, we assume that the magnetic ﬁeld is applied along the z-axis, the main axis of the crystal ﬁeld anisotropy tensor D. For pure axial anisotropy this tensor is diagonal, deﬁned as Dxx = −1/3 D, Dyy = −1/3 D and Dzz = +2/3 D. The effects of an in-plane anisotropy could be considered in the anisotropy tensor by the parameter E with E = 12 (Dxx − Dyy ). Usually |D| ≥ 3E holds. Therefore in our study we neglect

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Fig. 2 Temperature dependence of the magnetic susceptibility of powder samples of STOX . The deduced high temperature magnetic moments (in units of µB in a Curie analysis) are shown in comparison with the spin only value calculated for the free ions (in brackets)

the effect of a possible in-plane anisotropy. To calculate the energy levels for applied ﬁelds along different directions, we rotate the axis of the axial anisotropy tensor by an angle γ about a perpendicular axis. Then the spin Hamiltonian can be expressed as H = J S1 S2 + gγ µB B(S1z + S2z ) +

∑

Si UTγ D Uγ Si

(2)

i=1,2

with the rotation matrix Uγ and an angle dependent g-factor gγ . The magnetization in units of µB per dimer can then be calculated by using the equation ∂Z ∂F = g µB ∂ B ∂B Z ∑i eEi /T ψ i (J, D, g, γ )Sz ψ i (J, D, g, γ ) =g ∑i eEi /T

Mdimer =

(3) (4)

where F = −kB T lnZ is the free energy, Z is the partition function and ψ i (J, D, g, γ ) and Ei are the eigenvectors and eigenvalues of Eq. (2). The total magnetization

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Mtot then consists of the main contribution (4) and comprises also a temperature independent term M0 = Mdia + Mvv which includes a diamagnetic contribution Mdia and a Van-Vleck paramagnetic susceptibility Mvv , as well as a Curie contribution C/T with a Curie constant C owing to paramagnetic impurities: Mtot = Mdimer + C/T + M0 .

(5)

To analyse the magnetization data, we developed a ﬁt routine which numerically diagonalizes the Hamiltonian (2), calculates the magnetization Mtot using equation (5) and varies the parameters in the Hamiltonian to minimize the mean square deviation between the data and this model. The isolated dimer approximation may be improved by introducing an effective exchange interaction K along the legs treated in mean ﬁeld approximation in the calculation of the magnetic susceptibility χ :

χladder = χdimer /(1 + K χdimer ).

(6)

Furthermore we calculate the magnetic speciﬁc heat by calculating the partition function Z for the Hamiltonian (2) c p,mag = T ∂ /∂ T (lnZ − 1/T (∂ (lnZ)/∂ β )) = 1 ∂ (ln( eEi /T )) ∂ ∑i Ei /T ln ∑ e =T − ∂T T ∂β i

(7) (8)

with β = 1/kBT , leading to an analytical function for a given angle γ which can be used to analyse the experimental speciﬁc heat with a standard least square deviation ﬁt routine.

4 Na2 Ni2 (C2 O4 )3 (H2 O)2 Antiferromagnetic S = 1 chain systems are of particular interest in quantum magnetism since they show a non-magnetic singlet ground state with a spin excitation gap (Haldane chains) [24, 54]. The physical properties of two coupled S = 1 chains have been studied theoretically [51, 2, 55, 50] but no experimental realization of a Haldane ladder system has been identiﬁed to date. Therefore we start our investigations with the Na2 Ni2 (C2 O4 )3 (H2 O)2 (SNOX) compound. In this compound, the Ni(II) ions are in the 3d 8 conﬁguration with an orbitally non-degenerate ground state and a total spin of S = 1 per ion.

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4.1 Magnetic Susceptibility We examined several single crystals in magnetic susceptibility measurements using a Quantum Design Magnetic Properties Measuring System (MPMS) in external ﬁelds of Bex = 2 T and 5 T in the temperature range 2–300 K. In these experiments the external ﬁeld was oriented along the a-, b- and c-axes as well as along different intermediate angles with respect to the a-axis. Figure 3 shows the temperature dependence of the magnetization along the aaxis and at different angles with respect to the a-axis in an external ﬁeld of Bex = 5 T. Similar to the powder measurement presented in Fig. 2 and the single crystal measurements in Bex = 2 T [40] one can clearly see a pronounced maximum for all directions with a strong downturn below ≈50 K suggesting a non-magnetic spinsinglet ground state. This behavior in general is expected as well for an isotropic two-leg spin ladder as for the two limiting cases, an isolated S = 1 Haldane chain or a system of antiferromagnetically coupled dimers [24, 13, 2, 51, 55, 50]. The inset of Fig. 3 shows the behavior at very low temperatures where the magnetization along a is smaller and approaches zero for T → 0 in contrast to directions transverse to a. Since the a-axis is close to the axial anisotropy axis as we will see below this indicates an easy axis of the Ni spin moments. The difference between experiment and theory at very low temperatures is due to an overestimated temperature independent Van Vleck contribution in the theoretical ﬁt.

z z z

Fig. 3 Temperature dependence of the magnetization of a SNOX single crystal at an external ﬁeld of Bex = 5 T along the a-axis (circles) and at different angles with respect to the a-axis (triangles and squares). The solid line represents the ﬁt as described in the text

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To analyse the magnetization data, we performed a combined ﬁt of Mtot for Bex = 5 T along all measured directions using an isotropic magnetic exchange coupling constant J and a ﬁxed absolute value of the axial anisotropy of |D| = 11.5 K. This anisotropy value was determined by the S = 1 zero ﬁeld splitting measured by ESR as described below. From these measurements we obtain an intradimer coupling constant of J = 44 K, an interdimer coupling constant of K = 0 K and g-values of ga = 2.215, g45 = 2.300 and g90 = 2.330. The ﬁtted angles to the anisotropy axis are γa = 19◦ , γ45 = 57◦ and γ90 = 90◦ . In addition a temperature independent contribution χ0 = 0.03 µB /dimer has been used. The ﬁt results are shown in Fig. 3 as solid lines. They are in good agreement with the results obtained in [40] for Bex = 2 T. To determine the absolute value of the axial anisotropy constant D we have performed tunable high-ﬁeld ESR measurements of SNOX at frequencies up to ν ∼ 1 THz in magnetic ﬁelds up to B ∼ 40 T. Details of the experimental setup can be found in Ref. [23]. A complex spectrum comprising a main line and a number of weak satellites has been observed. The ESR-intensity shows a thermally activated behavior similar to the T -dependence of the static magnetization (see Fig. 4, inset) thus ensuring that the ESR response is determined by the bulk Ni-spins. The frequency vs. magnetic ﬁeld dependence of the ESR modes is shown in the main panel of Fig. 4. In our analysis of the ESR data we focus the attention on the strongest resonance line 1 which ν (B)-dependence exhibits the intercept with the frequency axis at ν0 ≈ 239 GHz (details of the ESR analysis can be found in Ref. [40]). This intercept implies a ﬁnite energy gap ∆ = ν0 h/kB = 11.5 K for this resonance excitation. This zero ﬁeld gap can be straightforwardly identiﬁed with the zero ﬁeld splitting of the ﬁrst excited S = 1 triplet state of the dimer and gives directly the magnitude of the axial anisotropy parameter ∆ = |D| = 11.5 K. The knowledge of γa , γb and γc [40] allows a determination of the orientation of the anisotropy axis in the crystal. It is oriented parallel to the connecting line of the opposite nearest neighbor oxygen ions along the a-axis, tilted by 18◦ with respect to the crystallographic a-axis mainly towards the b axis [40]. Note, that analyzing the data with a negative D = −11.5 K leads to a similar good agreement between the model and the data. However, the positive D value is strongly supported by the high ﬁeld magnetization data described in the next section. The relative energies of the spin states of SNOX calculated in the framework of the Hamiltonian (2) with the parameters yielding the best ﬁt to the temperature dependent magnetization are plotted in Fig. 5 for magnetic ﬁelds oriented parallel and perpendicular to the main axis of the anisotropy tensor. One can see that at zero ﬁeld the spin singlet ground state S = 0 is well separated from the S = 1 triplet and the S = 2 quintet state. The separation energy between singlet and triplet corresponds to the coupling constant J, whereas that between singlet and quintet corresponds to 3J. The splitting of the excited S = 1 and S = 2 states in zero magnetic ﬁeld is induced by the anisotropy D. Both triplet and quintet levels split with increasing magnetic ﬁeld due to the Zeeman effect. The interplay of the zero ﬁeld splitting and the Zeeman splitting in the determination of the high ﬁeld properties of SNOX will be discussed in detail in the next section.

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Fig. 4 Main panel: Frequency ν vs. resonance ﬁeld Bres dependences (branches) of the strongest line 1 and weak satellites 2, 3 and 4 comprising the ESR spectrum of SNOX. Note the intercept of branch 1 with the frequency axis which yields the magnitude of the anisotropy parameter D. Solid line is a theoretical ﬁt according to Hamiltonian (2) with |D| = 11.5 K. Dashed lines are a linear approximation of weak satellite branches. Grey bar indicates the spread of the overlapping satellite resonances in the high ﬁeld regime. Inset: T -dependence of the strongest ESR line 1. Note its similarity to the T -dependence of the static magnetization

4.2 High Field Magnetization As can be seen from the Breit-Rabi diagram of the spin states of SNOX (Fig. 5) the splitting of the states in a magnetic ﬁeld yields a level crossing of the ground state with the lowest triplet state | − at a ﬁeld BC1 and a second level crossing of the | − state with the | − − quintet state at ﬁeld BC2 with BC1 < BC2 . This leads to a step-like behavior in high ﬁeld magnetization measurements. If the ﬁeld is applied along the z axis, the critical ﬁeld BC1 can be estimated using the equation BC1 = (J − D/3)/(g µB/kB ), depending strongly on the sign of D. For a ﬁeld perpendicular to the local anisotropy axis, a negative D (leading to higher critical ﬁelds for B z) pushes the critical ﬁeld to lower ﬁelds and vice versa for a positive D. Since in powder measurements the perpendicular situation dominates the spectrum of the spin states, the ﬁeld dependent magnetization, among others, can determine the sign of the anisotropy D. The critical ﬁeld strengths BCi depend on the magnetic exchange J. An exchange energy of J = 44 K corresponds roughly to BC1 = 30 T for BC1 . Therefore high magnetic ﬁelds are needed to observe the magnetization steps. We performed

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Fig. 5 Relative energies of the spin states, calculated for the magnetic ﬁeld applied parallel and perpendicular to the z axis of the uniaxial anisotropy tensor: the triplet S = 1 and the quintet S = 2 are well separated from the S = 0 ground state by an activation energy of |J| = 44 K and 3 |J|, respectively. Both multiplets exhibit a zero ﬁeld splitting due to crystal ﬁeld anisotropy. The shaded areas mark the ﬁeld ranges of ground state level crossings around 29 and 60 T in powder magnetization measurements (for details see text)

measurements at several temperatures (1.47 K, 4.2 K, 10 K and 26 K) on a powder sample with a mass of 23.2 mg in magnetic ﬁelds up to 55 Tesla. The results are shown in Fig. 6. In the low temperature experiments the ﬁrst magnetization step is clearly seen, whereas it smoothes at higher temperatures due to the thermal averaging process. The critical ﬁeld BC1 can be obtained from the derivation of the magnetization dM/dB for T = 1.47 K (see inset of the upper panel of Fig. 6), T = 4.2 K and T = 10 K. At all temperatures a sharp peak is observed at 29 Tesla. Additionally a second increase of the magnetization at much higher ﬁelds of about 50 Tesla can be anticipated. This step is clearly observed at T = 1.47 K and

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Fig. 6 High ﬁeld magnetization of a SNOX powder sample. Upper panel: Measurements at 1.47 K and 4.2 K, the inset shows the differential magnetization ∂ M/∂ B at 1.47 K. Lower panel: measurements at 10 K and 26 K, the inset shows the simulations for T = 0.01 K up to 80 T ﬁeld strength. The solid lines represent the simulation as described in the text

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T = 4.2 K and strongly broadened in the 10 K measurement. At 26 K the magnetization steps are not observed and a linear ﬁeld dependence of the magnetization is found. The solid lines in Fig. 6 describe simulations using the dimer model with the calculated g values g = 2.201 (see above) and g⊥ = g90 . For the simulation of a powder measurement in this case a weighted average over a full set of angles between the anisotopy axis and the ﬁeld direction from 0◦ to 90◦ in steps of 1◦ is performed with varying g from g to g⊥ . Since high magnetic ﬁelds lead to strong nonlinearities in the magnetization curve this approach is more realistic than the two step average M = (M + 2M⊥ )/3. For a good description of the high ﬁeld magnetization we need to use a coupling constant of J = 44.5 K instead of J = 44 K. A small deviation is found in the ﬁeld range of 30–33 T in the T = 1.47 K measurement, which may be caused by a little shift of the sample induced by the external ﬁeld. The simulations for a negative D = −11.5 K did not describe the data reasonably well for any J. In this case the simulated magnetization step is clearly broader than the step of the experimental data set. The enhanced broadening of the magnetization step for D = −11.5 K can be explained by the resulting switching of the m = 0 state with the |m| = 1 states which leads to different level crossing ﬁeld strengths BC1 . From the differential magnetization ∂ M/∂ B we can determine the critical ﬁeld BC1 of the powder sample to BC1 = 29 T. In our simulations, we get BC1 = 27.8 T for B parallel to the magnetic anisotropy axis and BC1 = 30.2 T for B perpendicular to the magnetic anisotropy axis, resulting also in BC1 = 29 T in a powder average. Notice, that the powder averaging leads to a broadening of the magnetization steps. For the second step, we get BC2 = 67.3 T and BC2 = 51.1 T for B parallel and perpendicular, respectively. This difference leads in a powder sample at the lowest temperature of our experiment, T = 1.47 K, to a continuous double sigmoid-like increase of the magnetization in the ﬁeld range of 51–67 Tesla, of which only the ﬁrst increase is observed in the measurement.

4.3 Speciﬁc Heat As shown in Fig. 5 the energy gap ∆ between the singlet ground-state and the | − state of the magnetic S = 1 triplet is reduced by an external ﬁeld from ∆0T ≈ 41.1 K to zero for a ﬁeld perpendicular to the anisotropy axis at a level crossing ﬁeld of BC1 ≈ 30.2 T. Then at higher ﬁeld the gap increases with further increasing ﬁeld until the | −− state of the S = 2 triplet anti-crosses the S = 0 singlet at ≈42 T. At higher ﬁelds the gap decreases again towards zero at the second ground state change at BC2 ≈ 51.3 T. This behaviour can be directly observed in the calculated magnetic speciﬁc heat according to Eq. (8) (Fig. 7). At low ﬁelds only one broad maximum is found. Increasing the ﬁeld leads to a decrease of the gap. Approaching BC1 a low temperature peak is separated from the main component. At the critical ﬁeld this low temperature peak sharpens and shifts towards zero temperature whereas the maximum of the main component shifts to slightly higher temperatures. For ﬁelds above BC1 the low temperature peak shifts to higher temperatures.

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Fig. 7 Calculated temperature dependence of the magnetic speciﬁc heat of SNOX for high external ﬁelds

Measurements of the speciﬁc heat c p (T) were performed on a single crystal of mass 2.04 mg using a Quantum Design PPMS in zero-ﬁeld and an external ﬁeld of B = 9 T perpendicular to the a-axis in the temperature range 2–300 K (Fig. 8). The measurements differ only in the low temperature regime where the magnetic speciﬁc heat is dominant [41]. Due to the large coupling constant of the dimer, the magnetic contribution to c p (T) is strongest in the temperature range around T ≈ 15–50 K. Since the Debye temperature TD of such metal-organic compounds is typically in the range of 100– 200 K, the magnetic contribution to c p cannot be extracted with a standard T3 -ﬁt for the lattice contribution which is possible only for T TD . By analyzing the difference of both measurements c p (9T) - c p (0T) as shown in Fig. 9 we subtract the phonon contribution. A ﬁt using the isolated dimer model with variable parameters J and g in the temperature range T < 19 K results in a very good agreement for a coupling constant J = 43.7(5) K and a g-value of 2.41(5) [41]. In conclusion, the speciﬁc heat measurements on SNOX are consistent with the susceptibility and high ﬁeld magnetization measurements. All experiments show that the isolated dimer model in the spin hamiltonian approach provides a very good description of the system. This can be understood considering the different coordination and superexchange angles of the exchange mediating oxalate molecule along the rungs and legs of the ladder. A detailed discussion is presented in Section 8 below. The derived parameter values for D, J and the g-factor are

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Fig. 8 Temperature dependence of the speciﬁc heat of a SNOX single crystal (2.04 mg) at zero ﬁeld (open circles) and Bex = 9 T (ﬁlled squares) compared with the calculated magnetic speciﬁc heat for zero ﬁeld (thick line) and Bex = 9 T (thin line). The inset shows the measurements on an extended temperature range

typical for Ni(II) in a pseudo-octahedral environment and oxalate-bridged dimers [7, 39, 46, 58, 57, 17, 44].

5 Na2 Mn2 (C2 O4 )3 (H2 O)2 5.1 Magnetic Susceptibility In Na2 Mn2 (C2 O4 )3 (H2 O)2 the Mn(II) (3d 5 ) conﬁguration results in an orbitally non-degenerate ground state and a total spin of S = 5/2 per ion. This leads to a very speciﬁc behaviour compared with the Ni(II), Co(II) and Fe(II) ions: Whereas for the ions with a more than half-ﬁlled 3d shell the orbital magnetic moments are quenched and have to be considered as a perturbation, the orbital angular momentum L for the 3d 5 conﬁguration is zero following Hund’s rules. Therefore we can neglect the anisotropy term and the Hamiltonian simpliﬁes to H = J S1 S2 + g µB B(S1 + S2 ).

(9)

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Fig. 9 Speciﬁc heat difference c p (9T) - c p (0T) of SNOX. The inset shows the difference in an extended temperature range

We measured a powder sample of mass 24.8 mg in a home built vibrating sample magnetometer [30] in external ﬁelds of Bex = 1 T and 16.8 T in the temperature range 2–300 K. The measurements are shown in Fig. 10. For Bex = 1 T a clear maximum is observed at around 10 K whereas for Bex = 16.8 T the magnetization increases monotonously with decreasing temperature indicating that at this ﬁeld strength the antiferromagnetic intradimer interactions are suppressed by the strong Zeeman interaction of the individual spins in the external ﬁeld. To analyse the magnetization data, we performed a combined ﬁt of Mtot for Bex = 1 T and Bex = 16.8 T using an isotropic magnetic exchange coupling constant J and an averaged g-factor. Since without anisotropy the magnetizations for applied external ﬁelds parallel and perpendicular to the z-axis are identical we performed an analysis without powder averaging. We obtained an intradimer coupling constant of J = 3.5 K and a g-value of g = 2.01. The ﬁt is done without a temperature independent contribution and without considering paramagnetic impurities. The results are shown in Fig. 10 as solid lines. To prove the validity of the isotropic model (i.e. the 6 S5/2 ground state) we performed an analysis of the Bex = 1 T measurement including an anisotropy D and an anisotopic g-factor in a powder average. This approach did not improve the quality of the ﬁt. The same holds for including an interdimer coupling K on the mean ﬁeld level. The different behaviour of the 1 T and the 16.8 T magnetization curves can be explained considering the energy levels of the system. The Breit-Rabi diagram of SMOX shows 6 multiplets, one for each spin between S = 0 and S = 5. These

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Fig. 10 Temperature dependence of the magnetization of a SMOX powder sample at external ﬁelds of Bex1 = 1 T (triangles) and Bex1 = 16.8 T (squares). The 16.8 T measurement is divided by the ﬁeld. The solid line represents the ﬁt as described in the text

multiplets are degenerate for B = 0 reﬂecting the absent anisotropy D. In an external magnetic ﬁeld the multiplets split due to the Zeeman interaction and ﬁve ground state level crossings are observed: for J = 3.5 K and g = 2.01 at approximately 2.6, 5.2, 7.8, 10.4 and 13 T. A non-magnetic ground state (i.e. ∂ E/∂ B = 0) is only found for ﬁelds below 2.6 T. At higher ﬁelds a magnetic ground state exists with a ﬁnite magnetization for T → 0 which increases steplike at the level crossings. The different ground states can be observed in the low temperature behaviour of the temperature dependent magnetization curves. The temperature dependence of the magnetization for ﬁelds below 2.5 T shows a strong downturn to zero reﬂecting the depopulation of the magnetic levels whereas for higher ﬁelds a ﬁnite value of 2, 4, 6, 8 and 10 µB per dimer is found for the different ﬁeld ranges. For the highest ﬁeld regime above 13 T, the magnetization increases monotonously with decreasing temperature to the maximum value of 10 µB per dimer.

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5.2 High Field Magnetization Field dependent magnetization measurements were also performed in the vibrating sample magnetometer on a powder sample at temperatures 2.4, 6, 10 and 30 K in ﬁelds up to 16.8 T. The measurements are shown in Fig. 11. The analysis is done using the Hamiltonian (9). The eigenvalues can be calculated directly using En = J/2 (S(S + 1) − (S1(S1 + 1) − S2(S2 + 1)) − g muB B (S1z + S2z)

(10)

were the basis | S ms is given by total spin S = S1 + S2 = 0...5 with magnetic quantum numbers ms from −S to S leading to 36 eigenvalues. This basis allows to calculate the exact eigenvalues with parameters J, g and B leading to an analytical function for the magnetization Mdimer (J, g, B) using Eq. (3). With this function Mdimer (J, g, B) we performed a combined ﬁt for all temperatures leading to a coupling constant J = 3.5 K and g = 2.04 in excellent agreement with the temperature dependent measurements. The ﬁt results are shown in Fig. 11 as solid lines. The grey solid line shows the calculated magnetization for a very low temperature T = 0.1 K. Only at this temperature the magnetization steps at the level crossing ﬁelds are present. These steps are smoothed even at the lowest measured temperature of 2.4 K due to the thermal averaging process and the small coupling constant J. The deduced parameter value for J and the g-factor are typical for Mn(II) in a pseudo-octahedral environment and oxalate-bridged dimers [37, 21, 42, 38, 52].

6 Na2 Co2 (C2 O4 )3 (H2 O)2 6.1 Magnetic Susceptibility The Co(II) compound has been synthesized by Price et al. [47]. In SCOX, the Co(II) ions are in the 3d 7 conﬁguration with an orbitally degenerate ground state 4 T1g and a spin of S = 3/2 per ion in the high-spin state. The presence of an additional orbital moment is already evident from the high temperature Curie-Weiss analysis presented in the introduction. Therefore, a description of the magnetic properties using the spin hamiltonian approach will result in an anisotropic g-factor very different from g = 2 and a strong uniaxial anisotropy due to spin-orbit coupling. Alternatively interacting Co(II) ions in octahedral environments are described by a pseudo spin 1/2 and a strong Ising type anisotropy in the magnetic interaction. Since the purpose of this work is a comparison of the different spin multiplicities on the transition metal site we use the model hamiltonian Eq. (2). The ﬁrst magnetic susceptibility data on a single crystal parallel and perpendicular to the a-axis have been performed by Honda et al. [28]. There the data have

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Fig. 11 High ﬁeld magnetization of a SMOX powder sample. The ﬁgure shows measurements at 2.4 K, 6 K, 10 K and 30 K. The black solid lines represents the ﬁt as described in the text and the grey solid line is a calculated magnetization for T = 0.1 K

been analyzed using a phenomenological S = 1/2 two-leg spin ladder function in a limited temperature range only (7–20 K). From our systematic study of STOX it is evident that also the Co(II) system can be described as nearly uncoupled dimers of S = 3/2 spins on the rungs of the ladder. Therefore we present a new analysis of the single crystal magnetic susceptibility data based on this model similar to the Ni(II) system using hamiltonian (2). A combined ﬁt along both directions (solid lines in Fig. 12) results in an intradimer coupling constant of J = 13.8 K, an anisotropy of D = −6.3 K and g-values of ga = 2.60, and g⊥ = 2.27. For comparison with these single crystal measurements, we performed a susceptibility measurement at B = 1 T on the powder sample used for the high ﬁeld magnetization measurements. An analysis using Eq. (5) with a powder average (M = (M + 2M⊥ )/3) results in J = 11.8 K and D = −9 K and a large g-value anisotropy with g-values of g = 3.90, and g⊥ = 1.50. Note that J is smaller by a factor of ≈4 than in the Ni(II) system. The variation of the g-value is opposite to SNOX due to a different orbital contribution to the total magnetic moment. Since no high ﬁeld high frequency ESR measurements have been performed so far, the single ion anisotropy strength D had to be used as a free parameter. A satisfactory description of the susceptibility and high ﬁeld magnetization data is obtained only for a negative value of D. In view of the limited accuracy of the single crystal ﬁt the angles to the anisotropy axis γa = 28◦ and γ⊥ = 38◦ are consistent

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Fig. 12 Temperature dependence of the magnetization of a SCOX single crystal at an external ﬁeld of Bex = 0.01 T along the a-axis (circles) and perpendicular to the a-axis (diamonds) (the data are taken from Ref. [28]). The solid lines represent an isolated dimer ﬁt as described in the text

with an orientation of the anisotropy axis as in SNOX parallel to the connecting line of the opposite nearest neighbor oxygen ions along the a-axis. Figure 13 shows the low energy spin states of SCOX calculated for magnetic ﬁelds oriented perpendicular to the main axis of the anisotropy tensor. This orientation gives the dominant contribution in a powder magnetization measurement. For this plot we used the set of parameters J = 11.8 K, D = −9 K and g⊥ = 1.5 derived from the ﬁt of the powder measurements which is also used for the simulation of the high ﬁeld magnetization measurement as described below. Note the near degeneracy of the |m = 0 > state of the S = 1 triplet and the S = 0 singlet due to the particular combination of J and D. Therefore a series of ground state level crossings is predicted at critical ﬁeld strengths BCi of ≈11.6 T, 26.6 T and 41.7 T in this ﬁeld orientation.

6.2 High Field Magnetization We performed a pulsed high magnetic ﬁeld magnetization measurement at 4.2 K on a powder sample of SCOX in magnetic ﬁelds up to 52 Tesla (Fig. 14). It shows one broadened magnetization step from ≈0.5 µB to nearly 5.5 µB between 10 T and 20 T in addition to a strong ﬁnite magnetic susceptibility (gradient in M(B)) at

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Fig. 13 Low energy spin states in SCOX for a magnetic ﬁeld applied perpendicular to the z axis of the uniaxial anisotropy tensor: the ||m = 0 > state of the S = 1 triplet and the S = 0 singlet are nearly degenerate due to the particular combination of J and D. The arrows indicate the ground state level crossing ﬁelds

low ﬁelds. This behavior is in stark contrast to a three step function which can be expected for a dimer of S = 3/2 spins from the Breit-Rabi diagram (Fig. 13). In this case when the antiferromagnetic exchange is of similar strength as the single-ion anisotropy it is necessary for a qualitative and quantitative description of a powder magnetization measurement to perform a weighted average over a full set of angles between the anisotropy axis and the ﬁeld direction from 0◦ to 90◦ in steps of 1◦ with varying g from g to g⊥ since the Breit-Rabi diagram depends strongly on this angle. This is illustrated in the inset of Fig. 14 where we plot the calculated single crystal magnetization curves at T = 0.1 K for three different angles in comparison with the powder average. Due to the strong anisotropy for B || z the ﬁrst level crossing at 10 T leads from S = 0 to the |m = −3> of the S = 3 fully polarized sextet state resulting in a sharp step of the magnetization curve. This situation contributes signiﬁcantly to the powder average. The black solid line in Fig. 14 is calculated for T = 4.2 K using the set of parameters obtained by the powder susceptibility with an optimized value g = 3.0. Due to the dominance of the large step of the B||z direction, the inﬂuence of the steps in the B ⊥ z direction in the powder sample is noticeable only at very low temperatures. Note that in Fig. 14, at T = 0.1 K the magnetization shows two small kinks at the level crossing ﬁelds of the B ⊥ z direction (arrows on grey solid line). These kinks result from the powder averaging which leads to a smoothing of the steps (cf. inset of Fig. 14). Recently we became aware of experiments on Co-Oxalate by Y. Nakagiwa et al. which qualitatively agree with our ﬁndings [43].

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Fig. 14 Pulsed high ﬁeld magnetization of a powder sample of SCOX at 4.2 K. The solid lines represent the simulations at 4.2 and 0.1 K. The inset shows single crystal simulations for three different angles between the magnetic ﬁeld and the anisotropy axis in comparison with the powder average for T = 0.1 K

In the literature a very wide range of values for D and the g-factor are found for Co(II) complexes. See e.g. the review by R. Boca [5] or examples in [6, 8, 31, 29, 21, 36, 10].

7 Na2 Fe2 (C2 O4 )3 (H2 O)2 7.1 Magnetic Susceptibility The magnetic susceptibility of a powder sample of the Fe(II) compound has been measured by Kreitlow et al. [35]. In SIOX, the Fe(II) ions are in the 3d 6 conﬁguration with an orbitally degenerate ground state 5 T2g and a spin of S = 2 per ion in the high-spin state. The presence of an additional orbital moment is already evident from the high temperature Curie-Weiss analysis of the magnetic susceptibility. Comprehensive single crystal magnetic susceptibility data parallel and perpendicular to the a-axis have been published by Kikkawa et al. [33] showing a very strong anisotropy. Only the data parallel to the a-axis have been analyzed using a singlettriplet system including a uniaxial anisotropy resulting in a singlet-triplet gap of 51 K and a very large anisotropy D = 41 K. Here we present a new analysis of the

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Fig. 15 Temperature dependence of the magnetization of a SIOX single crystal at an external ﬁeld of Bex = 0.01 T along the a-axis (circles) and perpendicular to the a-axis (diamonds) (the data are from Ref. [33]). The solid line represents an isolated dimer ﬁt as described in the text

single crystal magnetic susceptibility data parallel and perpendicular to the a-axis based on a dimer model using Hamiltonian (2). The combined ﬁt along both directions (solid lines in Fig. 15) results in an intradimer coupling constant of J = 6.5 K, g-values of ga = 2.67, and g⊥ = 1.76. Note that J is reduced by a factor of ≈2 with respect to the Co(II) system. The single ion anisotropy strength D has been determined to D = −7.1 K. Similar to the SCOX compound, we performed magnetic susceptibility measurements on a SIOUX powder sample used for the high ﬁeld magnetization measurements and analysed these data in the same way. This results in a parameter set of J = 6.7 K, D = −8.7 K, g = 2.67, and g⊥ = 2.65 which we used to calculate the Breit-Rabi diagram (see inset of Fig. 16). Therefore in this system the isotropic magnetic exchange and the single ion anisotropy are of similar size leading to the strong anisotropy in the measured magnetic susceptibility. In view of the limited accuracy of the ﬁt the angles to the anisotropy axis γa = 35◦ and γ⊥ = 52◦ are again consistent with an orientation of the anisotropy axis parallel to the connecting line of the opposite nearest neighbor oxygen ions along the a-axis.

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7.2 High Field Magnetization We performed a pulsed high magnetic ﬁeld magnetization measurement at 4.2 K on a powder sample of SIOX in magnetic ﬁelds up to 52 Tesla (Fig. 16). It shows the same oveall features as the experiment on the Co analogue, namely a broadened magnetization step from ≈1.0 µB to nearly 10 µB per dimer between 7 and 20 T in addition to a strong ﬁnite magnetic susceptibility at low ﬁelds. The low energy part of the Breit-Rabi diagram for B perpendicular to the anisotropy axis using a set of parameters J = 6.7 K, D = −8.7 K, and g⊥ = 2.65 optimized to describe the high ﬁeld magnetization data is shown in the inset of Fig. 16. The |m = 0> state of the S = 1 triplet and the S = 0 singlet are very close and several level crossings occur between 5 and 40 T. In the literature similar to Co(II) a very wide range of values for D and the g-factor is found for Fe(II) complexes. See e.g. the review by R. Boca [5] or examples in [15, 27, 25, 18, 26, 56, 21, 3].

Energy (K)

0 40 20 0 20 40 60 80

10

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Fig. 16 Pulsed high ﬁeld magnetization of a powder sample of SIOX at 4.2 K. The solid lines represent simulations at 4.2 and 0.1 K. The inset shows the low energy states in a Breit-Rabi diagram for a magnetic ﬁeld applied perpendicular to the z axis of the uniaxial anisotropy tensor

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8 Magnetic Exchange Pathways An unexpected result of the present study is the very large difference between the strength of the magnetic exchange interaction along the rungs and legs of the spin ladder leading to the applicability of the dimer model in this system. This can be qualitatively understood considering the different coordination and superexchange angles of the exchange mediating oxalate molecules. On the rungs the oxalate forms a µ -1,2,3,4 bridge between two transition metal ions. This bridge provides two symmetric superexchange pathways. On each path the transition metal 3d x2 − y2 orbitals have an enhanced electron probability density extending directly towards the corresponding oxygen ions. There is a direct overlap of the transition metal 3d x2 − y2 and the O 2p wave functions forming σ bonds. The polarized 2p orbitals themselves are strongly overlapping. Therefore the intermediate carbon atom is not involved in the superexchange mechanism resulting in a strong antiferromagnetic superexchange interaction (see Fig. 17). In the literature several µ -1,2,3,4 oxalato-bridged Ni(II) dimer and chain systems are reported (see e.g. [17, 49] and references therein) with J values in the range of 20 to 42 K. For the oxalate bridge along the legs of the spin ladder the situation is completely different. In this case a µ -1,2,3 oxalato bridge is formed. As shown in the right panel of Fig. 17 only one transition metal ion forms two covalent bonds (one with the x2 − y2 orbital and one with the 3z2 − r2 orbital) with the oxalate molecule whereas of the second transition metal ion only the 3z2 − r2 orbital forms one covalent bond. The O 2p orbitals involved in the T-O bonds are not overlapping with each other and

Fig. 17 T-(ox)-T coordination and proposed magnetic exchange pathways on the rungs (left panel) and on the legs of the spin ladder (right panel). The involved T 3d and oxygen 2p orbitals are shown in black

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Fig. 18 Change of the isotropic magnetic exchange constant J with decreasing number of 3d electrons, i.e. increasing spin quantum number, in the series Na2 T2 (C2 O4 )3 (H2 O)2 . The solid line represents a ﬁt J = J1 1/S2 with J1 = 39(4) K assuming a magnetic exchange energy independent of the total spin S

therefore either the carbon atoms or orthogonal O 2p orbitals are involved which strongly suppresses the strength of the superexchange mechanism and may even lead to a weak ferromagnetic exchange. Similar µ -1,2,3 oxalato bridges are reported for Cu(II) systems with magnetic exchange strengths in the range of −0.2–0.3 K [45, 11] consistent with the estimate of |K| [40] being negligibly small in STOX. A second interesting result is the continuous decrease of the isotropic magnetic exchange strength J going from 3d8 (Ni(II)) to 3d5 (Mn(II)) shown in Fig. 18. The same systematics is present e.g. in the series of three dimensional antiferromagnets KTF3 with T = Ni, Co, Fe, Mn [16]. There the nearest neighbor exchange varies from 44–50 K in KNiF3 over 19 K in KCoF3 and 6 K in KFeF3 to 3.6 K in KMnF3 . This systematics can be regarded as a result of the description of the total magnetic exchange energy in terms of the product J S1 S2 . As described above the 3d x2 − y2 orbital contributes the most to the total spin exchange energy. In Co(II), Fe(II) and Mn(II) the additional spin density is located in the t2g orbitals. These orbitals do not point directly towards the oxygen ions. They may contribute only very little to the magnetic exchange energy but increase the total spin value S. As a consequence the numerical value of the exchange constant J is reduced proportional to 1/S2 as illustrated by the solid line in Fig. 18.

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9 Conclusion In this paper we demonstrate the application of high magnetic ﬁelds to study the magnetic properties of low dimensional spin systems. We present a case study on the series of 2-leg spin-ladder compounds Na2 T2 (C2 O4 )3 (H2 O)2 with T = Ni, Co, Fe and Mn. In all compounds the transition metal is in the 2+ high spin conﬁguation. The localized spin varies from S = 1 to 3/2, 2 and 5/2 within this series. The magnetic properties were examined experimentally by magnetic susceptibility, high ﬁeld ESR, pulsed high ﬁeld magnetization and speciﬁc heat measurements. The data are analysed using a spin hamiltonian description with Heisenberg exchange interaction. Although the transition metal ions form structurally a 2-leg ladder, an isolated dimer model consistently describes the observations very well. All compounds exhibit magnetic ﬁeld driven ground state changes which at very low temperatures lead to a multistep behaviour in the magnetization curves. In the Co and Fe compounds a strong axial anisotropy induced by the orbital magnetism leads to a nearly degenerate ground state and a strongly reduced critical ﬁeld. We ﬁnd a monotonous decrease of the intradimer magnetic exchange constant J proportional to 1/S2 which indicates that the additional spin density in the t2g orbitals does not contribute to the total magnetic exchange energy. Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft through SPP 1137 “Molecular Magnetism” grant KL 1086/6-1 and 6-2 is gratefully acknowledged. The work of R.K. in Toulouse was supported by the DFG through grant KL 1824/1-1. D.J.P. is grateful to the EPSRC of the UK for the award of an Advance Research Fellowship (Gr/A00836/02).

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Quantum Nanomagnets and Nuclear Spins: An Overview Andrea Morello

Abstract This mini-review presents a simple and accessible summary on the fascinating physics of quantum nanomagnets coupled to a nuclear spin bath. These chemically synthesized systems are an ideal test ground for the theories of decoherence in mesoscopic quantum degrees of freedom, when the coupling to the environment is local and not small. We shall focus here on the most striking quantum phenomenon that occurs in such nanomagnets, namely the tunneling of their giant spin through a high anisotropy barrier. It will be shown that perturbative treatments must be discarded, and replaced by a more sophisticated formalism where the dynamics of the nanomagnet and the nuclei that couple to it are treated together from the beginning. After a critical review of the theoretical predictions and their experimental veriﬁcation, we continue with a set of experimental results that challenge our present understanding, and outline the importance of ﬁlling also this last gap in the theory.

1 Introduction In the vast and diverse ﬁeld of quantum magnetism, the quantum properties of large magnetic molecules have enjoyed a strong and motivated research activity for more than a decade. Physicists and chemists, theoreticians and experimentalists, engineers and philosophers, all would ﬁnd at least one good reason to be interested in these systems. We shall refer here to “quantum nanomagnets” as the broad family of molecules containing a core of magnetic transition-metal ions, which interact by

A. Morello Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada Australian Research Council Centre of Excellence for Quantum Computer Technology, University of New South Wales, Sydney NSW 2052, Australia, e-mail: [email protected]

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strong superexchange and which possess magnetic anisotropy due to crystal ﬁeld effects [13]. At sufﬁciently low temperatures (typically ∼10 K), i.e. much lower than the typical intramolecular exchange interaction energy, the whole molecule effectively behaves as a nanometer-sized magnet. The total ground state spin value can be rather large, S ∼ 10, and is often called “giant spin”. The best studied examples are Mn12 [22], Fe8 [50] and Mn4 [3]. The magnetic core of each molecule is stabilized by organic ligands, and the molecules are typically bound to each other by van der Waals forces to form electrically insulating crystals (Fig. 1). With just a few exceptions [49, 16], the magnetic interaction between different molecules is only of dipolar origin, thus orders of magnitude weaker than the intramolecular exchange. Single-ion anisotropies, exchange interactions, point symmetry and crystal structure all contribute in a complicated way to the total magnetic anisotropy of the giant spin. The ﬁrst great success of molecular magnets consisted in the demonstration of magnetic bistability and hysteresis at the molecular level [35], i.e. not arising from long-range interactions but only from local ones. Molecules possessing this sort of bistability were named “Single Molecule Magnets” (SMMs). One could then think of 2D arrays of such molecules [30] as the ultimate magnetic recording medium [18], with information density ∼1012 bit/cm2 . For this purpose, the main research goal is to achieve the highest possible single-molecule anisotropy, to allow magnetic bistability up to–ideally!–room temperature. However, a proper description of bistability in nanomagnets would be incomplete without quantum mechanics. Since the anisotropy barrier for reorientation of the giant spin is large but not inﬁnite, there is in principle a non-vanishing probability for the spin to invert its direction by quantum-mechanical tunneling through the barrier [7]. That is, the magnetic memory would delete itself because of quantum mechanics! The tunneling probability can be estimated with the knowledge of the anisotropy parameters of the giant spin, obtained e.g. by electron paramagnetic resonance or neutron scattering. The resulting tunneling rate turns out to be extremely sensitive to the external magnetic ﬁelds applied to the molecule. In Mn12 for instance, one would naively expect the tunneling rate to become astronomically long if a stray ﬁeld of just 10−9 T is applied along the anisotropy axis. The experimental observation of quantum tunneling of magnetization in Mn12 [41, 10, 15, 12] represented a major breakthrough, but in some sense a puzzling one. The puzzle was solved by carefully considering the role played by the coupling between giant spin and surrounding nuclei, which are always present in the ligands (1 H, 13 C, 35 Cl, . . . ) or in the magnetic ions themselves (55 Mn, 57 Fe, 53 Cr, . . . ). The dynamics of the nuclear spins generates a ﬂuctuating magnetic ﬁeld on the giant spin, thereby sweeping its energy levels through the tunneling resonance and yielding a ﬁnite tunneling probability [32]. Once the giant spin is allowed to have (quantum) ﬂuctuations of its own, it exerts a back-action on the nuclei and essentially determines their dynamics. Thus, the system of giant spin + nuclei must be analyzed as a whole, and the resulting theoretical description is now known as “theory of the spin bath” [34]. It predicts, among other things, a square-root law for the relaxation of magnetization at very low-T [33], and a dependence of the tunneling rate on the isotopic composition of the sample. These predictions have

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∼ 1.5 nm

Fig. 1 Crystalline and molecular structure of a prototypical quantum nanomagnet, Mn12 -ac [22, 36]. The core contains 8 Mn3+ (s = 2) and 4 Mn4+ (S = 3/2) ions, which interact by superexchange to form a ferrimagnetic ground state with total spin S = 10. The molecules crystallize in a tetragonal structure

been promptly veriﬁed by measuring the electron spin relaxation in Fe8 crystals [46, 48], where it is possible to obtain samples that have stronger (by 57 Fe enrichment) or weaker (by replacing 1 H with 2 H) hyperﬁne couplings as compared to those with natural isotopic abundance, while leaving the giant spin unchanged. The picture has been completed by experiments looking at the “other side of the coin”, i.e. studying directly the dynamics of the nuclear spins by NMR experiments [17, 11, 14, 25, 27, 4], and ﬁnding that such dynamics is indeed profoundly entangled with the quantum ﬂuctuations of the giant spin. At the present stage, much of the efforts in the ﬁeld are directed towards using quantum nanomagnets as spin qubits for quantum information purposes, by pushing the giant spins into a regime where the tunneling can be made coherent [29]. The purpose of this mini-review is to give an accessible and somewhat pedagogical introduction to the crucial aspects of the coupled system “quantum nanomagnet + nuclear spins”, what makes it special, what has been understood, and what requires further attention. It will be shown that much of the common knowledge on nuclear and electron spin dynamics is entirely inappropriate to describe this system. Importantly, the same often applies to other quantum degrees of freedom in mesoscopic physics (SQUIDs, quantum dots, . . . ). The discussion given here on quantum nanomagnets should thus be taken as the “worked-out example” of a problem of much broader interest. The reason for choosing this speciﬁc example lies in the wonderful property of quantum nanomagnets to combine mesoscopic size and fascinating physics, with the cleanliness and reproducibility of a product of synthetic chemistry.

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2 Theoretical Framework At temperatures such that the internal magnetic excitations can be neglected, i.e. when the entire molecule behaves as a giant spin of value S, the minimal effective spin Hamiltonian that describes the quantum nanomagnet coupled to a bath of nuclear spins {Ik } is: H = −DSz2 + E(Sx2 − Sy2) − g µB S · B − ∑ Ik A˜ k S,

(1)

k

where zˆ is the easy axis of magnetization. The ﬁrst term of this Hamiltonian, −DSz2 , gives rise to an energy levels structure as shown in Fig. 2(a), with a parabolic anisotropy barrier separating pairs of degenerate states. Were this the only term in the Hamiltonian (1), its eigenstates would be the eigenstates of Sz , i.e. the |m projections of the spin along the easy axis. The second term in (1), E(Sx2 − Sy2 ), is the lowest-order anisotropy term that can break the uniaxial symmetry of the molecule, and represents a rhombic distortion with hard axis x. ˆ The Hamiltonian including this term no longer commutes with Sz , and its eigenstates are now symmetric and antisymmetric superpositions of the | + m, | − m states. At very low temperatures, such that only the two lowestenergy states are thermally occupied, the giant spin can be effectively described as a two-level system (that is, a qubit) with eigenstates |S , |A separated by a tunneling splitting 2∆0 (Fig. 2(b)). A giant spin prepared in one of the “classical states” |Z ± = 2−1/2(|S ± |A ), corresponding to the magnetization pointing along ±ˆz, would tunnel between | + Z and | − Z at a frequency h¯ /2∆0 in the absence of external ﬁelds. The third term, −gµB S · B, describes the coupling to a magnetic ﬁeld. If B ⊥ zˆ, this represents an extra non-diagonal term which has the effect of rapidly increasing the tunneling splitting 2∆0 (B⊥ ) (Fig. 2(c)). Conversely, B zˆ has the effect of breaking the degeneracy of the “classical” states ±|m. If the longitudinal bias ξ = gµB Sz Bz is much larger than 2∆0 , the spin is effectively localized on one side of the barrier, with vanishing probability to tunnel. Finally, the term − ∑k Ik A˜ k S represents the coupling to the nuclear spin bath. Although in some instances the coupling tensor A˜ may be isotropic (e.g. for the 55 Mn nuclei in Mn4+ ions), this is not true in general. Also, strictly speaking an external ﬁeld acts on the nuclei as well with a term −γk B · Ik (γ is the nuclear gyromagnetic ratio), but this can be usually neglected in comparison with the hyperﬁne coupling. If one were to assume that the nuclear spins are static, then the effect of the hyperﬁne ﬁeld on the giant spin would be the same as that of a static external ﬁeld, with typical strength 10−5 − 10−3 T, or 0.1 − 10 mK in terms of coupling energy. The spread of possible hyperﬁne couplings is incorporated in a parameter E0 . By inspecting Fig. 2(c) for the case B = 0, we see that a transverse component of the hyperﬁne ﬁeld would have hardly any effect on the nanomagnet’s tunneling splitting, whereas a longitudinal component gives a bias ξN that is easily several orders of magnitude larger than 2∆0 . Under these conditions, the giant spin should be frozen for the

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Bperp (T)

Fig. 2 (a) Energy levels scheme for the giant spin of Mn12 -ac in zero external ﬁeld, neglecting non-diagonal terms in the effective spin Hamiltonian. (b) Including the non-diagonal terms, the exact eigenstates become symmetric and antisymmetric superpositions of the “classical” localized states, separated by a tunneling splitting 2∆0 . (c) Tunneling splitting as a function of external ﬁeld applied perpendicular to the easy axis of anisotropy for Mn12-ac. The hatched area represents the typical range of hyperﬁne couplings

eternity. The case could be made even stronger by accounting also for the dipolar interaction between molecules in the crystal, which adds an extra (static) random ﬁeld of order 10−3 − 10−2 T. So why do we observe tunneling in the experiments? The answer comes from the dynamics of the nuclear spins. At the very least, the nuclear spin bath will have an intrinsic dynamics due to the mutual dipolar couplings, which can be described by an additional term of the form ∑k,k IkV˜k,k Ik . This effectively generates a bias on the giant spin that ﬂuctuates on a timescale of the order of the nuclear T2 (∼1 − 10 ms typically). The amplitude of this ﬂuctuation can be sufﬁcient to sweep the electron spin levels through the tunneling resonance, for that tiny minority of molecules that ﬁnds itself having ξ 0 at some time. The sweeping of hyperﬁne bias through the resonance yields a Landau-Zener – like incoherent tunneling probability. Once a giant spin has tunnelled, two crucial things happen: (i) the distribution of internal dipolar ﬁelds in the crystal suddenly changes, giving other molecules a chance to have ξ 0, etc. . . ; (ii) the hyperﬁne ﬁeld on the nuclear spins belonging to or surrounding the tunnelled molecule suddenly changes, stimulating further dynamics of the nuclear spins. The consequences of (i) are the formation of a “hole” in the distribution√of dipolar biases [42], with width related to the spread of hyperﬁne bias, and a ∝ t law for the short-term relaxation of the magnetization [33]. Here we concentrate on the aspect (ii), namely what happens to the nuclear spins as a consequence of a tunneling event. This point is extremely interesting and instructive, because it radically deviates from the framework under which the dynamics of nuclear spins is commonly analyzed [1, 37]. Practically all of the theory of nuclear magnetism is based on perturbation theory, since one typically has a large static magnetic ﬁeld, Bz , applied from the outside, plus some local ﬂuctuations of much smaller amplitude. For instance, one can estimate the longitudinal nuclear relaxation rate, T1−1 , by evaluating the magnitude of the transverse component of the ﬂuctuating ﬁeld, b⊥ , and assuming that

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the time-correlation of the ﬂuctuations decays exponentially with time constant τ , b⊥ (t1 )b⊥ (t2 ) b2⊥ exp(−(t2 − t1 )/τ ). Within perturbation theory, this yields:

τ 1 γ 2 b2⊥ , T1 1 + ωN2 τ 2

(2)

with γ and ωN the nuclear gyromagnetic ratio and Larmor frequency, respectively. Eq. (2) simply means that the nuclear relaxation rate is proportional to the power spectrum of the ﬂuctuating ﬁeld, taken at the nuclear Larmor frequency. The reader shall recognize that this is just the ﬂuctuation-dissipation theorem, a well-known result of linear response theory [20]. Expressions like (2) are ubiquitous in the NMR literature, since they relate an experimentally accessible quantity (the nuclear T1 ) to the dynamic properties of the environment where the nuclei are immersed (τ ), thereby giving NMR its status of “local probe” for the dynamics of complex systems. However, the nuclear spin dynamics in tunneling nanomagnets is one example where the use of expressions derived from perturbation and linear response theory is incorrect and unjustiﬁed. This is particularly true for the nuclei that “belong” to a magnetic ion in the molecule core, like 55 Mn or 57 Fe. Perturbation theory breaks down here because the hyperﬁne ﬁeld produced by the electron spins on their nuclei (typically ∼30 T in Mn and ∼50 T in Fe) is much larger than any externally applicable ﬁeld. Therefore, when a spin tunneling event occurs, the nuclear Hamiltonian suddenly changes by a large amount, and the effect of the “ﬂuctuation” of the electron spin cannot be treated as a perturbation. In addition, since the electron spin tunneling events are allowed in the ﬁrst place by nuclear spin ﬂuctuations (at least in the small transverse ﬁeld limit), it is not even possible to treat the nanomagnet as an independent source of “ﬁeld jumps”. The dynamics of the nanomagnet and its nuclei must be treated together, since the one drives the other and vice versa: again, this is a very uncommon situation in NMR. The problem is treated formally in the spin bath theory by writing a master equation for the probability P⇑⇑ (t) for the giant spin to remain in the | ⇑ until time t [31]. For a nanomagnet alone, that would be trivially:

∆2 P⇑⇑ (t) = 1 − 20 sin2 ε t, ε ε = ξ 2 + ∆02 .

(3) (4)

When coupled to the nuclear spin bath, the tunneling of S has the effect of “coﬂipping” some of the nuclei. Given the details of the spin-bath coupling, one can calculate what is the “natural” number of nuclei that would ﬂip upon a tunneling event, by considering two effects. (i) If the hyperﬁne ﬁeld acting on a nuclear spin does not exactly invert its direction by 180o upon tunneling (which can happen if the nucleus is subject to dipolar ﬁelds from different nanomagnets, only one of which tunnels at some instant, or if an external transverse ﬁeld is applied), then the quantization axis of the nucleus changes, i.e. the spin precesses around a new axis. Quantum mechanically, this is equivalent to a ﬂip of the nuclear spin. This mecha-

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nism depends only on the direction and not on the timescale of the hyperﬁne ﬁeld jumps. (ii) Even if the hyperﬁne ﬁeld changes direction by exactly 180o, the nuclear spin may still follow if S ﬂips slowly. The nuclear coﬂipping probability is then proportional to (ωN /Ω0 )2 , where Ω0 is the “bounce frequency” of the nanomagnet, i.e. the frequency of the oscillations of S on the bottom of each potential well. However, since typically Ω0 ∼ 1012 s−1 , this mechanism is usually unimportant. Knowing the strength and direction of the individual hyperﬁne couplings, one can calculate the average number, λ , of nuclear spins that would coﬂip with S at each tunneling event. Given a certain arbitrary state of the nuclear spin bath at some instant, a tunneling event may require a number M of nuclei to coﬂip with S in order to conserve energy. If M λ , such an event is essentially forbidden (“orthogonality blocking” [31]). This is accounted for in the theory by writing an effective tunneling matrix element, ∆M ∆0 λ M/2 /M!, which has its maximum value for M = 0 (and coincides with the half-tunneling splitting of the nanomagnet alone), and goes quickly to zero for M λ . Thus P⇑⇑ (t) may take any of the possible values PM (t) obtained by replacing ∆M for ∆0 in Eq. (3). In addition, since PM (t) depends on the bias ξ (Eq. (4)), which has a hyperﬁne component ξN that can ﬂuctuate over a range E0 , one must also average PM (t; ξ ) over the bias distribution, obtaining: 1 1 = exp(−ΓMN t), 2 2 2∆ 2 ΓMN = √ M , π h¯ E0

PM (t; ξ )ξ −

(5) (6)

where ΓMN represents the rate for S to tunnel accompanied by the coﬂip of M nuclear spins. Calling ξ0 the energy scale associated with the ﬂip of λ nuclei (the “natural” number of coﬂips), and considering that the highest tunneling probability is obtained for M = 0, the leading term for the global tunneling rate becomes: 2∆ 2 Γ N √ 0 exp(−|ξB |/ξ0 ), π h¯ E0

(7)

where ξB = gµB Sz Btot is the static component of the bias. Btot here is the sum of the longitudinal components of an externally applied ﬁeld and of the dipolar ﬁeld from neighboring nanomagnets. Since the latter is itself time-dependent as soon as tunneling events start occurring, this leads to an interesting collective dynamics, characterized by a square-root time relaxation [33]. Starting from a fully magnetized sample, M (0) = Msat , the short-time behavior of the magnetization is: M (t)/Msat 1 − −1 τshort ∼

t/τshort ,

ξ0 N Γ (ξ = 0), ED

(8) (9)

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where ED is the spread of dipolar biases. We see therefore that the microscopic properties of the spin bath enter directly in the macroscopic relaxation of an ensemble of nanomagnets, through the strength of the hyperﬁne couplings (E0 , in Eq. (7)) and the coﬂipping probability (ξ0 , in Eqs. (7), (9)). From the opposite perspective, we have seen that to each tunneling event one can associate a certain nuclear coﬂipping probability. However, for the purpose of comparing theory and experiments, it is convenient to translate this into more common NMR language by calculating the nuclear longitudinal (T1 ) and transverse (T2 ) relaxation times. The latter is relatively easy to estimate because, unless the tunneling rate is made extremely high by applying a strong transverse ﬁeld, one will generally have T2−1 Γ N , i.e. S remains static on the timescale of the transverse nuclear relaxation. On such a short timescale we therefore recover the applicability of the standard perturbative treatments, whereby T2 is determined by the nuclear dipole-dipole couplings and can be calculated using the van Vleck method [44, 1]. Conversely, T1 is determined precisely by the tunneling rate of the nanomagnet, which determines how often the local ﬁeld on the nuclear spins changes direction. Quite amusingly, this problem was ﬁrst treated by Abragam (Ref. [1], p. 478) as “. . . an example that has no physical reality but where the result can be obtained very simply . . . ”; 40 years later, that example has found physical reality in quantum nanomagnets! The simple result is that, since the local ﬁeld changes direction at intervals ∼τ , then [1, 2] T1−1 ∼ 2Γ N .

(10)

Once again, the point to bear in mind here is that Γ N is itself strongly dependent on the nuclear spin dynamics. Moreover, one should be careful before calling this the nuclear spin-lattice relaxation rate, as T1−1 is normally interpreted. We shall come back to this issue in the review of the experimental results, but here we simply note that no phonon bath has been introduced so far, in the context of nuclear-spin mediated tunneling. A phonon-mediated tunneling rate, Γ φ , can also be calculated [19, 38], yielding:

Γφ

∆02Wφ (T ) , 2 ∆0 + ξ 2 + h¯ 2Wφ2 (T )

(11)

where Wφ is the linewidth of the electron spin states due to intra-well phononassisted transitions [21]. At low temperatures this tunneling rate is orders of magnitude smaller than the nuclear-driven one.

3 Experimental Results From the theoretical treatment of the coupled nanomagnet-nuclear spins system, we have obtained several testable predictions. First of all, it is predicted that an initially magnetized sample would relax in a non-exponential way, given by Eq. (8).

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This has been veriﬁed very early on by magnetization measurement on Fe8 at very low temperatures, which show T -independent relaxation below T ∼ 360 mK, and indeed, a square-root time relaxation function [46] (Fig. 3(a)). Second, but perhaps more important for the present discussion, the nuclear-driven tunneling rate Γ N should depend on the details of the hyperﬁne couplings, through the spread of hyperﬁne bias, E0 , and the energy scale associated with the “natural” number of nuclear spins that coﬂip with the nanomagnet upon each tunneling event, ξ0 . Both can be changed by isotopic substitution of some constituents. Iron-based nanomagnets, like Fe8 , are particularly suited for this type of study because the strength of the spin-bath couplings can be either increased by replacing 56 Fe (I = 0) with 57 Fe (I = 1/2), or decreased by replacing 1 H (I = 1/2) with 2 H (I = 1 but 6.5 times smaller gyromagnetic ratio). When repeating the low-T magnetization relaxation experiments on isotopically substituted samples, it was found that the relaxation rate would increase with 57 Fe enrichment, and decrease with 2 H substitution [48] (Fig. 3(b)). Notice that in both cases the mass of the molecule is increased, so that no phonon isotope effect could explain the observed trend. Finally, by measuring the magnetization decay while applying a special sequence of longitudinal ﬁelds, it was conﬁrmed that the ﬂuctuating hyperﬁne bias creates a “tunneling window” within which the nanomagnets can undergo quantum relaxation [46, 48, 42]. The width of this tunneling window was found to be dependent on the isotopic composition of the sample. Thus, the effect of nuclear spins on the quantum tunneling of an ensemble of nanomagnets has been veriﬁed very early on and in a rather uncontroversial way. The opposite effect, i.e. the inﬂuence of quantum tunneling of the nanomagnets on the nuclear spin dynamics, was observed short thereafter for 1 H [43] and 57 Fe in Fe8 [4], and 55 Mn in Mn12 [25, 27] (Fig. 3(c)). In each experiment, a T -independent nuclear T1 was found below a certain temperature, and both for Fe8 and Mn12 this was comparable to the temperature at which, for instance, the magnetic hysteresis loops also became T -independent [45, 6, 5]. This can be qualitatively understood from Eq. (10), which obviously implies that when the T -independent Γ N is the dominant electron spin ﬂuctuation rate, also T1−1 should be T -independent. At higher temperatures, thermally-assisted transitions in the nanomagnet start to play a role, and the situation becomes much more complicated. This has led to some controversy in the interpretation of the nuclear relaxation data in the thermally activated regime, on which we shall not dwell. Focusing instead on the low-T quantum regime, it’s worth noting that one could look for the other obvious signature of resonant tunneling, i.e. that it should be strongly suppressed by a longitudinal external ﬁeld. Indeed, it has been observed that the nuclear T1 in Mn12 at very low temperatures becomes almost two orders of magnitude slower with the application of a small longitudinal ﬁeld [25, 27], as compared to the zero-ﬁeld case (Fig. 3(d)). The interpretation of the experiments on Mn12 -ac is slightly complicated by the fact that this particular compound contains a minority of fast-relaxing molecules [40, 47] which remain dynamic down to the lowest temperatures, whereas the majority species have a negligibly small tunneling rate in zero ﬁeld. On the other hand, the fact that a large fraction of the sample can remain fully magnetized for long times (months), has

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allowed to measure the nuclear dynamics in Mn12-ac as a function of the sample magnetization, and revealed the effect of dipolar coupling between nuclear spins (i.e. the nuclear spin diffusion described by the term ∑k,k IkV˜k,k Ik [31, 34]). Since spin diffusion can occur only between nuclei subject to the same local magnetic ﬁeld, a demagnetized sample where half of the spin are “up”√and half of the spins are “down” should have slower spin diffusion by a factor 2, as compared to a fully magnetized sample. This has been indeed veriﬁed by measuring the transverse relaxation rate T2−1 [25, 27]. So far, we have discussed experiments that essentially conﬁrmed the predictions of the theory of the spin bath. Not surprisingly, the situation becomes more puzzling when looking at the thermal properties of quantum nanomagnets and spin bath, because the theory does not deal with them. Speciﬁc heat is obviously the main experimental tool to investigate thermal equilibrium (or lack thereof) in an ensemble of nanomagnets. It was found early on that the magnetic speciﬁc heat in Fe8 would be unmeasurable at very low temperatures and zero external ﬁeld, because the spin-lattice relaxation time of the electron spins becomes much longer that the typical timescale of the experiment (∼103 s at most). However, by applying a large transverse magnetic ﬁeld, the tunneling rate could be made large enough to recover the equilibrium magnetic speciﬁc heat [23, 24]. Importantly, the case of Mn12 -ac is different in that one could observe the contribution of the nuclear spins, which have a large speciﬁc heat at millikelvin temperatures. The magnetic speciﬁc heat of Mn12-ac reveals a hyperﬁne contribution which approaches the full equilibrium value when a transverse ﬁeld is applied, but remains at least partially visible even in zero external ﬁeld. This observation seems to imply that the nuclear spins in Mn12 ac ﬁnd a way to thermalize to the phonon bath even at very low-T , and even without large transverse ﬁelds. The deﬁnitive experimental proof of the thermal equilibrium in the nuclear spins of Mn12 -ac was then found by directly measuring the 55 Mn nuclear spin temperature by NMR techniques [25, 27]. This is a crucial observation because the nuclear spins have essentially no direct link to the phonon bath, therefore their thermal equilibration must proceed via the coupling to the electron spins and their subsequent spin-lattice relaxation process. However, we are dealing here with a temperature regime where the only electron spin transitions are quantum tunneling ones. The thermalization rate τth−1 is found to be orders of magnitude faster than expected from the known phonon-assisted tunneling rate (Eq. (11)), and is actually very close to the observed nuclear-spin mediated tunneling rate. The precise value appears to be a matter of sample size, cooling power and thermal contact to the refrigerator, and it’s quite plausible that τth−1 Γ N in the limit of small sample and perfect contact to the thermal bath [25]. If the nuclear spins are found to be in thermal equilibrium, one may argue that the electron spins should be in thermal equilibrium as well. Since the latter are mutually coupled by dipolar interactions of the order ξD ∼ 0.1 K, at very low-T we may expect the ensemble of nanomagnets to undergo a transition to a magnetically ordered state, provided the timescale involved in ﬁnding the collective ground state is short enough. Long-range magnetic ordering in molecular magnets was ﬁrst found in the Mn6 compound [26, 28], which has negligible anisotropy and therefore

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maintains a very fast electron spin-lattice relaxation time down to the lowest temperatures. The electron-spin transitions involved in ﬁnding the ordered state are somewhat trivial (no tunneling), but an additional important result was the conﬁrmation that the nuclear T1 measured by NMR precisely coincides with the nuclear spin-lattice relaxation time as measured by speciﬁc heat [28]. The ﬁrst quantum nanomagnet to show long-range magnetic ordering was a special type of Mn4 [8] (Fig. 3(e)), characterized by fast tunneling rate in zero ﬁeld. In this case, the collective ordered state is found through quantum relaxation, and the ferromagnetic phase transition causes a peak in the speciﬁc heat at the temperature where the nanomagnets undergo long-range ordering. Also in this system the nuclear spins were found to be in thermal equilibrium. That the nuclear spins should play a role in this process is clear from the discussions above, and it has been experimentally veriﬁed in Fe8 , again by playing with isotopic substitutions. While a “natural” Fe8 sample falls out of thermal equilibrium at low-T in zero ﬁeld, a sample enriched with 57 Fe shows a much larger magnetic speciﬁc heat that almost reaches the full equilibrium value [9] (Fig. 3(f)). At the lowest temperatures, the hyperﬁne contribution to the speciﬁc heat is also revealed. These results prove once again that the thermalization time, the electron spin tunneling rate and the nuclear T1 are closely related and essentially belong to the same physical phenomenon, i.e. the inelastic tunneling of the nanomagnet spin, driven by the dynamics of the nuclear spin bath, where the phonon bath acts as a thermostat for both electrons and nuclei.

4 Open Questions and Future Directions After discussing the experiments involving thermalization of the nuclei and electron spins, it should be clear that the missing ingredient in the present description of nanomagnet + spin bath is the role of the phonon bath in inﬂuencing the tunneling transition in the presence of a ﬂuctuating hyperﬁne bias. To make the point absolutely clear, let us ask the question: “How does a nanomagnet know what is its most energetically favorable spin direction at the instant when a tunneling event can occur, so that it can participate to a long-range ordered state?” In general, each nanomagnet is subject to some bias originating from the dipolar ﬁeld of its neighbors, plus the hyperﬁne bias from the nuclei. Only when the two compensate each other, can a tunneling event occur. At that moment, however, the total energy of the nanomagnet + nuclei is the same regardless of the direction of the nanomagnet spin. The difference between the two possible orientations is that one will have the electron spin in a favorable direction with respect to the local dipolar ﬁeld, but the nuclear spins pointing against the hyperﬁne ﬁeld, and vice versa for the other orientation. It seems that the creation of long-range order in the nanomagnets should go at the expenses of the nuclear energy, but we know this is not the case since both electron and nuclear spins are found to be in thermal equilibrium down to the lowest temperatures, and the equilibrium nuclear speciﬁc heat is well visible precisely when

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NMR

(e)

(c)

cM

√t 57Fe

(d)

T

T1-1

(f)

natural

57

Fe

cM

(b)

T

2H

1/T

Bz

na tur al

M (t)

T1-1

(a)

τshort-1

Specific heat hyp erfi ne ord erin g

Magnetization

T

Fig. 3 Sketches of the qualitative features of some spin-bath related phenomena in quantum nanomagnets. (a) Square-root time relaxation of magnetization in Fe8 [46], Eq. (8). (b) Isotope effect in the short-time magnetic relaxation rate in Fe8 , Eq. (9), and T -independent relaxation at lowT [48]. (c) Crossover to T -independent nuclear relaxation rate T1−1 driven by quantum tunneling in Fe8 [43, 4] and Mn12 -ac [27], Eq. (7). (d) Longitudinal ﬁeld suppression of quantum tunneling in Mn12 -ac as seen by nuclear T1−1 [25, 27]. (e) Long-range magnetic ordering in Mn4 and equilibrium hyperﬁne speciﬁc heat at low-T [8]. (f) Isotope effect in the magnetic speciﬁc heat of Fe8 [9]

long-range magnetic ordering is observed. Thus, the theory needs to be improved to include the role of phonons in this process [25]. One reason to stress this point is that the attention of the community is progressively being shifted towards coherent tunneling processes in quantum nanomagnets, for the purpose of quantum information processing. Then we shall be interested in modelling, and ultimately measuring, the decoherence rate of the electron spins under the most favorable conditions. In particular, it has been shown that when considering the effect of nuclear spins and phonons on the decoherence rate, one expects a “coherence window” where the coupling to the environment is the least destructive [39]. Crudely speaking, at low tunneling frequencies the nanomagnet will couple strongly to the nuclear spins, because their Larmor frequencies are comparable, whereas higher tunneling frequencies would increase the coupling to phonon modes, leaving an optimal low-coupling point between these regimes. For dense (i.e., undiluted) and crystalline ensembles of nanomagnets, the dipolar couplings are actually more important than the hyperﬁne ones, and the (strongly T -dependent) optimal operation point is determined by the crossing between dipolar and phonon decoherence rates [29]. However, in view of the above discussion on the shortcomings of the present description of tunneling in the presence of spin and phonon baths, one may wonder to what extent are these predictions accurate. But this is precisely the beauty and the strength of the research on quantum nanomagnets: theoretical predictions

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can be tested qualitatively and quantitatively against the experiment, as has been done in the past ten years to understand the incoherent tunneling regime. It should be possible to single out each contribution to decoherence by virtue of its dependence on temperature, ﬁeld, and isotopic substitution. While the experiments needed to demonstrate coherent control of quantum nanomagnets are extremely demanding, the precious information harvested by studying relaxation and dephasing in these systems has the potential to push our understanding of nanometer-sized quantum systems to unprecedented levels. Acknowledgements The author is deeply indebted with many colleagues and coworkers who have supported, stimulated or challenged his activity in this ﬁeld of research. In approximate chronological order: F. Luis, L. J. de Jongh, R. Sessoli, H. B. Brom, G. Arom´ı, P. C. E. Stamp, I. S. Tupitsyn, B. V. Fine, W. Wernsdorfer, M. Evangelisti, W. N. Hardy, Z. Salman, R. F. Kieﬂ, J. C. Baglo, A. L. Burin.

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Quantum Dimer Models and Exotic Orders K.S. Raman, E. Fradkin, R. Moessner, S. Papanikolaou, and S.L. Sondhi

Abstract We discuss how quantum dimer models may be used to provide “proofs of principle” for the existence of exotic magnetic phases in quantum spin systems. The material presented here is an overview of some of the results of Refs. [8] and [9].

1 Introduction Consider a system of quantum spins on a lattice with antiferromagnetic interactions. The energy of a given pair of spins is minimized by forming a singlet but as a spin can form a singlet with only one other spin, the system is unable to simultaneously optimize all of its interactions. In this sense, the system is said to be frustrated. The most common solution to this problem is for the spins not to form singlets but instead to magnetically order, as in the classical Neel state. However, one may also

K.S. Raman Department of Physics and Astronomy, University of California at Riverside, Riverside, CA 92521, e-mail: [email protected] E. Fradkin Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, e-mail: [email protected] R. Moessner Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford, OX1 3NP, UK, e-mail: [email protected] S. Papanikolaou Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, e-mail: [email protected] S.L. Sondhi PCTP and Department of Physics, Princeton University, Princeton, NJ 08544, e-mail: [email protected]

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envision non-magnetic states where the order is contained in how the singlet pairs organize. In this paper, we concentrate on phases where the singlets form between nearest-neighbor spins. Such phases are relevant in systems having a spin gap. A nearest-neighbor singlet, also called a short-range valence bond, may be represented by drawing a dimer across the corresponding link of the lattice. Arrangements of singlet pairs correspond to dimer coverings of the lattice, where the hardcore condition of one dimer per site captures the frustration. This picture may be abstracted to a low energy description where the fundamental degrees of freedom are the dimers themselves. Quantum dimer models [1, 2] describe how exotic phase diagrams can arise from the competition between quantum ﬂuctuations of the dimers and various potential energies and are, perhaps, the simplest models containing the physics of frustration. The suggestion that microscopic frustration may lead to nontrivial phases such as incommensurate crystals [3] or liquids of resonating singlets [4] is a rather old one. The topic is of current interest due to proposals that such phases may be relevant to understanding high Tc superconductors and other strongly correlated systems [5, 6, 7]. In this context, we may ask how a particular exotic phase can, in principle, arise from a microscopic Hamiltonian that is local and does not break any lattice symmetries. One way to answer such questions of principle is to construct an explicit model having the phase in question. Quantum dimer models are well suited for this purpose because of the relative simplicity of the dimer Hilbert space. This leads to the question of how these effective models can arise from models with more physical degrees of freedom, such as SU(2) invariant spin models. In Section 2, we review some facts about the simplest dimer models. In Section 3, we outline the construction of a dimer model that contains a devil’s staircase of crystalline phases, including states with arbitrarily long (and hence incommensurate) periods. In Section 4, we discuss how dimer models may be mapped onto SU(2) invariant spin models on decorated lattices. In particular, the existence of a liquid phase of resonating dimers in the simplest triangular lattice dimer model implies the existence of a liquid phase of resonating singlets on a decorated triangular lattice. This paper gives an overview of some of the results in Refs. [8] and [9].

2 Basic Models The Hilbert space of quantum dimer models is deﬁned by taking each dimer covering of the lattice as a basis vector. The inner product is usually deﬁned by taking different dimer coverings to be orthogonal. The dimer Hilbert space may be partitioned into topological sectors labeled by a pair of winding numbers (Fig. 1). The winding numbers are global invariants in that they are not affected by local rearrangements of dimers. In particular, local operators (including the Hamiltonian) will not have matrix elements between states in different sectors. For bipartite lattices, the number of sectors is extensive while for non-bipartite lattices, there are four sectors corresponding to each winding number being either even or odd.

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Fig. 1 Winding numbers for the square lattice case. The thick horizontal line extends around the lattice. If the vertical lines of lattice are labeled alternately A and B, then the winding number is deﬁned as NA − NB where NA,B is the number of dimers crossing the thick line along A,B lines. For the section of the lattice shown, NA = 3 and NB = 0. One may verify that local dimer rearrangements will not change this number. In the triangular lattice case, the relevant quantity is deﬁned by just counting the number of dimers intersecting this reference line. Whether this number is odd or even is a topological invariant but the value of the number can change with local rearrangements of the dimers

The original quantum dimer model of Rokhsar and Kivelson [2] was deﬁned on the square lattice with the Hamiltonian: (1) The two terms are projection operators that ﬂip (the t term) or count (the v term) the dimers if the plaquette in question has parallel dimers, which we call a “ﬂippable” plaquette, and annihilate the state otherwise. The sum is over all plaquettes in the lattice. The model may be generalized to other lattices where the ﬂippable plaquettes now correspond to the dimers occupying alternate bonds of the minimal (even-length) resonance loop in the lattice. For the triangular lattice [10], these loops are still length 4 but there are now six distinct ways of having parallel dimers instead of just two for the square lattice. For the honeycomb lattice, the loops are length six and the projection operators are three-dimer moves analogous to the benzene resonance. For the pentagonal lattice [8], the loops are length eight, the operators are four-dimer moves and so on. Models of the Rokhsar-Kivelson type have certain generic properties. When v > t, Eq. (1) is positive deﬁnite so the system seeks to minimize the number of ﬂippable plaquettes. In particular, many lattices, including the ones mentioned above, may be covered without having any ﬂippable plaquettes and such “staggered” states will be zero energy ground states in this limit. When −v t, the system seeks to maximize the number of ﬂippable plaquettes so selects an analog of the “columnar” state. When v = t, Eq. (1) is again positive deﬁnite, but in addition to the staggered states, there are a number of liquid states which also have zero energy [2]. The liquid wavefunctions may be written as equal amplitude superpositions |ψ = ∑c |c where the sum is over all states that may be connected by repeatedly applying the ﬂip (kinetic) term in Eq. (1). There will be at least one such state in every topological sector. For the square lattice, there is only one state in each sector because the ﬂip

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term is believed to be ergodic within the sectors but additional subtleties may appear in other lattices [10]. Because these are zero energy states, any combination of them will be equally valid as a ground state, including the equal amplitude superposition of all dimer coverings. The liquid states at v = t, which is called the RK point, have no local order but they do carry the global quantum numbers of their topological sectors. However, the nature of the liquid states depends intimately on the lattice geometry. For some bipartite lattices, including the square and honeycomb, these states have algebraically decaying correlations, are gapless, and the RK point is a critical point separating crystalline phases with minimal and maximal winding number. Field theoretic studies [11, 12] suggest that perturbations about this critical point can stabilize crystalline phases with intermediate winding number and, generically, a devil’s staircase of such states will occur in between the states of minimal and maximal winding. In particular, it was noted that incommensurate crystals [13] could be obtained by suitable tuning of the perturbation. For some non-bipartite lattices, including the triangular and kagome, the correlations decay exponentially, the liquid states are gapped, and the RK point is the edge of a liquid phase [10]. This was the ﬁrst example of a model containing a liquid phase of resonating bonds and validated earlier suggestions of a physical mechanism by which such a phase could be stabilized, namely geometric frustration enhanced by strong ring exchange [14]. For the kagome lattice, a simpler dimer liquid phase, with exactly zero correlation beyond one lattice spacing, was obtained in Ref. [15] using a different construction with only ﬂip terms. In the next section, we discuss how a different kind of exotic structure can be realized in a dimer model. Then, in Section 4, we show how these proofs of principle in dimer models may be extended to SU(2)-invariant spin systems.

3 Modulated States and the Devil’s Staircase We now outline a dimer model that shows a devil’s staircase of crystalline phases where the period becomes arbitrarily large [9]. The following construction is valid in the limit of strong coupling and weak (quantum) ﬂuctuations, in contrast to the RK point which corresponds to the opposite limit of strong ﬂuctuations. In this sense, the connection between the following devil’s staircase and the one predicted by ﬁeld theory to occur near bipartite RK points (mentioned above) is tempting but speculative. Similarly, we work on the square lattice for convenience but we do not believe this choice is so important in the strong coupling regime. Our strategy is to perturb a model that contains a degenerate point separating crystalline states of different winding number. In particular, we consider the following diagonal Hamiltonian:

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(2)

where the dots denote terms related by lattice symmetries to ones shown, which will appear with the same coefﬁcient. Note that Eq. (2) is local and does not break lattice symmetries. The coefﬁcients satisfy p, q > c, d > a, b > 0 but ﬁne tuning is not required. Terms a and b are competing attractive interactions while the remaining terms are repulsive. This Hamiltonian is designed to favor states where every dimer participates in only two attractive bonds of the same type. Terms c and d penalize arrangements with dimers participating in more than two attractive bonds [16]. Terms p and q serve a technical purpose discussed in Ref. [9]. The quantum system described by the Hamiltonian in Eq. (2) has the zero temperature phase diagram shown in Fig. 2. As the ﬁgure shows, the system prefers exclusively a or b bonds depending on which coefﬁcient is larger. When a = b, there is a large degeneracy where the preferred states involve thin staggered domains separating columnar regions of opposite orientation; the staggered domains come in two orientations as illustrated in the ﬁgure. We perturb this model with a resonance term that is equivalent to two actions of the ﬂip term in Eq. (1):

Fig. 2 Ground state phase diagram of the parent Hamiltonian H0 as a function of the parameter a− b (see Eq. (2)). When a− b = 0, there are a number of degenerate ground states. The maximally staggered conﬁguration is commonly called the “herringbone” state

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(3) where 0 < t a, b. We now describe how the existence of the devil’s staircase follows by perturbing in t at successively higher orders. Before doing this, we remark that the main reason for choosing a somewhat complicated form for Eq. (2) and a non-standard resonance term (i.e. Eq. (3)) is that these choices simplify the calculational details. As discussed in Ref. [9], we anticipate that this construction can be made to work for simpler diagonal Hamiltonians and with the familiar two-dimer resonance. Referring to Fig. 2, we see that the operator 3 does not affect the phases on either side of the degenerate point but will partially lift the degeneracy at a = b. In particular, this term causes the staggered domain walls to ﬂuctuate which stabilizes them at second order in perturbation theory. In this sense, our model is a quantum analog of the Pokrovsky-Talapov model of ﬂuctuating domain walls [17]. In that classical model, thermal ﬂuctuations give entropy to the domain walls and the competition between this and their energy cost induces a commensurateincommensurate transition [3] to a striped phase where the walls arrange periodically with spacing that can be large compared to the interaction scale. In the present case, quantum ﬂuctuations play the role of temperature and cause modulated phases to appear. A technical issue is that the extent to which a staggered domain is stabilized by second order processes depends on its environment. One consequence of this is the favoring of states, such as the [1n] sequence (Fig. 3), where the staggered domains have the same orientation. Also, while the [11] state clearly has the maximum number of resonances, these resonances are individually weaker than those in the [12] state because of the higher energy virtual states that are involved (due to term c in Eq. (2)). If the [11] phase is selected when a = b, then by slightly detuning from this point (by an amount of order t 2 ), we will reach a value of b where the “self-energy” of a columnar line [18] becomes degenerate with the energy of a resonance-stabilized staggered line in the [11] state. However, b must be further increased before the advantage of [12] staggered lines is similarly compensated. Therefore, both the [11] and [12] phases appear in regions of width ∼t 2 between the initial phases and to second order in t, we obtain the phase diagram in Fig. 5(a). Note that the winding number increases as we move from left to right in the phase diagram.

(a)

(b)

(c)

Fig. 3 The [11], [12], and [13] states involve staggered domains separated by one, two, and three columnar units respectively. In these states, the staggered domains have the same orientation. We can similarly deﬁne [14], [15] etc.

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On the [11]–[12] and [12]-columnar boundaries, the phases and states with intermediate winding are degenerate to order t 2 but these degeneracies will be lifted at higher orders. Considering the [12]-columnar boundary at fourth order in perturbation theory, the relevant resonances are shown in Fig. 4. Process (a) destabilizes the [12] state due to the high energy virtual state involving term q in Eq. (2) (there is an analogous process involving term p) and its contribution is larger than process (b) which is stabilizing. In the [13] state, the analog of process (a) will not contribute because the ﬂipped clusters are disconnected which means that the energy and wavefunction terms in the perturbation series exactly cancel. However, process (b) will occur and stabilize a [13] phase in a region of width ∼t 4 between the [12] and columnar states. The argument may be applied inductively. Along the [1n]-columnar boundary, at (2n)th order in the perturbation, the analog of Fig. 4(a) will destabilize the [1n] state relative to states of lesser winding while the analog of Fig. 4(b) selects the [1, n+1] state from this set. The new [1, n+1] phase will occupy a region of width ∼ t 2n in the phase diagram and so on. While higher orders in the perturbation theory involve increasingly complicated resonances, including ﬂuctuations causing the staggered lines to effectively “break apart”, the calculation is designed so that these terms amount to self-energy corrections that simply move the boundaries. The stabilization of phases in the [1n] sequence will always be governed by the analogs of the straight resonances in Fig. 4. The [11–12] boundary can similarly open at higher orders and in Ref. [9], we verify that a [11–12] phase (the notation means that the repeating unit is a staggered domain followed by one columnar unit followed by another staggered domain followed by two columnar units) is stabilized at sixth order between the [11] and

Fig. 4 The fourth order resonances driving the transition. We emphasize that these ﬁgures represent terms that appear in fourth order perturbation theory, not additional terms added to the Hamiltonian. The circled cluster in (a) refers to term q in Eq. (2)

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t [12]

t

[11] [14]

columnar

b-a

herringbone

highly degenerate

(a)

[13]

[12]

[11]

columnar

b-a

herringbone

highly degenerate

(b)

Fig. 5 (a) Ground state phase diagram of H = H0 +tV from second order perturbation theory. The width of the [11] and [12] phases are order t 2 . (b) Sketch of the ground state phase diagram to all orders in perturbation theory. While the opening of the [11–12] boundary is explicitly indicated, the other boundaries will also open. The collection of phases forms a devil’s staircase. Note: These ﬁgures are not the result of a simulation but a sketch of the general properties of the phase diagram

[12] states. The ﬁner boundaries can also open and we have also veriﬁed that a [11 – (12)2 ] phase occurs at eighth order between the [11–12] and [12] phases. We are less certain about this ﬁne structure because the resonances involved are more complicated than in the primary [1n] sequence. However, we may speculate that the opening of boundaries will continue to ﬁner scales, at least for some range of parameters. In this sense, the phase diagram will most generically be described by an incomplete devil’s staircase, as shown in Fig. 5(b). Moving from left to right in Fig. 5, the system passes through a series of crystalline states with increasing winding number via ﬁrst-order phase transitions. We note that this phase diagram and the structure of the calculation is similar to the 3D classical ANNNI model [19, 20, 21].

4 SU(2)-Invariant Realizations Having discussed some examples of how exotic phases can arise in dimer models, we now turn to the question of what this implies for systems with more realistic degrees of freedom. The interpretation of a dimer as a nearest-neighbor singlet or valence bond involves two simpliﬁcations. The dimers have no orientation and different dimer coverings are orthogonal by deﬁnition. In contrast, valence bonds are oriented [22] and different valence bond coverings are not orthogonal. There is also the issue of projecting the spin Hilbert space onto the much smaller dimer Hilbert space. In this section, we discuss one way in which these issues may be resolved allowing the dimer model phase diagrams to be realized in SU(2)-invariant systems. We concentrate on the square lattice but the arguments may be generalized to

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Fig. 6 Decorated square lattice for the case N = 4

other lattices and higher dimensions. The interested reader may consult Ref. [8] for details. We would also like to point out a complementary approach by Fujimoto [23]. We construct the spin model on a modiﬁed square lattice where the links are decorated with an even number N of additional sites (Fig. 6). On each lattice site i, we deﬁne an operator hˆ i that projects the cluster of that spin and its (two or four) neighbors onto its highest spin state [24, 25, 26] (either 3/2 or 5/2). The parent Hamiltonian is a sum of these operators, one for each lattice site: H0 = ∑ αi hˆ i

(4)

i

where αi is a positive constant that, in principle, may vary with i. The SU(2)invariance of Eq. (4) may be seen by explicitly writing down the operators. Referring to the ﬁgure, hˆ a1 = S2 − ( 12 )( 32 ) where S = Sa1 + S1 + Se and likewise for the other link sites. Similarly, hˆ 1 = [S2 − ( 12 )( 32 )][S2 − ( 32 )( 52 )] where S = S1 + Sa1 + Sb1 + Sc1 + Sd1 and likewise for the other corner sites. As these expressions indicate, hˆ i is a product of terms that annihilate the lower spin sectors so only the highest spin state remains. Among the conﬁgurations annihilated by hˆ i is the case where two spins in cluster i form a singlet. This is because the highest spin state must be symmetric under interchange. A consequence of all this is that any conﬁguration where every spin is in a singlet with one of its neighbors, which we call a valence bond state, will be annihilated by Eq. (4). Because H0 is a sum of positive deﬁnite operators, its eigenvalues are positive so such annihilated states will be zero energy ground states. Even a minimal decoration of N = 2 ensures that the valence bond states (and their superpositions) are the only zero energy states. In the valence bond states, the chains forming the links of the decorated square lattice are in one of two possible dimerizations. One of the dimerizations involves

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the sites of the original square lattice while the other involves only the decorated sites. In the original square lattice, we may represent the former case by drawing a dimer across the link and the latter by an empty link. Therefore, the valence bond states correspond exactly with dimer coverings of the lattice but an important technical difference is the non-orthogonality of the valence bond basis. However, the decoration ensures that the magnitude of the overlap between different valence bond states will always be exponentially small in the decoration N. This allows us to treat the overlap in an expansion that becomes asymptotically exact for large enough N. In contrast to other large-N approaches [27], this procedure occurs entirely within the class of SU(2) models. The scale of the spin gap is determined by the minimum of the {αi }. If this scale is sufﬁciently large, then perturbations of Eq. (4) will generate effective operators in the valence bond manifold. In particular, consider the perturbation: δ H = J ∑ si · s j + v ∑ (s1 · sb1 )(s2 · sb2 ) + (s1 · sa1 )(s3 · sa3 ) (5) i j

where the ﬁrst sum is over nearest neighbors and the second is over square plaquettes (the symmetric terms are not explicitly written). The degenerate perturbation theory involves accounting for the non-orthogonality of the valence bond manifold. In particular, if Si j = i| j is the overlap matrix, then we may consider the orthog−1/2 onal basis |α = ∑i Sα i |i. Because the overlap between different states is small, we may label |α by its order unity component. In terms of this basis, the operator Eq. (5) becomes: Hαβ = (S−1/2δ HS−1/2)αβ =

(6)

∑(S−1/2)α i i|δ H| j(S−1/2) jβ ij

= −Jx4(N+1) αβ + vn f l,α δαβ + O(vx4(N+1) + Jx6(N+1) ) = −tαβ + vn f l,α δαβ + O(vx4(N+1) + tx2N ) (7) where x = √12 . i j is a matrix that is 1 if states |i and | j differ by the (minimal) length 4 loop and zero otherwise; n f l,i counts the number of ﬂippable plaquettes in state |i; and t = −Jx4(N+1) . Therefore, up to small corrections, Eq. (5) acts like the RK quantum dimer Hamiltonian in the orthogonalized basis and by decorating the lattice with a sufﬁcient number of sites, the matrix elements beyond the dimer model can be made arbitrarily small. While this procedure will not capture the ﬁne-tuned aspects of the dimer model, such as the v = t critical point for bipartite lattices, without taking N extremely large, the various gapped phases that appear can be obtained for a ﬁnite decoration. In particular, we may repeat this procedure on a decorated triangular lattice to obtain a model with a stable, SU(2)-invariant spin liquid phase. A similar

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construction may be used to realize the modulated phases discussed in the previous section.

5 Conclusion We have shown that incommensurate structures and spin liquids can, in principle, arise from purely local Hamiltonians that do not break any symmetries The usefulness of quantum dimer models in answering this kind of question was hopefully conveyed. Along with the proofs of principle comes an understanding regarding physical mechanisms by which such structures may form. In the spin liquid ground states, the key elements were geometric frustration and strong quantum ﬂuctuations through ring exchange. In the devil’s staircase construction, the key ingredients were the ﬂuctuating domain wall picture and competing interactions in a strong coupling limit. While the constructions involved rather complicated lattices and/or Hamiltonians, it is likely that the ideas apply for simpler, though perhaps less (analytically) tractable, models where frustration enters in a less rigid way than a hard-core dimer constraint. For example, we may consider models with longer bonds [28] or doped dimer models [29]. Therefore, we may hope the ideas presented here will provide a starting point for constructing physically realistic models with exotic phases. Acknowledgements This review is based on a contributed talk by KSR at the PITP-Les Houches School for Quantum Magnetism 2006. KSR would like to thank the organizers for the opportunity to attend. This work was supported by the National Science Foundation through the grants NSF DMR0213706 and NSF DMR 0442537.

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22.

23. 24. 25. 26. 27. 28. 29.

Imaging Transverse Electron Focusing in Semiconducting Heterostructures with Spin-Orbit Coupling Andr´es A. Reynoso, Gonzalo Usaj, and C.A. Balseiro

Abstract Transverse electron focusing in two-dimensional electron gases (2DEGs) with strong spin-orbit coupling is revisited. The transverse focusing is related to the transmission between two contacts at the edge of a 2DEG when a perpendicular magnetic ﬁeld is applied. Scanning probe microscopy imaging techniques can be used to study the electron ﬂow in these systems. Using numerical techniques we simulate the images that could be obtained in such experiments. We show that hybrid edge states can be imaged and that the outgoing ﬂux can be polarized if the microscope tip probe is placed in speciﬁc positions.

1 Introduction During the last decade, a tremendous amount of work has been devoted to manipulate and control the spin degree of freedom of the charge carriers [1]. It was quickly recognized that the spin-orbit (SO) interaction may be a useful tool to achieve this goal. This is due to the fact that the SO coupling links currents, spins and external ﬁelds. Using intrinsic material properties to control the carrier’s spin would allow one to build spintronic devices without the complication of integrating different materials in the same circuit [1]. The challenging task of building spin devices based purely on semiconducting technology, requires one to inject, control and detect spin Andr´es A. Reynoso Instituto Balseiro and Centro At´omico Bariloche, Comisi´on Nacional de Energ´ıa At´omica, 8400 San Carlos de Bariloche, Argentina, e-mail: [email protected] G. Usaj Instituto Balseiro and Centro At´omico Bariloche, Comisi´on Nacional de Energ´ıa At´omica, 8400 San Carlos de Bariloche, Argentina C.A. Balseiro Instituto Balseiro and Centro At´omico Bariloche, Comisi´on Nacional de Energ´ıa At´omica, 8400 San Carlos de Bariloche, Argentina

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polarized currents. During the last years a number of theoretical and experimental papers were devoted to the study of the effect of SO coupling on the electronic, magnetic and magnetotransport properties of 2DEGs (see [2] and references therein). The nature of the SO coupling in these systems is due to the Dresselhauss and the Rashba mechanisms, the latter being the dominant effect in several cases [3]. In addition, the Rashba coupling has the advantage that its strength can be changed when a gate voltage is applied to the heterostructure, opening new alternatives for device design [4]. In many transport experiments in 2DEG with a transverse magnetic ﬁeld, including quantum Hall effect and transverse magnetic focusing, the SO coupling plays a central role. The transverse focusing consists basically in injecting carriers at the edge of a 2DEG and collecting them at a distance L from the injection point. The propagation from the injector I to the detector D is ballistic and the carriers can be focalized onto the detector by means of an external magnetic ﬁeld perpendicular to the 2DEG. The ﬁeld dependence of the focusing signal is essentially given by the transmission from I to D. In a semiclassical picture, the trajectories that dominate the focusing signal are semicircles whose radius can be tuned with the external ﬁeld. The new scanning technologies developed in [5, 6] can be used to map these trajectories. The scanning probe imaging techniques consist in perturbing the system with the tip of a scanning microscope and plotting the transmission as a function of the tip position. The transmission change is a map of the electron ﬂow. In this paper we ﬁrst revisit the theory of transverse electron focusing in systems with strong SO coupling and interpret the results in terms of a simple semiclassical picture [7]. Then, we use numerical techniques to simulate the images that could be obtained with scanning probe microscopy experiments. We show that hybrid edge states can be visualized and that the outgoing ﬂux can be polarized if the microscope tip probe is placed in speciﬁc positions.

2 Transverse Electron Focusing in Presence of Strong Spin-Orbit Coupling The Hamiltonian of a 2DEG with Rashba spin-orbit coupling is given by H=

1 1 α (P2 + Py2) + (Py σx − Px σy ) − g µB σz Bz + V (x) 2m∗ x 2 h¯

(1)

here Pη = pη +(e/c)Aη with pη and Aη being the η -component of the momentum and vector potential respectively, α is the Rashba coupling parameter, g is the effective g-factor, {ση } are the Pauli matrices and V (x) describes the potential at the edge of the sample. In what follows, we use a hard wall potential: V (x) = 0 for x ≥ 0 and inﬁnite otherwise. For convenience we choose the vector potential in the Landau gauge A = (0, xBz , 0).

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Far from the sample edge (x 0) the eigenvalues and eigenfunctions of Hamiltonian (1) are well known [8]. The SO coupling breaks the spin degeneracy of the Landau levels. The spectrum is given by En± = h¯ ωc n ∓

E02 +

2 α 2n, lc

(2) 1

where n ≥ 1, ωc = e |B| /m∗ c is the cyclotron frequency, lc = (¯h/mωc ) 2 is the magnetic length, and E0 = h¯ ωc /2 − gµB Bz /2 is the energy of the ground multiplet corresponding to n = 0. The eigenfunctions for n ≥ 1, written as spinors in the z-direction, are [9]

1 φn−1 (x − x0) + iky Ψn,k (x, y) = e (3) −Dn φn (x − x0) An Ly and − Ψn,k (x, y) =

1 eiky An Ly

Dn φn−1 (x − x0) , φn (x − x0 )

(4)

where Ly is the length of the sample in the y-direction, φn (x − x0 ) is the n-th harmonic oscillator wavefunction centered at the coordinate x0 = lc2 k , An = 1 + D2n and √ (α /lc ) 2n . (5) Dn = E0 + E02 + (α /lc )2 2n − The wave functions of the ﬁrst Landau level are given by Ψn,k (x, y) with n = 0. These eigenstates have a cyclotron radius given by

rc2 = 2 Ψn± |(x − x0 )2 |Ψn± ,

(6)

that for large n gives rc2 2n(¯h/m∗ ωc ). We see from Eq. (2) that states with different n, and consequently different cyclotron radius, coexist within the same energy window. Additionally, in the limit of strong Rashba coupling or large n, Dn ∼ 1 and the spin lies in the plane of the 2DEG. Equivalent results are found in a semiclassical treatment of the problem [11, 10]. In this approach, the spin is described by a vector [11] S = h¯ /2(n1 (t), n2 (t), n3 (t)) and the classical orbits are given by q = r± (cos ω±t, sin ω±t) h¯ S = sign(Bz ) (∓ cos ω±t, ∓ sin ω±t, 0), 2 here q is the coordinate measured from the centre of the circular orbit of radius

(7)

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r± =

α h¯ ωc

2 +

2E m∗ ωc 2

±

α , h¯ ωc

(8)

and the corresponding cyclotron frequencies are

ω± = sign(Bz )(ωc ∓ α /¯hr± ).

(9)

In agreement with the quantum results obtained for large n, the spin is found to be in-plane pointing outwards for the smaller orbit and inwards for the bigger one when a positive perpendicular magnetic ﬁeld Bz is applied. Moreover, the BornSommerfeld quantization [12] of these periodic orbits reproduces the exact quantum results of Eq. (2) for large n. The calculation of the exact edge states with the hard wall potential requires a numerical approach. We have shown that the semiclassical approximation can be extended to describe edge states in which electrons bounce at the sample edge [10]. Due to the continuity of the wave function and the spin conservation at the edge, the two orbits with radii r+ and r− are mixed as schematically shown in Fig.1. The agreement between the Born-Sommerfeld quantization of the semiclassical edge states and the quantum results is excellent for states composed of semicircles centered in the edge (normal incidence). In what follows, we use these semiclassical orbits to interpret the numerical results for transverse focusing experiments. The transverse focusing experiments collect electrons or holes coming from a point contact [13, 14] into another point contact acting as a voltage probe. The carriers are focused onto the collector by the action of an external magnetic ﬁeld as schematically shown in Fig. 1. The signal measured in transverse focusing experiments is related to the transmission T between the two point contacts located at a distance L from each other (see Fig. 1). Typical experimental setups also include two ohmic contacts at the bulk of the 2DEG which are used to inject currents and measure voltages. The details of different conﬁgurations with four contacts have been analyzed in [15]. The main features of the magnetic ﬁeld dependence of the focusing peaks are contained in T [16]. Consequently, from hereon we will refer to the focusing signal or to T indistinctly. In the zero temperature limit we only need to evaluate T at the Fermi energy EF . For the numerical calculation of T the system was discretized using a tight-binding model in which the leads or contacts are easily attached. In this approach the Hamiltonian is given by H = H0 + HSO with H0 = ∑ εσ c†nσ cnσ − n,σ

∑

,σ

tnm c†nσ cmσ + h.c..

(10)

Here c†nσ creates an electron at site n with spin σ (↑ or ↓ in the z direction) and energy εσ = 4t − σ gµBBz /2, t = − h¯ 2 /2m∗ a20 and a0 is the lattice parameter which is always chosen small compared to the Fermi wavelength. The summation is made x + ny y where x and y are unit vectors on a square lattice, the coordinate of site n is nx in the x and y directions, respectively. The hard-wall potential V (x) is introduced by taking nx > 0. The hopping matrix element tnm connects nearest neighbors only and includes the effect of the diamagnetic coupling through the Peierls substitution

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155

O2

(a)

(b)

O1

x y

D

I O2 O1

D

I O1

D

(c)

O2

(d) I

Fig. 1 Panel (a) Transmission coefﬁcient between the contacts I and D as a function of the applied perpendicular magnetic ﬁeld in the presence of strong Rashba spin-orbit coupling (qualitative). Relevant semiclassical orbits for three different focusing conditions are shown in panels (b), (c) and (d)

[17]. For the choice of the Landau gauge tn(n+y) = t exp (inx 2πφ /φ0 ) and tn(n+x) = t, φ = a20 Bz is the magnetic ﬂux per plaquete and φ0 = hc/e is the ﬂux quantum. The second term of the Hamiltonian describes the spin-orbit coupling, HSO = ∑ λx c†n↑ c(n+x)↓ − λx∗ c†n↓ c(n+x)↑ n

+ einx 2πφ /φ0 λy c†n↑ c(n+y)↓ − λy∗ c†n↓ c(n+y)↑ + h.c.

(11)

where λx = α /2a0 and λy = −iα /2a0 . In what follows we use the following values for the microscopic parameters: a0 = 5 nm, m∗ = 0.055 m0—here m0 is the free electron mass—and EF = 23 meV. These parameters correspond to InAs based heterostructures with a moderate doping. We use different values of the SO coupling parameter α as indicated in each case. The two lateral contacts, I (injector) and D (detector) are attached to the semiinﬁnite 2DEG described by Hamiltonian (11). Each contact is an ideal (with α = 0) narrow stripe of width N0 a0 . They represent point contacts gated to have a single active channel with a conductance 2e2 /h, for details see [7].

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To obtain the transmission between the two contacts we calculate the Green functions between the sites of the injector and the sites of the detector. As the spin is not conserved, the Green function between two sites i and j has four components Giσ , jσ . First the propagators of the system without the contacts are obtained by Fourier transforming in the y-direction and generating a continuous fraction for each k. Having these propagators, the self energies due to the contacts can be easily included using the Dyson equation [17]. The transmission is then obtained as T=

e2 (2) R (1) A Tr Γ G Γ G h ω = EF

(12)

here G R and G A are the retarded and advanced Green function matrices with elements GiRσ , jσ and GiAσ , jσ . The matrices Γ (l) are given by the self-energy due to

contact l, Γ (l) = i[ΣlR − ΣlA ] where ΣlR and ΣlA are the self-energies matrices of the retarded and advanced propagators respectively. Note that the deﬁnition of T includes the spin index. A typical T vs. Bz signal for strong spin-orbit coupling is shown in Fig. 1(a). A splitting of the ﬁrst focusing peak is clearly observed [7]. Notably, there is no splitting of the second peak. These results can be easily interpreted in terms of the semiclassical picture given above. From all the semiclassical orbits that connect the I and D contacts, the ones that give the largest contribution to T are those with 2r± = L [7, 15]. When the applied magnetic ﬁeld Bz is increased the cyclotron radii are reduced as B−1 z and the ﬁrst maximum in the transmission is found when r− (Bz ) = L/2 as schematically shown in Fig. 1(b). There O1 is the electron path between I and D, this path is a semicircle of radius r− . For this ﬁeld, indicated as Bz = B1,1 , the electrons that ﬂow out of the injector in the O2 orbit do not arrive to the detector since r+ (B1,1 ) > L/2. Furthermore, the two orbits O1 and O2 correspond to electrons injected with spin down or up in the y-direction, respectively. Note that due to the SO coupling, the spin rotates along the orbit. It is convenient to split the total transmission in the four contributions Tαβ corresponding to electrons injected with spin α and collected with spin β . The total transmission can be put as T = Tuu + Tud + Tdu + Tdd and for Bz = B1,1 the total transmittance is dominated by the contribution Tdu . When Bz is increased over B1,1 , r− (B1,1 ) < L/2 and T decreases. The next maximum is reached for Bz = B1,2 when r+ (B1,2 )= L/2 and the relevant orbit is O2 as shown in Fig. 1(c). For this focusing ﬁeld the transmission is dominated by Tud . The next maximum in T is found when Bz = B2 and corresponds to the situation shown in Fig.1(d). This focusing condition is due to the semiclassical trajectories with one intermediate bounce at the edge of the sample. In this case the two possible paths O1 and O2 contribute to the transmission. Electrons leaving the injector with a given spin arrive at the detector with the same spin projection. Accordingly, the total coefﬁcient T is dominated by Tuu +Tdd . Clearly, B2 is the magnetic ﬁeld for which 2(r− +r+ ) = L holds. In agreement with the exact numerical result, by

Imaging Transverse Electron Focusing in Semiconducting Heterostructures

B1,1 B1,2

157

B2

(c)

1,2 (a)

T

T

0,6 0,0

(b)

Tuu Tud

T

0,6 0,3 0,0

200

Bz[mT]

Tdu Tdd

300

Fig. 2 (a) Total focusing transmission coefﬁcient T versus applied perpendicular magnetic ﬁeld Bz . (b) Spin resolved transmission coefﬁcients versus Bz . We used EF = 23 meV, m∗ = 0.055 m0 , α = 7 meVnm, β = 0, L = 1.5 µm and the width of the contacts is 70 nm. (c) Schematic of an SPM imaging procedure

extrapolating the semiclassical picture shown in Fig. 1, one ﬁnds that the peaks that are split due to Rashba interaction are those in which the number of bounces is even (or zero).

3 Imaging Techniques in Transverse Focusing with Spin-Orbit Coupling Scanning probe microscopy (SPM) techniques have been recently used for imaging the electron ﬂow in a variety of 2DEG ballistic systems [5, 6]. With this technique, the negatively charged tip of a scanning microscope is positioned above the 2DEG as schematically shown in Fig. 2(c). The tip position can be changed to sweep a given area of the explored 2D device. The electrons under the tip are repelled and consequently a zone of lower electron density (or divot) is formed under the tip. In the simplest case the transmission (and then the conductance) between two contacts of the device is measured as the tip position changes. If the tip is located in a region that affects the electron path between the contacts, the conductance changes providing a map of the electron ﬂow in the device. The resolution of these images is smaller than the divot size [5, 6], making this technique a powerful tool for studying nano-scale ballistic systems. Here, we propose the use of this technique to explore the transverse focusing in the presence of spin-orbit interaction [18]. We simulate the effect of the tip potential by perturbing (increasing) the site energies εi,σ in an area of the order of 100 nm2 centered at the tip position. The Dyson equation is used to introduce the perturbation

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x[ ]

200

(a)

0,70 0,62

100

B1,1

0,54

0

x[ ]

200

(b)

0,46 0,38

100

B1,2

0

x[ ]

200

0,30

T

(c)

0,38 0,30

100 0 300

0,22

B1,V 200

100

0,14 0

y[ ] Fig. 3 Total transmission coefﬁcient T between the injector contact I (centered at y = 0), the detector contact D (centered at y = 300 a0 ) as a function of the probe position for (a) Bz = B1,1 , (b) Bz = B1,2 and (c) Bz = B1,v = (B1,1 + B1,2 )/2. We used α = 7 meVnm and the parameters given in Fig. 2

and the exact propagators between the contacts I and D are calculated for each position of the tip. Figure 3(a) shows T vs the tip’s position when the perpendicular magnetic ﬁeld is ﬁxed to obtain the ﬁrst maximum (Bz =B1,1 ) for a SO coupling α =7 meVnm. The semicircular electron path is clearly observed. In this case the T map is dominated by a drop in T along the O1 path. A similar pattern is found for the second transmission maximum (Bz = B1,2 ) as shown in Fig. 3(b). In this case, the drop in T is due to the scattering induced by the tip of electrons travelling along the O2 path. A slightly different situation is found when Bz is ﬁxed in between B1,1 and B1,2 as shown in Fig. 3(c); although the variation is also dominated by a drop (dark area), T increases at the two sides of the minimum. The observation of these two lobes shows that the tip, when placed at those positions, modiﬁes the electron ﬂow making a partially focalized electron path—O1 or O2 in Fig. 1(b)–(c)—to increase its contribution to the transmission.

Imaging Transverse Electron Focusing in Semiconducting Heterostructures

x [a ]

100

0,70 0,56

50

0 100

x [a ]

T

(a)

159

0,42

T

(b)

0,28 0,14

0 100

1,40

x [a ]

50

T

(c)

0,93

50

0 100

x [a ]

0

0,47 0 1,00

P

(d)

0,33

50 0

-0,33 -1,00

300

200

100

y [a ]

0

Fig. 4 We plot (a) Tuu , (b) Tdd , (c) T and (d) P as a function of the probe position for Bz = B2 . We used α = 7 meVnm and the parameters given in Fig. 2. Panel (d) shows the existence of a zone, located near the edge (in the bounce region), where the SPM tip drastically modiﬁes the polarization of the electrons arriving at the detector

More interesting are the imaging results obtained when the ﬁeld is ﬁxed at the second focusing condition: Bz = B2 . As mentioned above, in this case T is dominated by the electron’s orbits with one bounce at the sample’s edge. For this ﬁeld the largest contributions to the transmission coefﬁcient are Tuu and Tdd , and the corresponding focusing peak is unsplit. In Figs. 4 and 5 the results for this case are shown for α = 7 meVnm and 15 meVnm, respectively. Panel (a) shows Tuu as a function of the position of the microscope probe. The change in the transmission in this case clearly shows that the electrons injected with spin up (in the y-direction) leave the injector in the bigger orbit, rebound and then arrive to the detector in the smaller orbit with spin up—see O2 in Fig. 1(d). In panel (b) the transmittance Tdd is shown —see O1 in Fig. 1(d)—and in panel (c) the total transmission coefﬁcient is presented. As Tud and Tdu are very small, the total transmission is essentially given by the sum of the contributions shown in panels (a) and (b). Experimentally these two contributions could be distinguished by selecting the spin of the injected carriers.

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x [a ]

100

0,70 0,56

50

0 100

x [a ]

T

(a)

0,42

T

(b)

0,28 0,14

0 100

1,40

x [a ]

50

T

(c)

0,93

50

0 100

x [a ]

0

0,47 0 1,00

P

(d)

0,33

50 0

-0,33 -1,00

300

200

100

y [a ]

0

Fig. 5 We plot (a) Tuu , (b) Tdd , (c) T and (d) P as a function of the probe position for Bz = B2 . We used α = 15 meVnm and the parameters given in Fig. 2. Panel (d) the regions of P = 1 and P = −1 are better resolved than in Fig. 3. This is a direct consequence of the increasing of r+ − r− with the SO coupling strength

In fact, a combination of an external in-plane magnetic ﬁeld in the y-direction and an appropriate gate voltage in the point contacts can be used to ﬁlter spins in the injector or detector [13]. Selecting the spin of the injected electrons would make it possible to separate the two trajectories—(a) and (b) in Figs. 4 and 5—and obtain a direct visualization of the two orbits split by the spin-orbit coupling. Conversely, selecting the spin in the detector D, the transmissions T+ = Tuu + Tdu and T− = Tud + Tdd of carriers arriving at D with spin up and down, respectively, could be measured. In terms of these quantities, we deﬁne the polarization P of the transmitted particles as P=

T+ − T− T

Panel (c) and (d) of Figs. 4 and 5 show the total transmission coefﬁcient T and the polarization P as a function of the tip position. The two semicircular electron paths including the rebound at the edge are visualized in the T map. In our simulations

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the smaller and the bigger electron paths are not easily resolved in total transmission coefﬁcient map except for the largest SO coupling case and for the tip close to the bounce position—see Fig. 5(c). There, an appreciable fall (about 50%) of the transmission in the two rebound positions indicates that, when the probe is positioned there, the contribution to T of one of the two possible electron paths (O1 or O2 ) is being suppressed. If O1 is being suppressed, the electrons arriving to the detector will have spin up. On the other hand, if O2 is suppressed only spin down electrons will arrive to the detector. This means that one can select the spin polarization of the outgoing carrier ﬂux by changing the tip position a few nanometers as shown in Fig. 4(d) and Fig. 5(d). Notably, the effect is also clearly observed in the case of the smaller SO coupling despite of the fact that the total transmittance T does not resolve the two orbits.

4 Summary and Conclusions We have discussed a microscopy imaging technique for the case transverse electron focusing in 2DEGs with strong Rashba coupling. The main results can be summarized as follows: (i) The existence of two different cyclotron radii splits the ﬁrst focusing peak onto two sub-peaks, each one corresponds to electrons arriving to the detector with different spin polarization along the direction parallel to the sample’s edge. (ii) The images of the electron ﬂow for focusing ﬁelds corresponding to the ﬁrst two sub-peaks are very similar and consequently, for this case, the technique can not clearly distinguish the two type of orbits. (iii) When the external ﬁeld is ﬁxed between the focusing ﬁelds of the two subpeaks, Bz = (B1,1 + B1,2 )/2, the transmission map shows a structure that indicates the presence of the two orbits. (iv) For the second focusing condition, and for the case of strong Rashba coupling, the technique can resolve the two orbits when the microscope tip is placed close to the rebound position. (v) For the case described in the previous point, the microscope tip can be used to polarize the electron ﬂux arriving at the detector. The direction of the polarization can be reversed by changing the tip position a few nanometers. Finally, we would like to emphasize a few points: (a) Interference fringes, characteristic of the quantum ballistic transport regime, are observed in all the T maps. (b) For the properties studied here, replacing the hard wall potential V (x) by a more realistic parabolic potential does not change the main properties of the system [19, 20]. Therefore, our results should correctly describe the images that could be obtained in heterostructures deﬁned by gates. (c) The competition between the Rashba and the Dresselhauss couplings leads to interesting features in the focusing signal and needs to be considered for interpreting imaging results in systems where these two SO interactions are present. These results will be presented elsewhere.

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Acknowledgements This work was supported by ANPCyT Grants No 13829 and 13476 and CONICET PIP 5254. AR acknowledge support from PITP and CONICET. GU is a member of CONICET.

References 1. D. Awschalom, N. Samarth, and D. Loss, eds., Semiconductor Spintronics and Quantum Computation (Springer, New York, 2002). 2. A. Reynoso, G. Usaj, and C. A. Balseiro, Phys. Rev. B 73, 115342 (2006). 3. R. Winkler, Spin-orbit coupling effects in two-dimensional electron and hole systems (Springer, 2003). 4. J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997). 5. M. Topinka, B. LeRoy, S. Shaw, E. Heller, R. Westervelt, K. Maranowski, and A. Gossard, Science (2000). 6. M. Topinka, B. LeRoy, R. Westervelt, S. Shaw, R. Fleischmann, E. Heller, K. Maranowski, and A. Gossard, Nature 410, 183 (2001). 7. G. Usaj and C. A. Balseiro, Phys. Rev. B 70, 041301(R) (2004). 8. Y. A. Bychkov and E. I. Rashba, JETP Letters 39, 78 (1984). ± 9. These are the solutions for positive Bz . For negative Bz the eingenstates change: Ψn,k |−|Bz | = ± iσyΨn,k ||Bz | . 10. A. Reynoso, G. Usaj, M. J. Sanchez, and C. A. Balseiro, Phys. Rev. B 70, 235344 (2004). 11. M. Pletyukhov, C. Amann, M. Mehta, and M. Brack, Phys. Rev. Lett. 89, 116601 (2002). 12. M. Gutzwiller, Chaos in Classical and Quantum Mechanics (Spring, New York, 1991). 13. R. M. Potok, J. A. Folk, C. M. Marcus, and V. Umansky, Phys. Rev. Lett. 89, 266602 (2002). 14. L. P. Rokhinson, V. Larkina, Y. B. Lyanda-Geller, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 93, 146601 (2004). 15. H. van Houten, C. W. J. Beenakker, J. G. Willianson, M. E. I. Broekaart, P. H. M. Loosdrecht, B. J. van Wees, J. E. Mooji, C. T. Foxon, and J. J. Harris, Phys. Rev. B 39, 8556 (1989). 16. C. W. Beenakker and H. van Houten, in Solid State Physics, edited by H. Eherenreich and D. Turnbull (Academic, Boston, MA, 1991), vol. 44, pp. 1–228. 17. D. K. Ferry and S. M. Goodnick, Transport in Nanostructures (Cambridge University Press, New York, 1997). 18. Imaging of cyclotron orbits using this imaging technique was reported at ICPS 28, Vienna (2006) by K. Aidala. See also [22, 23]. 19. G. Usaj and C. A. Balseiro, Europhys. Lett. 72, 631 (2005). 20. A. O. Govorov, A. V. Kalameitsev, and J. P. Dulka, Phys. Rev. B 70, 245310 (2004). 21. J. Schliemann and D. Loss, Phys. Rev. B 68, 165311 (2003). 22. K. E. Aidala, R. E. Parrott, E. Heller, and R. Westervelt, Int. J. Mod. Phys. A 21, 4407–4424 (2006). 23. K. E. Aidala, R. E. Parrott, Tobias Kramer, E. J. Heller, R. M. Westervelt, M. P. Hanson, and A. C. Gossard, Nat. Phys. 3, 464–468 (2007).

Spectroscopic Analysis of Finite Size Effects Around a Kondo Quantum Dot Pascal Simon and Denis Feinberg

Abstract We consider a simple setup in which a small quantum dot is strongly connected to a ﬁnite size box. This box can be either a metallic box or a ﬁnite size quantum wire. The formation of the Kondo screening cloud in the box strongly depends on the ratio between the Kondo temperature and the box level spacing. By weakly connecting two metallic reservoirs to the quantum dot, a detailed spectroscopic analysis can be performed. Since the transport channels and the screening channels are almost decoupled, such a setup allows an easier access to the measure of ﬁnite-size effects associated with the ﬁnite extension of the Kondo cloud.

1 Introduction The Kondo effect occurs as soon a magnetic impurity is coupled to a Fermi sea. It is characterized by a narrow resonance of width TK0 , the Kondo temperature, pinned at the Fermi energy EF [10]. This resonance is related to the many-body singlet state which is formed between the impurity spin and the spin of an electron belonging to a cloud of spin-correlated electrons. This cloud of electrons has been termed as the socalled Kondo screening cloud. The Kondo screening cloud length is therefore related to the spatial extension of this multi-electronic, spin-correlated, wave function. The size of this screening cloud may be evaluated as ξK0 ≈ h¯ vF /TK0 where vF is the Fermi velocity. Nonetheless, the Kondo screening cloud has never been detected experimentally and has therefore remained a rather elusive prediction.

P. Simon Laboratoire de Physique et Mod´elisation des Milieux Condens´es, CNRS and Universit´e Joseph Fourier, 38042 Grenoble, France, e-mail: [email protected] D. Feinberg Institut N´eel, CNRS and Universit´e Joseph Fourier, 38042 Grenoble, France, e-mail: denis. [email protected]

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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However, the remarkable recent achievements in nano-electronics may offer new possibilities to ﬁnally observe this Kondo cloud. Indeed, the Kondo effect appears as a rather robust and versatile phenomenon. It has been observed by various groups in a single semi-conductor quantum dot [7, 5, 22, 33], in carbon nanotubes quantum dots [17, 2, 12], in molecular transistors [19, 14] to list a few. In this respect, the observation of the Kondo effect may be regarded as a test of quantum coherence of the nanoscopic system under study. One of the main signatures of the Kondo effect is a zero-bias anomaly and the conductance reaching the unitary limit 2e2 /h at low enough temperature T < TK0 . In a semi-conducting quantum dot, the typical Kondo temperature is of order 1 K which leads to ξK0 ≈ 1 micron in semiconducting heterostructures. Finite size effects (FSE) related to the actual extent of this length scale have been predicted recently in different geometries: an impurity embedded in a ﬁnite size box [31, 9], a quantum dot embedded in a ring threaded by a magnetic ﬂux [1, 24, 11, 30, 27], a quantum dot embedded between two open ﬁnite size wires (OFSW) (by open we mean connected to at least one external inﬁnite lead) [25, 26, 3] and also around a double quantum dot [23]. In the ring geometry, it was shown that the persistent current induced by a magnetic ﬂux is particularly sensitive to screening cloud effects and is drastically reduced when the circumference of the ring becomes smaller than ξK0 [1]. In the wire geometry, a signature of the ﬁnite size extension of the Kondo cloud was found in the temperature dependence of the conductance through the whole system [25, 26, 3]. More speciﬁcally, in a one-dimensional geometry where the ﬁnite size l is associated to a level spacing δ ∼ h¯ vF /l, the Kondo cloud fully develops if ξK0 l, a condition equivalent to TK0 δ . On the contrary, FSE effects appear if ξK0 > l or TK0 < δ . In a one dimensional geometry, one can equivalently use the ratio ξK0 /l or δ /TK0 . In higher dimensions, we should rather use the latter ratio or introduce another typical box length scale [9]. In the aforementioned two-terminal geometry, the screening of the artiﬁcial spin impurity is done in the OFSWs which are also used to probe transport properties through the whole system. This brings at least two main drawbacks: ﬁrst, the analysis of FSE relies on the independent control of the two wire gate voltages and also a rather symmetric geometry. This is rather difﬁcult to achieve experimentally. In order to remedy to these drawbacks, we propose and study here a simpler setup in which the screening of the impurity occurs mainly in one larger quantum dot or metallic box1 or OFSW and the transport is analyzed by help of one or two weakly coupled leads. In practice, a lead weakly coupled to the dot by a tunnel junction allows a spectroscopic analysis of the dot local density of states (LDOS) in a way very similar to a STM tip. The geometry we study is depicted in Fig. 1. We note that this geometry has also been proposed by Oreg and Goldhaber-Gordon [18] to look for signatures of the two-channel Kondo ﬁxed point or by Craig et al. [4] to analyze two quantum dots coupled to a common larger quantum dots and interacting via the RKKY interaction. In the former case, the key ingredient is the Coulomb interaction of the box whereas in the latter, the box is largely open and used simply as a metallic reservoir mediating Note that for a 2D or 3D metallic box, the level spacing ∆ is not simply related to the Fermi velocity and the length scale of the box. In this case, we shall compare directly TK0 to ∆ .

1

Spectroscopic Analysis of Finite Size Effects Around a Kondo Quantum Dot Fig. 1 Schematic representation of the device we analyze in this paper. When the box Coulomb blockade energy is not neglected, we assume that the box potential can be controlled by a voltage gate e0

165

t e0

0 tL L

t0 dot

tR R

both the Kondo and RKKY interactions. Here we are interested in the case where ﬁnite-size effects in the larger quantum dot or metallic box do matter whereas in the aforementioned experiments the level spacing was among the smallest scales. Nevertheless, we emphasize that by increasing the box level spacing, the regime discussed in this paper should be accessible by these type of experiments. The plan of the paper is as follows: in Section 2, we present the model Hamiltonian and derive how the FSE renormalizes the Kondo temperature in our geometry. In Section 3, we perform a detailed spectroscopic analysis. In Section 4, we show how FSE affect the transport properties of the quantum dot. In Section 5, the effect of a ﬁnite Coulomb energy in the box is discussed. Finally Section 6 summarizes our results.

2 Model Hamiltonian and Kondo Temperature The geometry we analyze is depicted in Fig. 1. In this section we assume that the large dot is connected to a third lead. From hereon, the Coulomb interaction in the box is neglected except in Section 5. The Coulomb interaction does not affect the main results we discuss in this section. In order to model the ﬁnite-size box connected to a normal reservoir, we choose for convenience a ﬁnite-size wire characterized by its length l or equivalently by its level spacing ∆ ∼ h¯ vF /l. In fact, the precise shape of the ﬁnite-size box is not important for our purpose as soon as it is characterized by a mean level spacing ∆ separating peaks in the electronic density of states. We assume that the small quantum dot is weakly coupled to one or two adjacent leads (L and R). On the Hamiltonian level, we use the following tight-binding description, and for simplicity model the leads as one-dimensional wires (this is by no means restrictive): H = HL + HR + H0 + Hdot + Htun with

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HL = −t H0 = −t

∞

∑ (c†j,s,L c j+1,s,L + h.c.) − µLn j,s,L

(1)

j=1,s ∞

∑ (c†j,s,0 c j+1,s,0 + h.c.) − µ0n j,s,0

j=1,s

+ (t − t ) ∑(c†l,s,0 cl+1,s,0 + h.c.)

(2)

∑ εd nd,s + Und↑nd↓

(3)

∑ ∑

(4)

s

Hdot =

s

Htun =

s α =L,R,0

(tα c†ds c1,s,α + h.c.).

HR is obtained from HL by changing L → R. Here c j,s,α destroys an electron of spin s at site j in lead α = 0, L, R; cd,s destroys an electron with spin s in the dot, n j,s,α = c†j,s,α c j,s,α and nds = c†ds cds . The quantum dot is described by an Anderson impurity model, εd ,U are respectively the energy level and the Coulomb repulsion energy in the dot. The tunneling amplitudes between the dot and the left lead, right lead and box are respectively denoted as tL ,tR ,t0 (see Fig. 1). The tunneling amplitude amplitude between the box and the third lead is denoted as t (see Fig. 1). Finally t denotes the tight binding amplitude for conduction electrons implying that the electronic bandwidth Λ0 = 4t. Since we want to use the left and right leads just as transport probes, we assume in the rest of the paper that tL ,tR t0 . We are particularly interested in the Kondo regime where nd ∼ 1. In this regime, we can map Htun + Hdot to a Kondo Hamiltonian by help of a Schrieffer-Wolff transformation: HK = Htun + Hdot =

∑

α ,β =L,R,0

Jαβ c†1,s,α

σ ss · Sc1,s ,β , 2

(5)

where Jαβ = 2tα tβ (1/|εd | + 1/(εd + U)). It is clear that J00 J0L , J0R JLL , JRR , JLR . In Eq. (5), we have neglected direct potential scattering terms which do not renormalize and can be omitted in the low energy limit. The Kondo temperature is a crossover scale separating the high temperature perturbative regime from the low temperature one where the impurity is screened. There are many ways to deﬁne such scale. We choose the “perturbative scale” which is deﬁned as the scale at which the second order corrections to the Kondo couplings become of the same order of the bare Kondo coupling. Note that all various deﬁnitions of Kondo scales differ by a constant multiplicative factor (see e.g. Ref. [27] for a comparison between the perturbative Kondo scale with the one coming from the Slave Boson Mean Field Theory). The renormalization group (RG) equations relate the Kondo couplings deﬁned at scales Λ0 and Λ . They simply read: ⎡Λ ⎤ 0 −Λ ρ (ω ) 1 ⎦ γ dω (6) Jαβ (Λ ) ≈ Jαβ (Λ0 ) + ∑ Jαγ (Λ0 )Jγβ (Λ0 ) ⎣ + 2 γ |ω | Λ

−Λ0

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where ργ is the LDOS in lead γ seen by the quantum dot. Since J00 JLL , JRR , the Kondo temperature essentially depends on the LDOS in the lead 0. Using the RG equations in Eq. (6), the Kondo temperature can be well approximated as follows: ⎤ ⎡Λ 0 −T K J00 ⎣ ⎦ ρ0 (ω ) d ω = 1 + (7) 2 |ω | TK

−Λ0

When the lead 0 becomes inﬁnite (i.e. when t = t), ρ0 (ω ) = ρ0 = const and we recover TK = TK0 the usual Kondo temperature. It is worth noting that including the Coulomb interaction in the box does not affect much the Kondo temperature. The box Coulomb energy EB slightly renormalizes J00 in the Schrieffer-Wolff transformation since EG U, |εd | [18]. The LDOS ρ0 can be easily computed for a ﬁnite one-dimensional wire. In general, for a ﬁnite size open structure, ρ0 corresponds in the limit of a weak coupling to a continuum to a sum of resonance peaks. The positions of these peaks is to a good approximation related to the eigenvalues ωn of the isolated ﬁnite size structure, while their width γn is proportionnal to t 2 |ψn |2 where the ψn are the eigenvectors of the isolated structure. The LDOS ρ0 is very well approximated by a sum of Lorentzian functions [25] in the limit t t:

πρ0(ω ) ≈ ∑ |ψn |2 n

γn . (ω − ωn)2 + γn2

(8)

For a 1D ﬁnite size wire of length l, ψn = 2 sin(kn )/(l + 1) with kn ≈ π n/(l + 1). This approximation is quite convenient in order to estimate the Kondo temperature TK through (7). When the level spacing ∆n ∼ h¯ vF /l is much smaller than the Kondo temperature TK0 , no ﬁnite-size effects are expected. Indeed, the integral in (7) averages out over many peaks and the genuine Kondo temperature is TK ∼ TK0 . On the other hand, when TK0 ∼ ∆n , the Kondo temperature starts to depend on the ﬁne structure of the LDOS ρ0 and a careful calculation of the integral in (7) is required. Two cases may be distinguished: either ρ0 is tuned such that a resonance ωn sits at the Fermi energy EF = 0 (labeled by the index R) or in a non resonant situation (labeled by the index NR). In the former case, we can estimate

γn ∆n TKR = ! (∆n2 + γn2 ) exp J (∆ 2)ρ R (0) − ∆n2 00

≈ γn exp − In the latter case, we obtain,

n

1 J00 (∆n )ρ0R (0)

0

≈ γn

TK0 dn

πγd n n

.

(9)

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TKNR

= ∆n exp −

π∆n2 4J00(∆n )|ψn |2 γn 0 π8γdn

n 1 T ≈ ∆n K , ≈ ∆n exp − NR dn J00 (∆n )ρ0 (0)

(10)

where πρ0NR (0) ≈ 4γn |ψn |2 /∆n2 . These two scales are very different when t 2 t 2 . By controlling ρ0 , we can control the Kondo temperature (at least when TK ≤ δ ). The main feature of such geometry is that the screening of the artiﬁcial spin impurity is essentially performed in the open ﬁnite-size wire corresponding to lead 0. Now let us study what are the consequences of FSE on transport when one or two leads are weakly coupled to the dot. This is the purpose of the next section.

3 Spectroscopy of a Kondo Quantum Dot Coupled to an Open Box In this section, we consider a standard three-terminal geometry as depicted in Fig. 1. Since the leads are weakly connected to the dot, they allow a direct access to the dot density of states in presence of FSE in the box. We have used the Slave Boson Mean Field Theory (SBMFT) [10] in order to calculate the dot local density of states (LDOS). This approximation describes qualitatively well the behavior of the Kondo impurity at low temperature T ≤ TK when the impurity is screened and especially capture well the exponential dependence of the Kondo temperature. Furthermore, this method has been proved to be efﬁcient to capture ﬁnite size effects in Refs. [25, 26, 27]. Let us now compute the dot density of states ρd (ω ). We assuming ΓL/R Γ0 (ω ) such that for low bias the dot Green functions weakly depend on the chemical potential in the left and right leads. Under such conditions, the differential conductance reads as follows: e

2e2 dI 4ΓL ≈ d µL h

∞

−∞

−d f (ω ) πρd (ω + µL )d ω . dω

(11)

Varying µL allows a direct experimental access to ρd (µL ) at T TK . Note that a similar approximation is used for STM theory with magnetic adatoms [21]. We have plotted ρd (ω ) in Fig. 2 for both the non-resonant case and the on-resonance case for three different values of ξK0 /l. We took the following parameters in units of t = 1: t0 = 0.5, tL = tR = 0.1 (therefore tL2 << t02 ), t = 0.5, and l ∼ 1000a (a the lattice constant) or equivalently ∆ ∼ 0.006. The Kondo energy scale ξK0 can be varied using the dot energy level εd which is controlled by the dot gate voltage. One has to distinguish between two cases: TK0 δ and TK0 δ . When TK0 δ , no ﬁnite-size effect is to be expected. Nevertheless two non trivial features should be be noticed: (i) The various peaks appearing in ρ are included in an envelope of width O(TK0 ) ∆ (which has a broader range than the Fig. 2 actually

Spectroscopic Analysis of Finite Size Effects Around a Kondo Quantum Dot 5

5

0

0

2 (b0) ρd (ω)

K

3

ξ K / l = 0.15

3

2

2

1

1

0

-0.004

-0.002

0

ω

0.002

0.004

0.006

0

ξK/l =1 0 ξ /l =5 K

4 2 (b0) ρd (ω)

ξ K / l = 0.15 0 ξK / l = 1 ξ0 / l = 5

4

-0.006

169

0

-0.006

-0.004

-0.002

0

ω

0.002

0.004

0.006

Fig. 2 Dot density of states ρd (ω ) for both the non resonant case (upper panel) and the resonant case (lower panel). We took ∆ ∼ 0.006 and plot ρd for ξK0 /l = ∆ /TK0 ∼ 0.15, ξK0 /l = ∆ /TK0 ∼ 1 and ξK0 /l = ∆ /TK0 ∼ 5. Note that ρd has been scaled by b20 , the slave boson parameter for an easy comparison between both cases

covers for ξK0 /l = ∆ /TK0 ∼ 0.15). In this respect, the Kondo resonance plays the role of an energy ﬁltering device which ﬁlters the box high energy states that are not in the range of width O(TK0 ) around EF = 0. (ii) The LDOS ρd (ω ) mimics the density of state in the lead 0 but is shifted such that an off-resonance peak in the lead 0 corresponds to a dot resonance peak and vice versa. This can be simply understood from a non-interacting picture valid at T = 0. The non-interacting dot Green function reads Gdd (ω ) ≈

1 , ω − εd − δ ε (ω ) + iΓ0(ω )

(12)

where δ ε is the real part of the dot self-energy and Γ0 its imaginary part. The minima’s of Γ0 thus correspond to the maxima of −Im(Gdd ). We also note that the resonance peak is slightly shifted from ω = 0 in this limit. Therefore the 2-terminal conductance does not reach its unitary limit (i.e. its maximum non interacting value). This is due to the fact that we took εd = −0.68 and we are not deep in the Kondo regime. Particle-hole symmetry is not completely restored in the low energy limit. On the other hand, when ξK0 l (TK0 δ ), ρδ changes drastically. The ﬁne structure in the box density of states no longer shows up in ρd . This is expected since only the energy states which are within a range of order O(TK ) ∆ appears in the dot LDOS. In the off-resonance case, only the narrow Kondo peak of width TKNR TK0 mainly subsists for ξK0 /l = 5 (upper panel of Fig. 2). We can also show that the position of the small peaks at ω ∼ ±∆ /2 for ξK0 l are related to the resonance peaks in lead 0. We also note that the narrow peak is this time almost at

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ω = EF = 0 i.e. particle-hole symmetry is restored. In order to reach large value of ξK0 , we took small values of εd such that we are deep in the Kondo regime where nD ∼ 1. The most surprising result occurs for the resonant case where the Kondo peak is split for ξK0 l. Usually a splitting of the Kondo LDOS is associated with the destruction of the Kondo effect as this would be the case for a Kondo quantum dot under a magnetic ﬁeld or interacting with another quantum dot via the RKKY interaction [4, 28, 32]. Here the splitting cannot at all be attributed to the destruction of the Kondo effect but instead to a subtle non interacting destructive interference phenomenon. Let us use some renormalization group argument. When we integrate out high energy electronic degrees of freedom from the bandwidth down to an energy scale of order δ > TK0 , we start building the Kondo resonance at the Fermi energy EF . If we continue integrating out electronic degrees of freedom from d down to γ , this reasoning would tell us that we end up with a Kondo resonance pinned at EF . However, this does not take into account that the box has also a (non interacting) resonance pinned at EF and these two resonances are coupled via the tunnel amplitude t0 . By analogy to a molecular system with two degenerate orbital states, where a tunneling amplitude leaves the degeneracy and creates bonding and anti-bonding states, a strong t0 (as is our case here) may lead to a splitting of the Kondo resonance. Therefore this splitting is more related to a destructive interference between the two resonances -an interacting one in the dot and a non-interacting one in the box. The study of the splitting of the Kondo resonance as a function of t0 has been extensively studied recently in a slightly different geometry by Dias da Silva et al. [6]. One can also quantify this splitting within the SBMFT method. By approximating Γ0 (ω ) ≈ t02 |ψn |2 ω 2γ+n γ 2 at a resonance n, one can well understand analytically n

the structure of ρd . When TKR γn , one can show that the peak splitting is of order ∼2

γn TKR and the peaks width is of order γn .

4 Analysis of Transport Properties At T = 0, it is straightforward to show, using for example the scattering formalism [15], that the conductance matrix GUα ,β is simply given by GUα ,β =

4Γα Γβ 2e2 h (ΓL + Γ0 + ΓR)2

(13)

where Γα = π tα2 ρα (0), and α = l, 0, r. Since the SBMFT aims at replacing the initial Anderson Hamiltonian by a non-interacting one, one may easily access the conductance by directly applying the Landauer formula or equivalently by using

2e2 −∂ f 4Γα (ω )Γβ (ω ) Im(−Grdd )(ω ). dω (14) Gαβ = h ∂ω (∑α Γα (ω ))

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Using the SBMFT, one can extensively study the conductance as function of temperature for various case TK0 δ , TK0 δ . This has been reported in details in [29]. Actually, it turns out that ﬁnite size effects, which are related to the ﬁnite size extension of the Kondo cloud, are clearly visible at some intermediate temperature TK0 > T > TKR , TKNR . In this temperature range, signiﬁcant deviations from the noninteracting limit are obtained. Let us assume that the box can be gated (see Fig. 1). We took the same parameters as in Fig. 2 except that l = 200a. The ﬁnite size effects are much more spectacular when one look at the conductance, at ﬁxed temperature, as a function of the box gate voltage e0 . We have therefore ﬁxed the temperature at T = 2.10−3 and plotted in Fig. 3, the conductance GL0 as a function of e0 for different values of ξK , controlled here by the parameter εd . The other parameters are unchanged. The upper (plain style) curve corresponds to ξK = 50a < l. We observe large oscillations of the conductance corresponding to e0 being on a resonance or off a resonance. At large ξK ∼ 1000a l, the conductance (dashed-dashed-dotted style) has a completely different shape. The minima’s and the maxima’s of the conductance at ξK ∼ 50a become now respectively maxima’s and minima’s. Furthermore the conductance at these minima’s is very small close to 0. This regime corresponds to the high temperature for the non resonant case. The intermediate values of ξK show how the conductance crosses over in between these two extreme cases. This dramatic change of the conductance in the regime in which ξK l is a direct consequence of interactions effects and should be directly observable in experiments. Similar results can be obviously obtained by analyzing the conductance GLR despite the fact √ that the amplitude of GLR will be in general smaller than GL0 by a factor ∼ ΓLΓR /Γ0 . In fact, the conductance GLR can be made signiﬁcantly larger by adjusting the chemical potential µ0 such that the current in lead 0 veriﬁes I0 = 0. We have essentially used the SBMFT to analyze the spectroscopic and transport properties. Nevertheless, one can also use analytical calculations in two limiting cases. When T TK , one can safely rely on renormalized perturbative calculations while at T TK the Nozi`eres Fermi liquid approach [16] can be applied. Using the latter theory, one can for example show that the conductance in the on resonance case is a non monotonous function of temperature. We refer the reader to Ref. [29] where these analytical calculations are detailed and conﬁrm the present analysis.

5 Finite Box Coulomb Energy In this section, we discuss whether a ﬁnite box Coulomb energy modiﬁes or not the results presented in this work. As we already mentioned in Section 2, the Kondo coupling J00 , is almost not affected by the box Coulomb energy EB (since EB

U) and therefore the Kondo temperature remains almost unchanged. As shown in [18, 20, 8], a small energy scale EB changes the renormalization group equation in Eq. (6). The off-diagonal couplings J0L (Λ ), J0R (Λ ) tend to 0 for Λ EB . At energy Λ EB the problem therefore reduces to an anisotropic 2−channel Kondo problem. The strongly coupled channel is the box 0, the weakly coupled one is

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G (2e²/h)

0.15

0.1

0.05

0 0

0.02

e0

0.04

0.06

Fig. 3 Conductance (in units of 2e2 /h) as a function of εW for different values of ξK . From top to bottom ξK ∼ 50a (plain style), ξk ∼ 150a (dotted style), ξK ∼ 300a (short dashed style), ξK ∼ 500a (dot-long dashed style), ξK ∼ 600a (long dashed style), ξK ∼ 750a (dot-dot-dashed style) and ξK ∼ 1000a (dot-dashed-dashed style)

the even combination of the conduction electron in the left/right leads. At very low energy, the ﬁxed point of the anisotropic 2−channel Kondo model is a Fermi liquid. It is characterized by the strongly coupled lead (here the box) screening the impurity whereas the weakly coupled one completely decoupled from the impurity. The dot density of states depicted in Fig. 2 should remain therefore almost unaffected. The problem is to read the dot LDOS with the weakly coupled leads since they decouple at T = 0. Nevertheless, for a typical experiment done at low temperature T , such a decoupling is not complete and the dot LDOS should be still accessible using the weakly coupled leads but with a very small amplitude. We up to now analyze the situation in which a box or a ﬁnite size wire is also used as a third terminal i.e is coupled to a continuum. In some situations, like the theoretical one presented in Ref. [18], no terminal lead is attached to the box and the geometry is a genuine 2-terminal one. In order to analyze this system, we have to take into account both a ﬁnite level spacing and a ﬁnite box Coulomb energy. One can make progress if we assume TK0 ∆ EB U D0 where D0 is the bandwidth. One proceeds with a RG treatment in three steps: EB << Λ << U, ∆ << Λ << EB et TK0 << Λ << ∆ . When EB << Λ << U, the RG equations derived in Eq. (6) remain valid and are the usual ones. When ∆ << Λ << EB , the couplings JL0 and JR0 stop renormalizing because of the Coulomb energy EB . The other couplings keep on growing. When TK0 << Λ << ∆ , two situations are to be considered. Let us start with the case where no resonance sits at the Fermi energy in

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the box. The coupling J00 no longer renormalizes since there is no state available. On the other hand, the couplings Jα ,β with α , β = L, R keep on renormalizing and may eventually reach a strong coupling regime. Therefore, the screening of the impurity ultimately occurs in the L, R leads though we may have started with J00 JLL , JRR . This situation is quite different than the one we explored previously. However, when a resonance sits at the Fermi energy of the box, we are left with a double dot problem with one electron in each dots. They form a singlet and we expect a split LDOS in the dot. If we assume EB δ , the problem gets more complicated since we already reach a strong coupling regime at the scale EB . A step toward this direction was recently achieved in Ref. [13] in a slightly modiﬁed geometry.

6 Conclusions In this paper, we have studied a geometry in which a small quantum dot in the Kondo regime is strongly coupled to a large open quantum dot or open ﬁnite size wire and weakly coupled to other normal leads which are simply used as transport probes. The artiﬁcial impurity is mainly screened is the large quantum dot. Such a geometry thus allows to probe the dot spectroscopic properties without perturbing it. We have shown using the SBMFT how ﬁnite size effects show up in the dot density of states and in the the conductance matrix. We also analyzed how these results are modiﬁed in presence of Coulomb interactions in the box. We hope the predictions presented here are robust enough to be checked experimentally. Acknowledgements This research was partly supported by the contract PNANO ‘QuSpins’ of the French Agence Nationale de la Recherche.

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13. R. K. Kaul, G. Zarand, S. Chandrasekharan, D. Ullmo, and H. U. Baranger, Phys. Rev. Lett. 96, 176802 (2006). 14. W. Liang et al., Nature 417, 725 (2002). 15. T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988). 16. P. Nozi`eres, J. Low Temp. Phys. 17, 31 (1974). 17. J. Nyg˚ard et al., Nature 408, 342 (2000). 18. Y. Oreg and D. Goldhaber-Gordon, Phys. Rev. Lett. 90, 136602 (2003). 19. J. Park et al., Nature 417, 722 (2002). 20. M. Pustilnik, L. Borda, L. I. Glazman, and J. von Delft, Phys. Rev. B 69, 115316 (2004). 21. A. Schiller and S. Herschﬁeld, Phys. Rev. B 61, 9036 (2000). 22. F. Simmel, R. H. Blick, U.P. Kotthaus, W. Wegsheider, and M. Blichler, Phys. Rev. Lett. 83, 804 (1999). 23. P. Simon, Phys. Rev. B 71, 155319 (2005). 24. P. Simon and I. Afﬂeck, Phys. Rev. B 64, 085308 (2001). 25. P. Simon and I. Afﬂeck, Phys. Rev. Lett. 89, 206602 (2002). 26. P. Simon and I. Afﬂeck, Phys. Rev. B 68, 115304 (2003). 27. P. Simon, O. Entin-Wohlmann, and A. Aharony, Phys. Rev. B 72, 245313 (2005). 28. P. Simon, R. Lopez, and Y. Oreg, Phys. Rev. Lett. 94, 086602 (2005). 29. P. Simon, J. Salomez, and D. Feinberg, Phys. Rev. B 73, 205325 (2006). 30. E. S. Sorensen, I. Afﬂeck, Phys. Rev. Lett. 94, 086601 (2005). 31. W. B. Thimm, J. Kroha, J. von Delft, Phys. Rev. Lett. 82, 2143 (1999). 32. M. Vavilov and L. Glazman, Phys. Rev. Lett. 94, 086805 (2005). 33. W. G. van der Wiel, S. De Franceschi, T. Fujisawa, J. M. Elzerman, S. Tarucha, and L. P. Kouwenhoven, Science, 289, 2105 (2000).

Effective Magnus Force on a Magnetic Vortex L.R. Thompson and P.C.E. Stamp

Abstract In classical hydrodynamics, a Magnus force exists between a vortex and the hosting ﬂuid acting transverse to their relative motion. There is a quantum Magnus force acting on vortices in superﬂuids and superconductors and an analogous force acting on magnetic vortices excited in spin systems. Couplings with the system quasiparticles can modify this to an effective Magnus force by introducing transverse damping forces. The existence and magnitude of transverse damping forces are highly controversial and have not been settled by experiment. We derive the various damping forces on a vortex in a magnetic system, in particular, damping forces acting longitudinally and transversely to current and past motion (memory effects). In a magnetic system, we expect experiments can more accurately study vortex motion for comparison with theory. Despite the simplicity of the spin system, the results are general and should reveal quantitative behaviour for the superﬂuid/superconductor systems.

1 Introduction Classically, a vortex moving relative to its hosting ﬂuid will experience a perpendicular force, the Magnus force. If the vortex moves horizontally in a plane perpendicular to its axis of circulation, then the local ﬂuid ﬂow above and below the vortex will be moving faster or slower depending on the direction of the vortex circulation. According to Bernoulli’s equation, this is equivalent to a pressure differential which in turn leads to the Magnus force. In a superﬂuid (or superconducting) L.R. Thompson Department of Physics & Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada, e-mail: [email protected] P.C.E. Stamp Department of Physics & Astronomy, University of British Columbia and the Paciﬁc Institute of Theoretical Physics, Vancouver, BC, V6T 1Z1, Canada

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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system, a quantized vortex moving relative to the superﬂuid ﬂow also experiences a Magnus force. If we consider a moving vortex in the complete absence of perturbing excitations, the bare Magnus force (per unit length), FM , is due to the accumulated Berry’s phase along a path enclosing the vortex FM = ρs K × (vV − vs)

(1)

where ρs is the superﬂuid density, and vV and vs are the vortex and superﬂuid velocities, respectively. In addition to the transverse Magnus force, a vortex is also subjected to a variety of damping forces acting both longitudinally and transversely to the vortex motion. Originally, damping was described phenomenologically [1]. A vortex is endowed with an effective mass that can be argued on purely dimensional grounds (ﬁrst noted by Duan and Leggett [2]) to be of order EV /c2s , where EV is the vortex energy and cs is the speed of sound of the quasiparticles in the system of interest (phonons in the superﬂuid and magnons in the magnetic system). There is much controversy about a microscopic derivation of damping forces, including the modiﬁcations to the bare Magnus force, upon the introduction of perturbations. In an early calculation, Iordanskii [3] observed that phonons were asymmetrically scattered by a vortex leading to a transverse force proportional to ρn and depending on the relative vortex and normal ﬂuid velocities. The actual magnitude of this additional term was not immediately obvious, nor was it clear whether it should reinforce or oppose the bare Magnus force. More recently, Sonin [4] derived a perpendicular Iordanskii force ρn K × (vV − vn ) to be added to the bare Magnus force. Thouless et al. [5, 6] argue that all perturbing effects cancel exactly and the bare Magnus force remains unmodiﬁed. Experiments in various superﬂuid and superconducting systems have not so far provided unambiguous evidence on the existence or magnitude of the effective Magnus force. Superﬂuid and superconducting systems are not the only systems in which vortex dynamics can be studied. There are a number of other linear defects in ordered condensed matter media exhibiting analogous motion to that of vortices in superﬂuid systems. These include dislocations in solids, Bloch lines and vortex lines in magnetic materials, disinclinations in liquid crystals, among others. In this paper, we will consider the motion of a 2D vortex (or, equivalently, we neglect motion along the length of the vortex line) in a magnetic system, quoting results for the multiple vortex conﬁguration. Early calculations of the bare Magnus force in a magnetic system (or gyrotropic force) were made by Huber [7] and Nikiforov and Sonin [8]. They estimated phenomenological damping forces according to the form derived by Thiele [9] using the Landau-Lifshitz magnetization equation. Slonczewski [10] extended the microscopic calculations by considering linear deformations of a moving vortex contributing quadratically to the vortex energy, essentially introducing an inertial energy from which one can write a vortex mass. By examining the extended deformations while in motion, he calculated the rate of energy transported from the moving vortex by magnons, giving an estimate of the damping rate. This method

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gives information of the longitudinal damping force only (the transverse damping force cannot produce work). The dynamics of the vortices in these various essentially 2D systems have much in common. The order parameter in each of them, the spin ﬁeld in the magnetic system for instance, has U(1) symmetry. We ﬁnd topological defects with integer winding number of the order parameter along paths surrounding the defect, a general result of topology, i.e. Π1 (S1 ) = Z. This winding number is the vortex circulation in a superﬂuid or superconductor and corresponds to the vorticity of a vortex in a magnetic system. We assume that couplings to other degrees of freedom (external to the main order parameter under consideration) are negligible compared to ﬂuctuations in the order parameter. For instance, in the magnetic system, the vortex will couple only weakly with phonons compared to couplings with magnons. In this case, the vortex and bath modes arise within the same background ﬁeld so that, the vortex being a proper stable excitation, there are no ﬁrst order couplings to the vortex position. However, once in motion, there can be ﬁrst order couplings of the bath modes to the vortex velocity. To lowest order in perturbation theory, such vortex velocity couplings introduce an inertial energy shift (from which we deduce a vortex mass), while on-shell scattering provide a damping mechanism. Various techniques, such as those examined in this paper, estimate effective damping forces arising from such a coupling to the bath. There are also second and higher order couplings between the vortex position and bath coordinates. These are in fact crucial if the bath modes are gapped (since the ﬁrst order coupling becomes ineffective below a threshold vortex speed, in particular for vanishing speeds, i.e. the low energy limit of interest here). Should the bath modes be ungapped, the higher order couplings may still prove more important, due in part perhaps to an increased scattering phase space or in order to introduce temperature dependence to damping terms (of particular interest for isolating damping effects in experiments). In this paper, we restrict considerations to ungapped bath modes and include only the ﬁrst order velocity term, neglecting entirely the higher order couplings. The vortices in the magnetic system differ from the superﬂuid or superconducting vortices by carrying a second topological index, the polarization, describing whether the vortex core has twisted out of plane (p = 0) either up (p = 1) or down (p = −1). In fact, non-zero polarization is crucial for introducing a Magnus force acting on a magnetic vortex, although the derivation of damping forces does not rely at all on these core spins (they introduce only exponentially small core corrections). Hence, we don’t expect this additional complexity to limit the application of the methodology in any way to the other systems. Our goal in this paper is to ﬁnd effective equations of motion of a vortex moving in a 2D magnetic system, accounting for interactions with the environmental spin modes. Beginning with the microscopic governing equations, we promote the vortex center coordinates to collective coordinates [11] and integrate over the 2D plane to obtain an energy functional of vortex position. If we ﬁrst expand about the vortex proﬁle to include perturbations, which are in fact the bath mode coordinates, after integration we have a functional of vortex position and bath degrees of freedom. We then average over the bath motion. This is analogous in the classical

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theory of Brownian motion in which we average over a ﬂuctuating force in order to obtain an effective damping force acting on our central particle. The process is done using a path integral representation of the system’s combined vortex and bath modes density matrix. The averaging corresponds to a partial trace over bath degrees of freedom. This method of introducing a quantum version of Brownian motion was ﬁrst discussed by Caldeira and Leggett [12] and modiﬁed for a soliton interacting with environmental modes arising within the same ﬁeld by Castro Neto and Caldeira [13]. The ﬁnal result is a microscopically derived equation of motion for the vortex including the bare Magnus force, an inertial mass term and a variety of damping forces, not necessarily directly opposing vortex motion. The damping forces found possess interesting memory effects, resulting from the non-Ohmic form of the vortex-bath couplings. Note that the ﬁrst order velocity coupling introduces damping forces that are temperature independent (survive to T = 0!). In the soliton work of Castro Neto and Caldeira, they found temperature dependent damping forces. In their system however, the lowest order coupling was second order in velocity, due to orthogonality of the soliton translation mode with the remaining bath modes. In our spin ﬁeld system, as opposed to the simple scalar problem, we do not have such a cancellation of the ﬁrst order terms. In order to introduce temperature dependence to our damping forces, for instance, for easier comparison with experiments, we would have to extend our analysis to second order in magnons.

2 System and Excitations Consider a circular 2D insulating lattice of ferromagnetically coupled spins with free boundary conditions. For small radii systems, this should model well the quantum dots in experiments. For bulk systems, the R → ∞ limit must be taken very carefully. We are interested in the low energy behaviour and hence approximate the system by a continuous ﬁeld of spins. The nearest neighbor ferromagnetic exchange coupling in a lattice Hamiltonian is approximated by a gradient term of the opposite sign: − ∑ Si ·S j → d 2 r(∇S)2 . We also include an easy plane anisotropy which reduces the symmetry of the order parameter, here the magnetic ﬁeld S(r), to a U(1) symmetry. The continuum Hamiltonian of our system is then H =

d2r J (∇S)2 + KSz2 a2 2

(2)

where J and K are the exchange and anisotropy constants, respectively, and a is the lattice spacing. In this notation, the gradient term expands as (∇S)2 = ∑i=x,y,z (∇Si )2 . We restrict ourselves to a perfectly rigid lattice, or, equivalently, we ignore phonons. There are three coordinate systems deﬁned in this paper. The ﬁrst denotes the 2D plane upon which the spins are ﬁxed, denoted by r, or, in polar coordinates, (r, ξ ). Next, the spin ﬁeld S(r) will be described in spin space using the spherical coordinates (φ , θ ). The vortex itself will be described by the position of its center

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X in the 2D plane of spins. In a spin system, using coordinate φ and conjugate momentum S cos θ we can verify directly the classical version of the commutation relations, [Si , S j ] = i¯hεi jk Sk , that is, {Si , S j }φ ,S cos θ = εi jk Sk where we deﬁne S = S(sin θ cos φ , sin θ sin φ , cos θ ) and {·, ·}q,p is the usual Poisson bracket. However, spin being an essentially quantum concept, we must remember that when speaking of spin directions given by (φ , θ ), we mean the spin state of highest probability to be found in that direction. In this description, S has units of h¯ . We can alternatively transform to a Lagrangian formulation where the corresponding Berry’s phase is

ωB =

dt

d 2r S cos θ φ˙ a2

(3)

Writing the Lagrangian in terms of angles φ and θ , we have

L =S

c 1 d2r ˙ 2 2 2 (∇ (∇ φ cos θ − θ ) + sin θ φ ) − a2 2 rV2

(4)

Varying with respect to the conjugate variables, we recover the spin ﬁeld equations of motion 1 c 1 c

∂φ ∂t ∂θ ∂t

=

∇2 θ sin θ

− cos θ (∇φ )2 + r12 cos θ

(5)

V

= − sin θ ∇2 θ − 2 cos θ ∇θ · ∇φ

wherec = SJa2, not to be confused with the magnon speed of sound cs = c/rV , and J is the typical vortex lengthscale. rV = 2K There are two families of excitations satisfying these equations of motion: the perturbative spin waves, or magnons, and the metastable vortex states. Magnons are the small amplitude excitations of our system, and it sufﬁces to describe their motion and properties by expanding in small deviations about the ferromagnetic in plane ground state, choosing the ground state φgs = 0 amongst the continuum of ground states without any loss of generality. The magnon spectrum is ω (k) = ckQ where Q = k2 + r12 . Note that it is not V

gapped (there is an overall factor of k), resulting from the continuous degeneracy Q of ground states. However, the density of states goes as k2 +Q 2 remaining ﬁnite as

ω → 0 in comparison to the isotropic ferromagnet with density of states 1k diverging for zero frequency. The form of the magnon density of states along with the form of the vortex-magnon coupling determines the dependence of the vortex dynamics on its current and former motion. The vortex is formed by an integer winding of the spin ﬁeld about a central defect (the vortex core). Notice that without the easy plane anisotropy, the spins could continuously deform out of plane with no energy penalty so that they would easily decay back to the ferromagnetic ground state. Allowing the spins to lie only in plane (or at least nearly so), the vortex gains topological stability against decay. A good

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discussion of topological excitations and their stability can be found in the reviews by Volovik and Mineev [14] and by Mermin [15]. The system has two symmetries: a continuous in-plane symmetry and a discrete up-down symmetry. The vortices are thus characterized by two topological indices, the vorticity q = ±1, ±2, . . ., sometimes also called the winding number, and the polarization p = 0, ±1, describing whether the vortex core lies out of plane and along which direction (up/down). The in-plane vortex can be described analytically. The spin conﬁguration of this solution has φV = qξ + δ and θV = 0, where δ is a uniform spin rotation. The vortex energy calculation within our continuum approximation requires both an infrared and ultraviolet cutoff, E = S2

J Rs d 2 r (∇φV )2 = S2 J π q2 ln 2 a

(6)

where Rs is the radial size of the system and a is a lower cutoff, the lattice spacing, required since the system is actually discrete (making r → 0 unphysical). This energy is independent of where the vortex center is within the circular integration region. The out of plane solution spin conﬁguration has the same polar angle dependence, φV = qξ + δ , while the azimuthal spin angle cannot be solved for analytically. The asymptotic behaviour is ⎧ 2 ⎨ 1 − c1 r , r → 0; rV cos θV = p (7) rV r ⎩c exp(− ), r → ∞. 2

r

rV

where c1 , c2 > 0 are free constants that can be set by imposing appropriate continuity conditions. The out of plane vortex has the same leading order energy as the in plane solution. Core corrections to the energy, as for many other quantities to be calculated, are ﬁnite and hence negligible in comparison to the log divergent contribution. A notable exception is the Magnus force that depends on the core behaviour via the core polarization and disappears entirely for in plane vortices. This is a differentiating feature of magnetic vortex dynamics from that of classical ﬂuid or superﬂuid vortices where the Magnus force in those systems depends only on the vortex circulation, the ﬂuid analogue to the magnetic vorticity. To derive the effective vortex forces from the microscopic spin system, we ﬁnd the system energy as a functional of the vortex position, X(t), by integrating over the spin ﬁeld conﬁguration across the 2D plane. The force acting on the vortex is related to the gradient of the energy taken with respect to the vortex center position: F = −∇X E. The bare Magnus force, often called the gyrotropic force in a magnetic system, arises from integrating over the Berry’s phase term related to the precession of the 2D plane of spins required to follow the shifting vortex proﬁle when we let the vortex center change as a function of time [7, 8, 10]. No deformations of the vortex proﬁle are needed to introduce the bare Magnus force, although such deformations

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181

Fig. 1 The gyrotropic force: left, a vortex with p = 1 and q = −1 traveling to the right experiences an upward force; right, a vortex with p = 1 and q = 1 traveling to the right experiences a downward force. Note zˆ is deﬁned out of the page

can, and indeed do, modify the force. Substituting the vortex proﬁle (7), with φ = φV (r − X) and θ = θV (r − X), into the Berry’s phase term (3), and then varying with respect to the vortex position, the Magnus force is

d2r (−S∂t X·∇φV cos θV ) a2 π Spq = 2 zˆ ×∂t X a

FM = ∂X

(8)

The gradient is taken without the usual negative sign since this kinetic term appears in the Lagrangian. This force is derivable from the equivalent Lagrangian term LM = −

π Spq zˆ × X · ∂t X = PM · ∂t X 2a2

(9)

Spq where PM = − π2a ˆ × X is a pure vector potential term analogous to the term 2 z added in the Peierls substitution, ec A. Instead, we could have directly evaluated the associated momentum of the spin ﬁeld

PM =

∂L =− ˙ ∂X

Sd 2 r cos θV ∇φV a2

(10)

with the same result. The vortex dynamics is analogous to that of a charged particle 2 in a perpendicular magnetic ﬁeld, B = 4Sε0 rJV pˆz, where the analogous charge is e → 4πε0 q (set by casting the inter-vortex force into a Coulomb-like form) and the speed of light is replaced by the magnon speed of sound, cs = SJa2/rV . The gauge freedom expected in the vector potential of this effective magnetic ﬁeld corresponds exactly to the gauge freedom in specifying the Berry’s phase.

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We could also allow the uniform polar spin angle to precess in time, δ → δ (t), giving rise to a radial force. Depending on the polarization and direction of rotation, we ﬁnd either a restoring force, drawing the vortex back to the center of our circular system, or a force expelling the vortex to the boundary. To excite a uniform spin precession of the entire system requires application of a external magnetic ﬁeld and hence will not be considered further in this paper. Couplings to magnons introduce an inertial mass and various damping forces. There is a one magnon coupling with the vortex velocity. Normally, considering a central system coupled to perturbative ‘bath’ modes, we would consider the lowest order coupling to be ﬁrst order in magnons, as considered in the treatment of Caldeira and Leggett [12]. There is no such coupling here because the vortex is itself a minimum action solution of the same system in which the magnons arise. Thus, there are no ﬁrst order variational terms. This assumes, however, that the vortex proﬁle is stationary. Allowing it to move about the system introduces a coupling ﬁrst order in the vortex velocity with the magnons via the Berry’s phase term. We truncate this coupling to ﬁrst order in magnons. In a gapped system, a ﬁrst order coupling is ineffective for slow moving vortices. In emitting a single magnon, we see immediately that in order to conserve energy the vortex must be moving with some minimum velocity to surmount the gap in the magnon spectrum. In such cases, the higher order magnon couplings must be considered in place of the ﬁrst order coupling. Consider small perturbations as before, but instead of expanding about the uniform ferromagnetic ground state, we expand about a vortex centered at X(t): θ = θV + ϑ and φ = φV + ϕ to ﬁnd the perturbed magnon excitations. Substituting into the microscopic Lagrangian (4), and expanding in magnon variables, we recover the uncoupled vortex term (9) and uncoupled magnon terms the same as when expanding about the aligned ferromagnetic ground state (although this was not necessarily so), and ﬁnd various coupled vortex-magnon terms. The kinetic term is exactly ﬁrst order in vortex velocity and any approximations come in expanding in magnon variables. The ﬁrst order interacting terms are Lint = −S

d2r (∂t X · ∇φV sin θV ϑ − ∂t X · ∇ϕ cos θV ) a2

(11)

The second term is localized to the vortex core and is hence negligible compared to the ﬁrst term. The second order corrections, both from the kinetic term and from the expanded energy terms will also be neglected here. In the next section, we detail passing from this microscopic interaction between the vortex and magnons to endowing the vortex with an inertial mass and creating effective damping forces.

Effective Magnus Force on a Magnetic Vortex

183

3 Methodology We will follow the prescription developed by Caldeira and Leggett [12] for quantizing Brownian motion using the path integral description of entangled systems of Feynman and Vernon [16]. Whereas they had the freedom of choosing a coupling between the central particle and bath excitations such that Ohmic dissipation ensued, the vortex-magnon coupling is as found in the previous section, dictated by the underlying microscopic equations of motion. We very brieﬂy review the results of their analysis of a linear coupling, Lint = −x ∑k ck rk where x is the coordinate of the central particle and the rk denote the bath modes. The combined system is described by its density matrix whose propagator is expressed in terms of path integrals. The systems are assumed initially separable with the bath excitations in thermal equilibrium (the central particle’s initial state is left unspeciﬁed). After allowing the two sub-systems to entangle after time t and averaging over the ensuing bath motion, the density matrix propagator written as a path integral is then J(x, y,t; x , y , 0) =

D[x]D[y] exp 1 h¯ 2

i

t

τ

0

0

S˜[x(t), y(t)]

ds(x(τ ) − y(τ )) × A(τ − s)(x(s) − y(s))

−

dτ

h¯

(12)

where A(τ − s) = F(τ )F(s) is the correlation of random forces resulting from coupling to the bath. The effective action S˜[x(t), y(t)] is a functional of the forward and backward paths x(t) and y(t) S˜[x(t), y(t)] = S0 [x(t)] − S0 [y(t)] t η ˙ τ ) + x(τ )y( ˙ τ) − d τ (x(τ )x( 2 0 ˙ τ ) − y(τ )y( ˙ τ )) −y(τ )x(

(13)

where S0 [x(t)] denotes the (potentially modiﬁed) uncoupled action of the central particle (ie. the integral over the effective Lagrangian including inertial terms and all potential energy terms) and η is the damping constant, determined by the strength of the couplings to the bath. The mixed terms xy˙ − yx˙ yield equations of motion of the central particle better described by the normal coordinates X = x + y and ξ = x − y, essentially the packet center of motion X and spread ξ . The damping terms introduce damped motion in X(t), as for a classical Brownian particle, and exponential packet spreading. Refer to the original paper for more details [12]. Complications arise when the central particle and bath modes are excitations arising from the same ﬁeld, as is the case for our vortex-magnon system. In the context of quantizing the Brownian motion of a soliton (the vortex is a topological

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L.R. Thompson and P.C.E. Stamp

soliton), Castro Neto and Caldeira [13] modiﬁed the path integral derivation above. Quantizing the motion of a soliton was a well examined topic amongst the particle theorists in the mid-1970’s. A popular method of performing the quantization while avoiding pesky divergences is by introducing so-called collective coordinates (for a good introduction, see the book of Rajaraman [11]). The degrees of freedom of the bath modes bound to the soliton motion that correspond to translation or rotation are absorbed instead by promoting the center coordinate or angular coordinate to a dynamical variable, the so-called collective coordinates. For instance, an expansion about a soliton solution can be written as ∞

σ (x,t) = σS (x) + g ∑ qn (t)ηn (x)

(14)

n=0

or alternatively, after promoting the soliton center coordinate to a collective coordinate, ∞

σ (x,t) = σS (x − X (t)) + g ∑ qn (t)ηn (x − X (t))

(15)

n=1

where, in particular, we note that the n = 0 mode, η0 (x) = σS (x), associated to translational motion of the soliton is omitted, its degrees of freedom having been absorbed by the dynamical variable X (t). Orthogonality of the ηn excitations ensures that the lowest order soliton-bath coupling goes as ∑ ∼ X˙ (t)gnm qn (t)q˙m (t). Castro Neto and Caldeira derive soliton damping forces resulting from this coupling that possess memory effects and are temperature dependent (a direct consequence of the coupling being second order in the bath modes). When we generalize from a scalar order parameter σ to a spin vector order parameter, the Berry’s phase term no longer cancels in ﬁrst order due to orthoganality conditions of magnon modes. Expanding about the vortex proﬁle after promoting the central coordinate to a dynamical variable ∞

φ (r,t) = φV (r − X(t)) + ∑ qn (t)ϕ (r − X(t)) n=1 ∞

θ (r,t) = θV (r − X(t)) + ∑ qn (t)ϑ (r − X(t))

(16)

n=1

where n indexes the magnon normal modes, excluding the translational symmetry mode. We’ve written our solution in the moving frame of reference following the vortex core. The results are identical if we instead refer the magnons to the ﬁxed frame in which the vortex moves instead. Substituting into the microscopic Lagrangian in (4), neglecting second and higher order terms, (0)

L = LV + Lm +

2

d r S (sin θV ϑ ∇φV − cos θV ∇ϕ )·∂t X a2

(17)

Effective Magnus Force on a Magnetic Vortex

185 (0)

where LV is the vortex Lagrangian, so far including only the Magnus term (9), Lm is the unperturbed magnon Lagrangian. The expansion to ﬁrst order in magnon variables has non-zero terms from the Berry’s phase term only and is exactly ﬁrst order in vortex velocity while the Hamiltonian presents no ﬁrst order magnon coupling. In the scalar ﬁeld problem, these ﬁrst order kinetic terms vanish, being an overlap integral of two orthogonal quasiparticle modes, drη0 (r)ηn (r). Our coupling does not reduce simply to an overlap integral so that the lowest order coupling that survives is ﬁrst order in magnons. In particular, we do not expect the damping forces resulting from these couplings to be temperature dependent. As already mentioned we will drop the ϕ linear term since it is much smaller than the other term and introduces no new damping forces. Consider the density matrix of the vortex-magnon combined system, ρ (t), evolving according to

ρ (t) = e−

iHt h¯

ρ (0)e

iHt h¯

(18)

where ρ (0) describes the system initially and we assume is decoupled in the vortex and magnon degrees of freedom; that is, ρ (0) = ρV (0)ρm (0). The magnons are assumed to be in thermal equilibrium initially, so that (0)

e−β Hm ρm (0) = Z

(19)

(0)

where β = 1/kB T , Hm is the vortex-free Hamiltonian and Z is the corresponding partition function. We are only interested in the vortex dynamics, so we trace over the magnon variables to deﬁne a reduced density matrix ρV (t) = trm (ρ (t)). In terms of the collective coordinates X(t), we can write

ρV (X ;Y ;t) = dX

dY J(X, Y,t; X , Y , 0) × ρV (X , Y , 0)

(20)

where J=

X X

D[X]

Y Y

i D[Y] exp (SV [X] − SV [Y])F [X, Y] h¯

is the propagator for the density operator and F =∏

d ϕk d ϕk d ϕ˜ k ρm (ϕk , ϕ˜ k , 0)

k

×

ϕk ϕ˜ k

D[ϕ˜ k ] exp

ϕk ϕk

D[ϕk ]

i Sm+int [ϕk , X] − Sm+int [ϕ˜ k , Y] h¯

(21)

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L.R. Thompson and P.C.E. Stamp

is the so-called inﬂuence functional [16] where Sm+int [ϕk , X] is the action related to the magnons plus ﬁrst order interactions, expressed in terms of the Fourier transformed magnon variables

T

c dt −ϕ˙ k ϑ−k − k2 ϕk ϕ−k 2 0 + Q2 ϑk ϑ−k + f−k ϑk

Sm+int =S

a2 d 2 k (2π )2

(22)

where fk [X] is the Fourier transformed forcing term fk [X] =

˙ · ϕˆ k ik·X 2iπ q X e 2 a kQrV

The magnon perturbed equations of motion derived from this action are 2 ∂ cl

ck ∂ t 0 ϕk = fk [X] ϑkcl − ∂∂t cQ2

(23)

(24)

We solve (24) subject to the boundary conditions ϕk (0) = ϕk and ϕk (t) = ϕk and substitute back into the action Sm+int for the classical contribution to the path integral. With this choice of boundary conditions, we must ﬁrst entirely integrate out the θk dependence. This yields a boundary term left over from completing the square in θk a SBC = = =

a2 d 2 k (2π )2

S π q2 2ca2 T 0

T

fk f−k cQ2

dt 0

T

dk

dt 0

1 ˙ 2 (X) kQ4 rV2

1 ˙ 2 dt MV (X) 2

(25)

where we’ve identiﬁed the effective vortex mass as MV =

RS π q2 rV2 ln 4 Ja a2 + r 2

(26)

V

The quantum ﬂuctuations about the classical solution offer only an unimportant prefactor. Performing the ﬁnal trace integrations, we ﬁnd the inﬂuence functional F = exp −

1 h¯

a2 d 2 k (2π )2

T

t

dt 0

0

ds ( fk [X(t)] − fk [Y(t)])

× (αk (t − s) f−k [X(s)] − αk∗ (t − s) f−k [Y(s)])

(27)

Effective Magnus Force on a Magnetic Vortex

where

αk (t − s) =

187

Sk 2 cos ωk (t − s) e−iωk (t−s) + 2Q eh¯ ωk β − 1 −i

We split the real and imaginary phase terms according to F = e h¯ (Φ −iΓ ) . Comparing with the analysis of Caldeira and Leggett, we expect Φ to modify the effective vortex action to include damping forces while Γ is a measure of the correlations of the corresponding ﬂuctuating forces. Notice that α deﬁned here cannot include the integration over k-states as in their original analysis because we include the non-linear phase terms attached to the vortex velocity operators arising from the two-dimensional nature of our system. The effective vortex action, including the Φ term, is now S˜[X(t), Y(t)] = SV [X(t)] − SV [Y(t)] −

T t

dt ds 0

0

×

2

Sq 2a2

(28)

ik·X(t) ik·Y(t) ˙ ˙ · ϕˆ k d 2 k X(t)e − Y(t)e

sin ωk (t − s) ˙ −ik·X(s) −ik·Y(s) ˙ X(s)e · ϕˆ k + Y(s)e kQ3 rv2

where the vortex uncoupled action SV [X(t)] has been modiﬁed to include inertial terms

T 1 ˙ 2 (29) SV [X(t)] = dt LM + MV (X) 2 0 where LM is given in (9). ˜ Careful variation of this action, δδS X = 0, yields equations of motion in X(t) (or in Y(t), obtained by exchanging X ↔ Y) FM −

d 2 keik·X(t) kQ3 rV2

t

˙ ds ωk cos ωk (t − s)Y(s) ¨ × e−ik·Y(s) · ϕˆ k ϕˆ k = MV X

(30)

0

We dropped all derivatives of the phase term since subsequent integration over k will cancel such terms by symmetry. The ωk cos ωk (t − s) term provides damping forces,

Fig. 2 Deﬁnition of angles for angular part of k integration

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opposing vortex motion. We must integrate the angular portion of the k integral to ﬁnd the direction of these damping forces. Deﬁning angles as in Fig. 2,

sin φk∆ eik∆ cos φk∆ d φk∆ sin φkv − cos φk∆

1 cos φv∆ − cos2φk∆ cos φv∆ eik∆ cos φk∆ = d φk∆ sin φv∆ + cos2φk∆ sin φv∆ 2 = π J0 (k∆ )vˆ − π J2(k∆ )vˆre f l where ∆ = |X(t)−Y(s)|, vˆ denotes the direction along Y(s) and vˆre f l is the reﬂection of vˆ along X(t) − Y(s). We’ve dropped all sin φk∆ terms since they integrate to zero by symmetry. This casts the damping forces into the following form FD = −

t 0

˙ − γ⊥ (t − s, ∆ )Y ˙ re f l (s) ds γ|| (t − s, ∆ )Y(s)

where the two damping coefﬁcients are

γ|| =

S 2 J π q2 2

γ⊥ =

S 2 J π q2 2

dk k

cos ω0 (t − s)J0 (k∆ ) Q2 rV2

(31)

dk k

cos ω0 (t − s)J2 (k∆ ) Q2 rV2

(32)

The equations of motion in X and Y are (nearly) decoupled by introducing the rotated variables R(t) = 12 (X(t) + Y(t)) and ξ (t) = X(t) − Y(t), ˙ ¨ MV R(t) = FM (R) t ˙ + γ||,⊥ (∆XY )Y(s) ˙ ∓ ds γ||,⊥ (∆Y X )X(s)

(33)

MV ξ¨ (t) = FM (ξ˙ ) t ˙ − γ||,⊥ (∆XY )Y(s) ˙ ± ds γ||,⊥ (∆Y X )X(s)

(34)

0

0

where ∆XY = |X(t) − Y(s)| and ∆XY = |Y(t) − X(s)| and we imply a sum over par˙ = allel and perpendicular damping forces. Recall the bare Magnus force, FM (R) π Spq ˙ zˆ × R. Due to the ∆ dependence of the damping coefﬁcients, we have not a2 fully decoupled the motion. However, considering the motion in a classical limit, for instance by letting the quantum packet spread described by ξ go to zero (a valid solution of the coupled equations of motion), the dynamics of the classical center of motion described by R(t) is given by

π Spq ˙ ¨ MV R(t) = 2 zˆ × R a t ˙ − γ⊥ (∆R )R ˙ re f l (s) − ds γ|| (∆R )R(s) 0

(35)

Effective Magnus Force on a Magnetic Vortex

189

Fig. 3 Longitudinal and reﬂected damping forces deﬁned

where ∆R = |R(t)−R(s)| and the re f l subscript now refers to reﬂecting the direction ˙ of R(s) about the line connecting R(s), the vortex position at the earlier time s, with the current vortex position, R(t) (see Fig. 3). Turning our attention now to the real term, Γ , within the exponential of the inﬂuence functional (27). Comparing with the analysis of Caldeira and Leggett, we expect Γ to contain the correlator of the ﬂuctuating forces responsible for the damping forces. After integration by parts in t and s, again dropping derivatives in k that disappear by symmetry after integration, we ﬁnd

Γ=

t ω 2 cos ω (t − s) h¯ ωk β T Sq2d 2 k k cosh dt ds k 2 3 2a 2 kQ rV2 0 0 × X(t)eik·X(t) − Y(t)eik·Y(t) · ϕˆ k × X(s)e−ik·X(s) + Y(s)e−ik·Y(s) · ϕˆ k

(36)

Taking the Markovian limit, that is, of no history dependence, or rather γ (t) → γδ (t) and Γ (t) → Γ δ (t), we ﬁnd that γ and Γ now satisfy the low temperature ﬂuctuation dissipation theorem [17].

4 Many Vortex Systems The analysis above can be generalized to a system of multiple vortices. The vortices are coupled via a 2D Coulomb-like force. Couplings to magnons introduce a nondiagonal mass tensor

π qi q j rv2 RS ln vˆ i · vˆ j (37) Mi j = 4 Ja ri j

1 + ((ˆvi · eˆ ∆ )(ˆv j · eˆ ∆ ) − (ˆvi · eˆ ⊥ )(ˆv j · eˆ ⊥ )) 2 where vˆ i is the unit vector along the velocity of vortex i, and for i = j, ri j → a2 + rv2 and the O(1) terms disappear, so that the diagonal terms are the same as for the isolated vortex. The equations of motion for vortex i will have an inertial term coupling to another vortex j: Mi j V j . The inertial terms are likewise non-diagonal in

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vortex coordinate. The shorthand notation from the single vortex treatment is now understood to denote ∆ = |Ri (t) − R j (s)|. The damping coefﬁcients are otherwise unchanged. The equations of motion for system with more than one vortex is ¨ j− Mi j R

π Sqi pi ˙ i − 8S2J π qi q j Ri j zˆ × R 2 a R2i j T ij ˙ i j ˙ re f l + ds γ0 R =0 j + γ2 R j

(38)

0

where the reﬂection line is now about the line connecting Ri (t) with R j (s). Imagine now we have a collection of vortices forming a two dimensional lattice. To investigate the phonon modes of this lattice, the Tkachenko modes, denote the perturbing lattice motion by x(R) where the position vector R is to be evaluated at the various lattice points. Assuming harmonic perturbations x(R) = ε ei(k·R−ω t)

(39)

where ε gives both the magnitude and polarization of the phonon mode. The equations of motion simplify to

π Sq0 p0 ε × zˆ − fkε a2 − iωγ0k (ω , ε )ε − iωγ20(ω , ε )ε = 0

−Mk ω 2ε + iω

(40)

where we note that the symmetry of any lattice of interest will cancel the inter-site second order damping terms (using that ε Ri j ). We ﬁnd the four-branch dispersion relation, a two-fold degeneracy being broken by introducing damping,

−1 π Sq0 p0 iγk (ω , ε ) ± (41) ω (k) = 2Mk a2 ⎞

π Sq0 p0 2 iγk (ω , ε ) ± ∓ − 4Mk fk ⎠ a2 where the ± appearing in the bracketed terms iγk (ω , ε ) ± π Sqa20 p0 are taken the same and the other ∓ is independent. The mass and inter-vortex force terms are R R π rv2 s s Mk = 4 q20 ln + ∑ q0 qR ln e−ik·R (42) Ja rv R=0 R fk = 8S2J π

q0 qR ∑

R =0

1 − e−ik·R R2

(43)

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191

while the combined damping constant is now (keeping only the highest order contributions) q20 κ πS J 2 (κε ) γk (ω ,ε ) = 2 (44) 2 a Qrv (κ 2 + Q2 ) 0

κR 4q0 qR κ −ik·R J e + ∑ 2 (κ 2 + Q2 ) 0 Qr 2 v R =0 cκ Q=ω

For a given lattice, this can be solved iteratively to suitable accuracy where in the ﬁrst iteration, γ ≈ 0.

5 Results and Conclusions We have derived the equations of motion of a vortex moving within a circular insulating easy plane ferromagnet. We ﬁnd that the damping forces arising from scattering spin waves act both longitudinally and transversely to the vortex motion. Hence, the transverse Magnus force is modiﬁed by the transverse component of the damping force. Although the vortex is a solitonic excitation of the spin ﬁeld, we ﬁnd temperature independent damping in contrast to the temperature dependent damping found for a scalar ﬁeld soliton. The crucial difference between these systems is that in the underlying microscopic system the conjugate momentum is not simply the time derivative of the ‘position’ variable (here we have conjugate variables φ and S cos θ ). The damping force acting on the vortex is the integrated weighted contribution of longitudinal and the so-called reﬂected damping forces over all past motion. These memory effects disappear in the limit of very slow vortex speeds as compared to the spin wave speed of sound cs = c/rV . For speciﬁc examples of vortex motion, consider a vortex moving in a circle (this is in fact the easiest motion to prepare experimentally when we consider the image vortex arising due to proper boundary conditions). At arbitrary time s < t, we see that the reﬂected damping force conspires to lie always exactly to reinforce

Fig. 4 For circular motion, the reﬂected damping force conspires to always reinforce current motion. The effective damping force will in general be dominated by the longitudinal force

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current motion (Fig. 4). The regular longitudinal damping force is much larger, and although the directions at previous times cancel one another, with the diminishing effects for earlier times, this force should still dominate and oppose current motion. For a vortex moving in a straight line, the longitudinal and reﬂected components will completely oppose one another, with the longitudinal of course dominating. In particular, there will be no transverse component to the effective damping for such motion. Numerical studies of vortex motion in a quantum dot applying various damping forces may differentiate motion with varying longitudinal/transverse damping. For comparison with experiment, to isolate these varying effects, it may be necessary to include higher order effects to introduce temperature dependent (and hence experimentally tunable) damping forces. Acknowledgements This work was partially supported by the Natural Sciences and Engineering Research Council of Canada.

References 1. H. E. Hall and W. F. Viven, Proc. R. Soc. London, Ser. A 238, 204 (1956); ibid. 238, 215 (1956) 2. J. M. Duan and A. J. Leggett, Phys. Rev. Lett. 68, 1216 (1992) 3. S. V. Iordanskii, Ann. Phys (NY) 29, 335–349 (1964); and JETP 22, 160–167 (1966) 4. E. B. Sonin, Phys. Rev. B 55, 485 (1997) 5. P. Ao and D. J. Thouless, Phys. Rev. Lett. 70, 2158 (1993) 6. D. J. Thouless, P. Ao and Q. Niu, Phys. Rev. Lett. 76, 3758 (1996) 7. D. L. Huber, Phys. Rev. B 26, 3758 (1982) 8. A. V. Nikiforov and E. B. Sonin, Sov. Phys. JETP 58, 373 (1983) 9. A.A. Thiele, Phys. Rev. Lett. 30, 230 (1973); J. Appl. Phys. 45, 377 (1974) 10. J. C. Slonczewski, from Physics of Magnetic Materials, Eds. J. Rauluszkiewicz et. al., World Scientiﬁc: Singapore (1984) 11. R. Rajaraman, Solitons and instantons, An introduction to solitons and instantons in quantum ﬁeld theory, North Holland Publishing: Amsterdam (1982) 12. A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983) 13. A. H. Castro Neto and A. O. Caldeira, Phys. Rev. E 48, 4037 (1993) 14. G. E. Volovik and V. P. Mineev, JETP 45, 1186 (1977) 15. N. D. Mermin, RMP 51, 591 (1979) 16. R. P. Feynman, F. L. Vernon, Ann. Phys. 24, 118 (1963) 17. R. Kubo, Rep. Prog. Phys. 29, 255 (1966)

Anisotropic Exchange in Spin Chains Dmitry Zakharov, Hans-Albrecht Krug von Nidda, Mikhail Eremin, Joachim Deisenhofer, Rushana Eremina, and Alois Loidl

Abstract Quasi-one-dimensional (1D) spin systems are highly unconventional materials, which exhibit a wide variety of phenomena, including spin-Peierls transition and charge ordering. In this paper, we show that electron spin resonance (ESR) is a very powerful tool to study spin relaxation mechanisms in these systems. We review the microscopic theory of superexchange and include a discussion of some recent experimental and theoretical developments concerning ESR in 1D solids. Furthermore, we evaluate the anisotropy and the temperature dependence of the ESR linewidth in three 1D systems (LiCuVO4 , CuGeO3 , α -NaV2 O5 ). Thus, we can determine the type and the magnitude of the anisotropic exchange interactions between spins and investigate the effects of ﬂuctuations of charge and lattice degrees of freedom in the vicinity of phase transitions in these systems.

D. Zakharov Experimental Physics V, Center for Electronic Correlations and Magnetism, University of Augsburg (EKM), University of Augsburg, 86135 Augsburg, Germany, e-mail: [email protected] H.-A. Krug von Nidda Experimental Physics V, EKM, University of Augsburg, 86135 Augsburg, Germany, e-mail: [email protected] M. Eremin Kazan State University, 420008 Kazan, Russia, e-mail: [email protected] J. Deisenhofer D´epartement de Physique de la Mati`ere Condens´ee, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland, e-mail: [email protected] R. Eremina E. K. Zavoisky Physical Technical Institute, 420029 Kazan, Russia, e-mail: [email protected] A. Loidl Experimental Physics V, EKM, University of Augsburg, 86135 Augsburg, Germany, e-mail: [email protected]

B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V. 2008

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1 Introduction Low-dimensional spin systems, intensively studied experimentally in the last decades, continuously send new challenges to theoretical physics. Despite the fact that the ﬁeld of low-dimensional magnetism developed into one of the most active areas of today’s solid-state physics, the number of unexplained phenomena tends to increase with time. Many of the conventional approaches well established in 3D case should be reconsidered in case of systems with lower dimensionality. In particular, the main source of spin relaxation is under heavy debates during the last couple of years. The original assumption of the dominant role of the antisymmetric anisotropic exchange interaction came strongly into doubt both for theoretical and experimental reasons: An analysis of the high-temperature line broadening in spin S = 1/2 chains showed that this contribution is only of the same level as the contribution of the symmetric anisotropic exchange [1, 2]. Electron spin resonance (ESR) measurements on the quasi one-dimensional antiferromagnet LiCuVO4 revealed a strong line broadening [3], although for symmetry reasons the antisymmetric interaction should vanish. Hence, another source of the line broadening had to be found. The solution of this problem was found only recently, by taking into account exotic effects like quantum interference [4, 3] and new types of anisotropic superexchange [5]. They strongly intensify the symmetric part of anisotropic exchange which was established to be the dominant source of spin relaxation in a number of low-dimensional systems. The aim of this review is to give an insight into the physics of exchange interaction in low-D transition-metal spin systems. In order to explain all quantum effects important for spin relaxation we will consider in detail a microscopic method which allows to deal with the exchange interactions in the most obvious (and correct) way. To give the ﬂavour of the topic of this work it is helpful to write now the general Hamiltonian of the exchange interaction of two spins S on the ions a and b [6] Hex = ∑ Ji j Sa,i Sb, j i, j = {x, y, z} (1) ij

and divide it into the following three parts: Hex = J(Sa · Sb ) +

1 Di j Sa,i Sb, j + Sa, j Sb,i + d · [Sa × Sb ] . ∑ 2 ij

(2)

The ﬁrst term describes the isotropic exchange, the second and third terms – symmetric and antisymmetric parts of the anisotropic exchange, respectively. The coupling constant (exchange integral) J is the largest exchange parameter and can attain values of about 0.01 − 0.1 eV. Therefore, the magnetic state of the system is determined mainly by isotropic exchange. If J > 0 the potential energy is minimal at antiparallel (antiferromagnetic) alignment of interacting spins, in case J < 0 – at parallel (ferromagnetic) ordering. The anisotropic part of exchange interaction plays an important role in the magnetic anisotropy of the system, especially

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in the case S = 1/2, where the single-ion anisotropy Hamiltonian (crystal ﬁeld) vanishes. The energy of the antisymmetric exchange becomes minimal for perpendicular orientation of the spins leading to a small ferromagnetic moment in antiferromagnetic compounds. The symmetric part of anisotropic exchange plays the most important role in the magnetic anisotropy of a number of low-dimensional systems and the exchange constant D can reach values of about 5% of the isotropic coupling J. The isotropic exchange constants are usually obtained by measuring the magnetic susceptibility or by inelastic neutron scattering. As the isotropic exchange is usually orders of magnitude larger than the anisotropic exchange, its contributions cannot be easily revealed by these techniques. However, especially in the case of low-dimensional (low-D) antiferromagnetic S = 1/2 spin systems, the anisotropic contributions can become accessible by means of ESR spectroscopy, because the spin-spin relaxation measured by the ESR linewidth is driven primarily by the corresponding effective local ﬁelds. Therefore, after introducing some theoretical basics in Section 2, we will discuss the main features and theoretical approaches to the ESR absorption necessary for the analysis of the obtained data in Section 3. An experimental overview of three prototypical low-dimensional systems LiCuVO4 , CuGeO3 and NaVO5 will be given in Section 4. We will show the close similarity of their relaxation behavior and identify the dominant source of spin relaxation on the basis of the explicit microscopical analysis of exchange mechanisms.

2 Microscopic Theory of Exchange Interaction 2.1 Exchange Mechanisms Spin-exchange correlations may only occur, if the wave functions of the electrons of interest have a non negligible overlap. In case of direct overlap of the orbitals of neighboring magnetic ions one speaks about direct exchange.

2.1.1 Direct Exchange Interaction The Hamiltonian of direct exchange between the ions a and b has generally the form [7] 1 H P+ 2

1 + PH + H | ψ ψ ψ ψ |P + P| ψ ψ ψ ψ |H . (3) 1 2 1 2 1 2 1 2 ∑ 2 ∑

Hdir = −

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Fig. 1 Schematical representation of two interacting electrons 1 and 2 belonging to the ions a and b, respectively

Here |ψ1 ψ2 ≡ |ψ1 |ψ2 denote the product of the wave functions of interacting electrons, the permutation operator P = ∑i j Pi j interchanges two electrons i and j. Let us discuss the case of one electron on each site (see Fig. 1). The total Hamiltonian then reads H =

p2 Za e2 Zb e2 Zb e2 Za e2 e2 p21 + 2 − − − − + . 2m 2m ra1 rb2 rb1 ra2 r12

(4)

In terms of spin operators the Hamiltonian of direct exchange between two electrons in the states η and ζ reads

ηζ dir 1 + 2 sη · sζ , Hdir = Jηζ (5) 2 dir is determined by [8] where the parameter Jηζ

e2 p2 |η2 ζ1 + 2Reη |ζ η | |ζ − (6) r12 2m

e2 − | η |ζ |2 εη + εζ − η |U2 |η − ζ |U1 |ζ + η1 ζ2 | |η1 ζ2 . r12

dir = −η1 ζ2 | Jηζ

Sηζ ≡ η |ζ represents the overlap integral. U1 and U2 are the Coulomb energies of electrons in the ﬁeld of the other atoms, εη and εζ – the Hartree-Fock energies of the electrons in the states η and ζ , respectively:

2 p1 Za e2 Zb e2 e2 − |η1 . + ζ2 | |ζ2 , εη |η1 ≡ (7) U2 = − rb1 r12 2m ra1 The ﬁrst term in Eq. (6) represents the potential exchange in terminology suggested by Anderson [9]. It is always negative and favors a ferromagnetic spin alignment. dir is negative only in case of zero overlap Detailed calculations show [10, 11] that Jηζ of the neighboring orbitals, i.e. when Sηζ = 0, otherwise the last two terms in Eq. (6) will dominate and lead to an antiferromagnetic coupling. The microscopical expression (6) corresponds to the exchange between a pair of spins. In case of several electrons on one site the exchange operator is usually dir depicted in Fig. 2: approximated by the sum of individual exchange integrals Jηζ ηζ

Hdir = ∑ Hdir . For detailed discussion of this ’channel model’ we refer to [12, 13]. Here we will consider only the case of interacting ions in the ground state what is

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197

Fig. 2 Channel model of exchange interaction [12]. The exchange coupling of each pair of spins Jαα is supposed to be mutually independent that allows their algebraical summation (9)

usually the case in magnetic resonance measurements. According to Hund’s rule they possess the maximal spin value sη = 2S1a Sa , sζ = 2S1 Sb , and, hence, the spin b dependent part of the Hamiltonian of direct exchange reads Hdir = J dir (Sa · Sb ) ,

(8)

where the effective exchange integral is given by J dir =

1 dir Jηζ . 2Sa Sb ∑ ηζ

(9)

The exchange integral (9) scales with the overlap squared and decreases exponentially with the distance r between the spins. Therefore, direct exchange plays a large role for large enough spatially distributed spin systems like radicals [14] and in case of 90◦ metal-ligand-metal exchange geometry. For 3d-ions it becomes negligi˚ In practice, magnetic ions a and b are always separated ble already at r ∼ 2.7 − 3 A. by diamagnetic ions c, so that no appreciable direct overlap is to be expected in this case. This fact led Kramers [15] to propose that a strong admixture of the cation’s and intermediate anion’s wave functions could be invoked to couple the cations indirectly. 2.1.2 Indirect Exchange Mechanisms It is tempting to assume, by analogy with the case of direct exchange, that exchange coupling in case of a three-center system a − c− b will be proportional to the product 2 S2 (κ is the state of the intermediate diamagnetic of the overlap integrals Jηζ ∝ Sηκ κζ ion c). This contribution referred as Yamashita-Kondo’s mechanism is, nevertheless, only marginal, because the overlap integrals Sηκ are usually small compared to the covalency parameters γηκ = tηκ /∆ηκ (tηκ denote hopping integrals (Eq. 67), ∆ηκ is the energy of charge transfer). The hopping integrals tηκ characterize the kinetic energy of electrons or their desire to delocalize. Whenever their repulsion predominates and prevents metallic

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Fig. 3 Schematic representation of the indirect exchange between two magnetic ions a and b via a diamagnetic ion c proposed by (i) Kramers in 1934 [15] and (ii) Pratt in 1955 [16]. The green arrows denote the virtual hoppings of electrons. Numeration corresponds to the sequence of the electron transfers

conduction, the tendency to delocalize can gain energy by spreading electrons into nonorthogonal overlapping orbitals, naturally leading to the antiferromagnetic spin ordering. This problem was ﬁrst considered by Kramers who developed a unique method of handling conﬁguration interactions [15]. The idea by Kramers is displayed in Fig. 3(i). The excited conﬁguration in which an electron has been removed from the nonmagnetic center c and placed on a leaves an unpaired spin on c and b leading to a spin coupling due to the direct overlap of these charge densities. The 2 S2 . magnitude of this type of indirect exchange can be estimated as Jηζ ∝ γηκ κζ Twenty years later Pratt [16] could show that such polarization effects cannot produce the actual antiferromagnetic spin-ordering and proposed another scheme shown in Fig. 3(ii): two electrons of the diamagnetic ion are simultaneously transferred to the magnetic ions providing an antiferromagnetic coupling. However, the energy of this virtual excitation is very large. An attempt to compare the effectiveness of all possible mechanisms of superexchange was made by Yamashita and Kondo [17], but it was not possible to single out the most efﬁcient one. In 1959, Anderson [9] realized that there is a virtual hopping process which requires smaller energy with respect to Pratt’s one. This superexchange process is illustrated in Fig. 4. The energy of the intermediate state with the ionic conﬁguration a+ b− is higher than the energy of the unperturbed one by an amount U = ∆ab . On the other hand, the system gains kinetic energy. This delocalization stabilizes the singlet conﬁguration, because the two electrons at site b must necessarily pair up. The corresponding exchange constant is proportional to the product of covalency parameters 2 γ 2 . Anderson’s approach to the theory of superexchange interactions has Jηζ ∝ γηκ κζ become classical and will be considered in more detail in the next section.

2.2 Isotropic Exchange Interaction The following consideration generalizes Anderson’s approach [9] by accounting directly for the states of an intermediate ion. It is based on the method of canonical transformations which in a natural way allows to derive the constants of exchange

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Fig. 4 Schematic representation of the “superexchange” between two magnetic ions a and b via a diamagnetic ion c proposed by Anderson [9]. The green arrows denote the virtual hoppings of electrons. The Greek letters denote the orbital states of an electron on the corresponding ion

interactions using perturbation theory. The detailed description of this method can be found in the Appendix. Here we will only use the results of these calculations, which are needed to discuss the underlying physical processes. + First, let us introduce some notations: a+ η (aη ) and bζ (bζ ) are the creation (annihilation) operators of electrons on the magnetic ions, c+ κ (cκ ) – on the intervening diamagnetic ions. With η , ϕ , ξ we denote the orbital and spin states of the magnetic ion a [η ≡ (nη lη mη sη )] and with ζ , θ , λ the corresponding states of the magnetic ion b. The letters κ , ρ signify the one-electron states of the diamagnetic ion c.

2.2.1 Classical Concept: Antiferromagnetic Spin Ordering The most effective mechanism of the isotropic superexchange between two magnetic ions a and b via the intervening ion c as ﬁrst described by Anderson [9] is illustrated in Fig. 4. By counting the involved electronic hopping processes, it becomes evident that this mechanism corresponds to the forth order of perturbation theory with respect to the energy of the electron hoppings. Consequently, this process can be described by the Hamiltonian (72), which is derived in the Appendix. The factor of two in the Hamiltonian

tηρ tρζ tζ κ tκη 1 + 2(s − · s ) , (10) Hkin = 2 · η ζ 2∆ ∆ac 2 ab accounts for electron transfer processes from site b to site a as well. Moreover, tηρ denotes the hopping (transfer) integral between the orbital states η and ρ , ∆ab ∼ U and ∆ac are the electron transfer energies from ion a to the cation b and to the anion c, respectively. The expression (10) represents the kinetic exchange in Anderson’s notation and was considered to be intrinsically antiferromagnetic. This type of exchange constitutes an isotropic antiferromagnetic exchange interaction ηζ

Hiso = Jηζ (sη · sζ ) −→ Hiso = J (Sa · Sb ),

(11)

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where the constant of isotropic exchange is given by Jηζ = 4 ·

tηρ tρζ tζ κ tκη 2∆ ∆ac ab

and J =

1 Jηζ . 4Sa Sb ∑ ηζ

(12)

The direct exchange (8) is negligible as compared to the kinetic exchange because of the large distance between the interacting ions. Usually, the antiferromagnetic isotropic exchange is most effective in case of 180◦ bond geometry (all three ions are lying on a straight line). The deviations of the bond angle θ from 180◦ can be described by introducing geometrical factors like, for example, t(θ ) ∝ −tσ cosθ − tπ sinθ . As we will discuss in the following, competition and quantum-interference effects between different exchange paths can occur in some cases and can lead to a strong ferromagnetic coupling of spins [3].

2.2.2 Ferromagnetic Superexchange The ferromagnetic coupling can be obtained considering two major processes: (i) potential exchange and (ii) transfer of the electron to the unoccupied orbital of the neighboring ion. The indirect potential exchange is, as a matter of fact, of the same nature as the corresponding direct process. It describes the electrostatic energy of the system as a spin-spin interaction in the basis of one-electron wave functions. Anderson’s treatment allows to separate the part of the interaction containing the spreading of the orbitals of the d-electrons into the neighboring ions by the modiﬁcation of the local wave functions. The exchange effect then appears as a consequence of the direct overlap of the new longer-ranging orbitals. But this process plays usually a minor role compared to the following one. The second mechanism is shown schematically on the left side of Fig. 5. This is a ﬁfth-order effect of transfer together with the internal exchange coupling (1) Vmm = JH (see Appendix). Therefore, its order of magnitude in comparison with the antiferromagnetic isotropic exchange (11) can be estimated as J FM ≈ −J ·

JH . ∆ab

(13)

Fig. 5 Two possible mechanisms favoring ferromagnetic exchange. Left: transfer of an electron to an empty orbital state of the neighboring ion, the spins will be aligned parallel in accordance with Hund’s rule. Right: Scheme of the electron transfer from the fully-ﬁlled orbital state of ion b into the half-ﬁlled orbital of ion a, the ferromagnetic conﬁguration will be minimized due to Pauli principle as well as Hund’s rule

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Here, JH represents the matrix element of the Coulomb interaction on the involved pair of orbitals. For example JH (dxy , dx2 −y2 ) = C, JH (dxy , dyz ) = JH (dxy , dxz ) = 3B + C, JH (dxy , d3z2 −r2 ) = 4B + C [18], where B and C are the Racah parameters. Anderson estimated JH /∆ab to be about 0.1 ÷ 0.2 and supposed that this interaction is negligible in comparison with the potential exchange [9]. However, the analysis of experimental data has shown the importance of this mechanism for formation of the magnetic structure in many dielectrics: Goodenough [19] has emphasized the key role of empty and ﬁlled orbitals, and proposed two processes depicted in Fig. 5 for the formation of the ferromagnetic ground state. The processes depicted in Fig. 5 dominate when the hopping integrals tκη or tρζ are equal to zero, i.e. the fourth-order exchange processes shown in Fig. 4 are not realized. In the following we will give some examples in which the exchange coupling between the excited states and the ground states exceeds the exchange between ground states by orders of magnitude, favoring a ferromagnetic coupling of spins.

2.2.3 Quantum Interference of Superexchange Interactions As described above the traditional approach to superexchange in dielectrics is based on the three-centre (metal-ligand-metal) model [20]. It is assumed tacitly that the resulting interaction is the algebraical sum of all contributions from the bridging ligands. In this section we would like to emphasize that, in general, superexchange is not additive. In fact, an interference occurs in superexchange via two or more intermediate ions, which can suppress as well as amplify Anderson’s conventional superexchange. This statement is easily to substantiate in terms of the ‘four centers - six electrons’ model. Various sequences of one-electron transfers responsible for superexchange between magnetic ions a and b via the diamagnetic ions c1 and c2 are shown in Fig. 6. The scheme (i) describes Anderson’s additive superexchange through several ligands [20]. The three other sequences correspond to essentially four-centre exchange mechanisms. All the mechanisms speciﬁed contribute naturally to the fourth-order perturbation and therefore their effect should be of the same order of magnitude. The looped exchange of the type (ii) is the interference process to the usual Anderson’s exchange (i). If the product of all transfer integrals is positive, it provides

Fig. 6 Electron-transfer schemes illustrating various superexchange mechanisms via anions c1 and c2 between cations a and b. (i), (ii), (iii) and (iv) refer to transfer sequences

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Fig. 7 Schemes of overlapping orbitals having (i) the same and (ii) different symmetries in the case of two bridging ligands. In the former case the product of transfer integrals for ring-like processes is positive favoring antiferromagnetic coupling, in the latter one it is negative providing a ferromagnetic contribution

an antiferromagnetic contribution. Correspondingly, the contribution becomes ferromagnetic, if the product is negative. Note that, if the ligands c1 and c2 are equivalent, this contribution doubles as well as completely suppresses the Anderson type of exchange [21]. These two extreme cases are shown in Fig. 7(i) and (ii), respectively. This effect is called quantum interference of superexchange interaction. The two other types of exchange processes shown in Fig. 6(iii),(iv) are not reducible to the exchange looped paths. They should be rather called the interference processes with regard to Pratt’s mechanism shown in Fig. 3(ii). However, with the transfer energy of two electrons from one ligand being higher than the transfer energy of electrons from different ligands, mechanisms (iii) and (iv) of Fig. 6 dominate the Pratt’s exchange mechanism. The effective Hamiltonian of this type of exchange has the form [22] H

(iii,iv)

=∑

1 |∆η − κ + ρ + ζ − |

t tηρ ζκ

∆ζ κ

+

t tζ κ tηρ tρζ κη

∆ηρ

∆κη

+

tκη tρζ

∆ρζ

+ · aη a+ η bζ bζ ,

(14) where ρ and κ belong to different anions. Using (14) one can conclude that the exchange in the case Fig. 7(ii) is ferromagnetic and estimate the exchange constant as Jxy,x2 −y2 ∼ −3(∆π + ∆σ )/(∆π ∆σ )2 · tπ2 (tσ )2 . Exact calculations showed that this contribution is considerable and amounts e.g. for the Cu2+ dimers [23] to Jxy,x2 −y2 /kB ≈ −475 K. 2.2.4 Exchange at Small Charge-Transfer Energy In the previous sections we focused on the cases, when the isotropic exchange interaction leads to ferromagnetic coupling of spins. Let us now discuss a further

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generalization of the classical Anderson’s one electron per site theory [20] and show that in multi-electron systems the kinetic exchange is essentially antiferromagnetic only, if the charge transfer energies exceed by far the inter-term splittings. If this condition is not fulﬁlled, the averaging of the transfer energies over all conﬁgurations is not justiﬁed anymore (in other words, the energy denominator in Eq. (68) cannot be factored out). That affects the spin-dependent part of the exchange Hamiltonian which from the standard Anderson’s form (Eq. 10) is transformed to [13, 24] (sm.∆)

Hex

∝ 1−

Sa (Sa + 1) − Sa(Sa + 1) − 34 Sb (Sb + 1) − Sb(Sb + 1) − 34 · (Sa · Sb ) , Sa (Sa + 1) Sb (Sb + 1) (15)

where Sk and Sk denote the spin of the ion k in the ground and excited (with charge transfer) states, respectively. This factor depends on the particular type of excitation. Since we are interested in one-electron transfer, possible excited spin states are Sk = Sk ± 1/2. Therefore, there are only four types of spin operators corresponding to the charge-transfer processes illustrated in Fig. 8. The ﬁrst process corresponds to Anderson’s exchange, and the spin operators in the last three terms have an unusual form. Now the type of magnetic ordering due to kinetic exchange depends now on competing contributions. The second and the third type of exchange processes favor a ferromagnetic coupling, whereas the other ones are responsible for antiferromagnetic spin alignment. The resulting

Fig. 8 Four possible types of one-electron charge transfer. Sa and Sb represent the change of the total spin of the ions a and b, respectively. In the fourth column the corresponding spin part (15) of the exchange Hamiltonian is given

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coupling is determined by the ﬁrst term only in the case of 180◦ exchange between equivalent ions with half-ﬁlled orbitals. Otherwise, couplings of other types may contribute, and the exchange may become ferromagnetic. This conclusion agrees with numerous experimental data. Let us consider the example of a Cu2+ -Mn2+ pair in KZnF3 in order to clarify the previous discussion. Exchange coupling between the ions in the ground state has been found to be antiferromagnetic J/kB ≈ 187 K, whereas Mn2+ excitation to the 4 A state results in ferromagnetic exchange J/k ≈ −144(±7) K [25]. Ferromag1 B netism in the excited state cannot be rationalized within the traditional exchange theory frame. In particular, the Goodenough-Kanamori rules are obviously violated in this case. The pair under consideration is characterized by a relatively low chargetransfer energy (∆ab ≈ 5 eV), which is of the same order of magnitude with the energy of the 4 A1 state (3 eV). Therefore, the exchange interaction should be analyzed using Eq. (15) rather than Eq. (10). The quantitative analysis [13] explained the negative J by showing that the Mn2+ ion being in a state of lower than maximum multiplicity gives rise to the exchange process SCu = SCu − 1/2, SMn = SMn + 1/2 (case 2 in Fig. 8), which is intrinsically ferromagnetic.

2.2.5 Double Exchange: Extreme Case of Free Charge Transfer Having considered the exchange coupling at low energy of transfer, let us discuss shortly the limiting case of free charge transfer. Zener [26] has shown for the ﬁrst time that the indirect coupling of incomplete d-shells via conducting electrons is ferromagnetic. This mechanism called double exchange is explained in Fig. 9 by the example of a Mn4+ -Mn3+ pair. The ﬁrst semiclassical consideration of double exchange was performed by Anderson and Hasegawa [27] in terms of a pure spin model. They concluded that this exchange results in the equidistant energy spectra ES ∝ −|JDE |(S + 1/2). The double exchange is kinematic and therefore cannot be written in form of an effective spin-Hamiltonian as an interaction of spin or orbital moments. But

Fig. 9 The double-exchange schema by the example of a Mn4+ -Mn3+ pair. Both manganese ions are in the spin state with highest multiplicity in compliance with Hund’s rule. Since the eg electron carries its spin unchanged as it wanders from one ion to another, the energy of the pair will be lower when the spins of both ions are parallel

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using the Eq. (64) we can easily write its Hamiltonian in the second quantization representation [28]:

1 1 1 t t b+ a + h.c. a + + (16) HDE = ∑ tζ η b+ ζ η 2 ∑ ∆κζ ∆κη ζ κ κη ζ η Here the ﬁrst term corresponds to the direct hopping of an electron from one magnetic ion to another, the second – to the cascade transfer via intermediate anions. 2 /∆ are usually about The typical values of the exchange constant JDE ≈ tαα αα 0.1 eV. This indirect coupling involving real electron transfer should not be confused with the superexchange. The double exchange is inherently degenerate owing to the presence of magnetic ions of two charges. Therefore, this mechanism is linked inseparably with the electrical conductivity, which is not the case for superexchange. Since we are interested in dielectric materials, we will not account for double exchange in the present work.

2.3 Anisotropic Exchange Interaction The isotropic exchange interaction, which was discussed so far, does not depend on the direction of spins with respect to the crystal axes. Exchange interaction becomes anisotropic, if the exchange Hamiltonian includes the spin-orbit (SO) coupling. Due to SO coupling the excited orbital states are always admixed to the ground state of the electron making it sensitive to the local crystal environment. This admixture can be accounted for in terms of the given perturbation expansion (for details see Appendix) by including an additional V (1) term acting on one site, e.g. (a), – the SO coupling (a) HLS = λa (la · sa ). (17) The SO coupling constant λ is of the order of 102 K [29]. The resulting ﬁfth- or higher order effect will be notably smaller as compared to the four-order isotropic exchange (11) but nevertheless makes a profound impact on magnetism and relaxation dynamics of many compounds. Taking into account a single on-site excitation due to the SO interaction results in the so-called Dzyaloshinsky-Moriya (antisymmetric) anisotropic exchange interaction. The effect of two excitations due to SO coupling yields a part which is symmetric in the spin variables and, therefore, is called symmetric (pseudo-dipole) anisotropic exchange. 2.3.1 Antisymmetric Part of Anisotropic Exchange The combination of the on-site SO coupling HLS and isotropic exchange Hiso can be expressed by the effective Hamiltonian

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Fig. 10 Possible pathways of the DM interaction between the spins in the η and ζ states on the ions a and b, respectively, via the excited orbital states ϕ and θ . The transition due to the SO coupling is represented by the blue dashed line, the electron hoppings with the transfer integral tαα – by the solid green lines (5)

Heff = − ∑ m

|HLS(a) + HLS(b) |m m |Hiso | + |Hiso |m m |HLS(a) + HLS(b) | , (18) ∆mm

where m and m are the ground and the excited states involved into the exchange process, respectively. The term of this Hamiltonian, which is quadratic with respect to the spin variables, can be expressed as a cross-product of spin operators 1 (la · sa )(sa · sb ) = (la · sb ) + i (la [sa × sb ]) . 4

(19)

The ﬁrst term is vanishing whereas the second one gives HDM = d · [Sa × Sb ] .

(20)

In general case, the magnitude of the Dzyaloshinsky-Moriya vector d can be estimated as [30]

1 1 i 1 dj = · (21) + 2Sa Sb ∆ηζ ∆κη ∆ρη ∆κζ ∆ρζ

η |λa l j |ϕ ζ |λb l j |θ · tζ ρ tρη tϕκ tκζ − tηκ tκζ tθ ρ tρη , ∆ηϕ ∆ζ θ where j = {x, y, z}, tαα are the effective hopping integrals between the states |α , |α and a sum over all states of an intermediate ion (|κ , |ρ ) is implied. We also assumed that the charge-transfer energy ∆ab ≡ ∆ηζ from site a to site b is large

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compared to the crystal-ﬁeld splittings ∆cf ≡ ∆ηϕ , ∆ζ θ and the energy of the charge transfer to the intermediate ion ∆ac ≡ ∆κη , ∆ρη ... These virtual hopping processes are displayed schematically in Fig. 10, where e.g. the frame (iii) corresponds to the ﬁrst term of Eq. (21): the electron on site b is transferred to the empty state ϕ at site a and interacts via SO coupling with the electron in the corresponding ground state η . Then, one of the electrons hops to the initial state ζ . The existence of this kind of anisotropic interaction was pointed out by Dzyaloshinsky [32] on the basis of symmetry analysis and the microscopic derivation of the interaction was done by Moriya [33]. Therefore, this interaction is called the antisymmetric exchange interaction of Dzyaloshinsky-Moriya (DM). If the point bisecting the straight line connecting two interacting ions is not a center of inversion, one can expect that the Dzyaloshinsky-Moriya vector d = 0. A very useful rule to determine the direction of the DM vector has been given by Keffer [31, 34]: dab is an axial vector perpendicular to the plane spanned by the interacting magnetic ions a, b and the intermediate diamagnetic ion c dab = dc · [nac × nbc ],

(22)

where the unit vectors nac and nbc connect the spins a and b with the bridging ion c, respectively, as it is shown in Fig. 11. Note that dc is a numerical parameter dependent on the distance between the ions but not on the bond angle. The information about the bond geometry is contained only in the so-called “geometrical factor” Gc = [nac × nbc ] which can be easily calculated. The rough estimation of the strength of DM interaction using ESR can be obtained in the following way: Let us consider e.g. Cu2+ ions placed in octahedra (with a strong tetragonal distortion along the z-axis, corner sharing in the xy-plane with [nac × nbc ] = 0) and estimate the magnitude of the DM vector dz along the z axis. Knowing the ground state η = dx2 −y2 and the corresponding exited one ϕ = dxy of a hole in such an environment, the expression (21) can be simpliﬁed in this case as dz 2i

tϕκ tϕκ 4λ 4λ 4λ λ η |lz |ϕ tϕκ tκζ tζ ρ tρη ·4 = · Jϕζ ζ η = · Jηζ ζ η = ·J , 2 ∆ηϕ ∆ac ∆ab ∆ηϕ ∆ηϕ tηκ ∆ηϕ tηκ (23)

Fig. 11 Direction of the DM vector for the interaction between the ions a and b via the intermediate ion c following Keffer [31]

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where we used η |lz |ϕ = dx2 −y2 |lz |dxy = −2i and the deﬁnition (12) of the exchange integral. The hopping integrals of an electron to the diamagnetic ion from the ground and excited orbital of the magnetic ion are denoted by tηκ and tϕκ , respectively (see Fig. 10). The value of the g factor along the z axis gives us the possibility to estimate λ /∆ηϕ by [29] gz = 2 − 8

λ ∆ηϕ

→

2 − gz 1 ∆ g λ , = ∆ηϕ 8 4 g

(24)

where we have used g = 2 as the g-factor for the free electron value and ∆ g denotes the deviation of the experimental g-value from the pure spin value. Hence, the magnitude of the Dzyaloshinsky-Moriya vector can be estimated as |d| = dz

∆ g tϕκ ·J . g tηκ

(25)

Then the well-known expression by Moriya [33] |d| ≈

∆g · J, g

(26)

(J is the integral of the usual isotropic exchange) can be obtained using the assumption of equal hopping integrals between the ground tηκ and excited tϕκ levels (see Fig. 10).

2.3.2 Symmetric Anisotropic Exchange Interaction Let us consider now the next order of perturbation theory taking into account four virtual electron transfers and twice the effect of SO coupling. As we will show later, in case of orbital order and reduced dimensionality this type of exchange can dominate all other anisotropic spin-relaxation mechanisms. Proceeding just as described in the previous section, we rewrite the general Hamiltonian (6)

Heff =

∑

mm

(a)

(a)

|HLS |m m |Hiso |m m |HLS | , ∆mm ∆mm

(27)

considering only the terms which are quadratic in the spin variables, 1 la · sa sa · sb la · sa −→ ∑ la,i la, j sa,i sb, j + sa, j sb,i 4 ij

(28)

λa2 η |li |ϕ tϕρ tρζ tζ κ tκξ ξ |l j |η Sa,i Sb, j + Sa, j Sb,i . 2∆ 8Sa Sb ∑ ∆ηϕ ∆ac ∆ ηϕ ab

(29)

as (i) = HAE

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Fig. 12 Possible paths for the symmetric anisotropic exchange between two sites a (with ground state |η , exited states |ϕ , |ξ ) and b (|ζ and |θ , |λ , respectively). Solid arrows correspond to the effective hopping integrals, dashed arrows indicate the matrix elements of the spin-orbit coupling. The process of type (v) is represented in more detail at the bottom, where all hopping integrals tαα and matrix elements of SO coupling α |li |α are explicitly indicated

Introducing the exchange parameter D of the AE this expression can be written in the form HAE = Sa · D · Sb.

(30)

Being a tensor of second rank, D does not allow for a simple graphical illustration. This process was considered ﬁrst by Bleaney and Bowers in 1952 [35] in case of one electron placed on each site. The schematic graphical representation of this process is given in Fig. 12(i). This picture represents the conventional symmetric anisotropic exchange (AE) process [35, 36]: electron at site a is excited via SO

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coupling from the ground state η into the state ϕ , is transferred to the ground state ζ at site b and then retraces its steps returning to the initial state. It is obvious that the SO coupling can take place on the ion b as well. That case is represented in Fig. 12(ii), the respective Hamiltonian can be obtained from (29) by substituting {a, η , ϕ , ξ } → {b, ζ , θ , λ }. All other processes shown in Fig. 12 were pointed out by Eremin et al. [5]. All such combinations of the interaction operators lead to the effective symmetric anisotropic exchange as well. Then the general expression for the exchange constant of AE between the ions a and b reads ξ |λa l j |η 1 η |λa li |ϕ Di j (η → ζ ) = tϕρ tρζ tζ κ tκξ + 2 2∆ab ∆ac ∆ηϕ ∆ηϕ + tϕρ tρλ

ζ |λb l j |θ η |λa li |ϕ λ |λb li |ζ ζ |λb l j |θ tθ κ tκη + tϕρ tρζ tθ κ tκη + ∆ζ λ ∆ζ θ ∆ηϕ ∆ζ θ

ϕ |λa l j |η ϕ |λa l j |η θ |λb li |ζ ζ |λb li |θ tζ κ tκϕ + tηρ tρζ tθ κ tκϕ + ∆ζ θ ∆ηϕ ∆ζ θ ∆ηϕ

θ |λb l j |ζ η |λa li |ϕ (31) + tϕρ tρθ tζ κ tκη . ∆ηϕ ∆ζ θ

+ tηρ tρθ

Here Sa = Sb = 1/2 and the sum over all possible excited states is implied. The exchange constant Di j (ζ → η ) for the reverse processes b → a has the same magnitude in case of equal ions a and b. The Moriya relation [33] for D ∼ (∆ g/g)2 J implies that the transfer integrals 2 . in the ground and excited states are of the same order of magnitude, i.e. D ∝ tηζ Here, we would like to emphasize that the actual transfer integrals which enter into expression (31) are given by 2 D(i-iv) ∝ tϕζ ,

D(v,vi) ∝ tζ η tθ ϕ .

(32)

These terms can deviate considerable from tηζ . For example, in case of two bridging ions (no 90◦ -exchange) shown in Fig. 7(ii) the transfer integral in the exited state tϕζ = txy,x2 −y2 exceeds the transfer integral in the ground state tηζ = tx2 −y2 ,x2 −y2 by orders of magnitude. Moreover, the exchange in the excited state is strongly ferromagnetic clearly showing that the approximate relation D ∼ (∆ g/g)2 J neither gives the correct order of magnitude nor the right sign of the interaction. The generalization of the classical AE schema is of particular importance in the case of spin-chain systems. This is related to the fact that in linear chain systems the overlap between the ground and excited states is negligible tϕζ ≈ 0. Therefore, the ‘ring-exchange’ processes (Fig. 12(v,vi)) containing the electron transfer between states of the same symmetry play a major role. The foregoing discussion shows that estimations of exchange parameters, especially in low-dimensional systems, demand special care. For a reliable estimation microscopic considerations of involved orbital states and exchange paths are

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necessary. In the following such an analysis will be presented for several lowdimensional systems, which evidence that AE constitutes one of the main sources of spin relaxation.

3 Electron Spin Resonance in Strongly Exchange Coupled Spin Systems The following short introduction to the magnetic resonance of exchange coupled spins is conﬁned to the basics necessary to understand the results presented in this work. A fundamental introduction to electron spin resonance is given for example by Pake and Estle [37]. The ESR of diluted transition-metal ions is described in detail by Abragam and Bleaney [29] as well as in the book of Altshuler and Kosyrev [38]. A very useful review on the development of ESR experiments and theory in exchange coupled systems before the year 1990 is provided by Bencini and Gatteschi [39].

3.1 Principle of Magnetic Resonance → An electron spin s interacts via its magnetic moment − µ = −gµB s with an external static magnetic ﬁeld H, resulting in an equidistant splitting of the 2s + 1 spin eigenstates, which is known as Zeeman effect. The corresponding Hamiltonian is given by → HZee = −− µ · H = gµB s · H = gµB Hsz (33) where µB = (e¯h)/(2mc) denotes the Bohr magneton with elementary charge e, electron mass m, Planck’s constant h¯ , and light velocity c. Here cgs units are used to be compatible with the majority of ESR literature. For a free electron the g value is g = 2.0023. Choosing the direction of the static ﬁeld along the z axis, the eigen energies read Ems = gµB Hms , −s ≤ ms ≤ s. (34) A magnetic microwave ﬁeld h cos(ω t) of frequency ν = ω /(2π ) and amplitude h H applied transversal to the static ﬁeld, e.g. along the x direction, results in a perturbation 1 HMW = gµB h cos(ω t)sx = gµB h cos(ω t) (s+ + s− ). 2

(35)

The spin operators s+ = sx + isy and s− = sx − isy increase respectively decrease the magnetic quantum number ms by 1 and, therefore, induce dipolar transitions ∆ ms = ±1 between neighboring Zeeman levels, if the microwave energy equals the

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corresponding energy difference. Resonance absorption takes place at the Larmor frequency ω0 = gµB H/¯h = γ H. (36) Another association of the Larmor frequency is obtained from the equation of motion of the spin operator in the Heisenberg picture: i ds = [HZee , s] = −γ [s × H]. dt h¯

(37)

The spin feels a torque from the magnetic ﬁeld and, hence, precesses with the Larmor frequency around the ﬁeld direction like a mechanical gyroscope in the gravitation ﬁeld.

3.2 Magnetic Resonance of Interacting Spins So far uncoupled spins have been considered, only interacting with the magnetic ﬁeld. If the spins are coupled by interactions, the physics becomes much more complicated, but generally resonance absorption still occurs also in the interacting system. For a system of exchange coupled spins the Hamiltonian can be written as H = HZee + Hiso + Hint ,

(38)

where Hiso denotes the sum over the isotropic Heisenberg exchange interaction J(sa · sb ) between every two spins at sites a and b. The third contribution Hint includes all other interactions like dipole-dipole interaction, symmetric and antisymmetric anisotropic exchange, crystal-electric ﬁeld and hyperﬁne interactions. Then the equation of motion for the magnetization, i.e. the total spin S = ∑i si reads i i dS = [H , S] = [HZee , S] + [Hint , S] . dt h¯ h¯

(39)

Note that the Heisenberg exchange interaction does not contribute to the equation of motion, because due to its isotropy it commutes with the total spin [Hiso , S] = 0. Thus, ESR is well suited to detect anisotropic exchange contributions. The ﬁrst commutator in Eq. (39) again describes the Larmor precession due to the torque of the magnetic ﬁeld. The second commutator contains stochastic torques, which every spin feels from spatially ﬂuctuating local anisotropic interactions, and, thus, results in a Gaussian distribution of resonance ﬁelds characterized by its second moment M2 . If the isotropic exchange √ interaction can be neglected, the corresponding linewidth is given by ∆ H = 2 ln 2M2 (so called slow-exchange regime). But if the isotropic exchange interaction is strong enough, it acts like a fast motion of the spins averaging out the local ﬁelds and, therefore, narrows the Gaussian shape spectrum into a Lorentzian line (fast-exchange regime). The so called exchange

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frequency ωex of the virtual hopping limits the spin-correlation time τ = 1/ωex . The corresponding theory of exchange narrowing was developed by van Vleck [40] and treated more generally by Anderson and Weiss [41], and by Kubo and Tomita [42]. Using time dependent perturbation theory [42] under the assumption HZee + Hiso = H0 Hint the general expression for the linewidth in high-temperature approximation reads

∆ H(T ) =

h¯ M2 (T ) gµB ωex (T )

(40)

with the second moment M2 due to the stochastic torques M2 (T ) =

1 [Hint , S+ ][S− , Hint ] , S+ S− h¯ 2

(41)

where S± = Sx ± iSy . The denominator consists of the thermal average of the transverse magnetization which is determined by the static susceptibility χ (T ) of the spin system via 2 χ (T ) S+ S− = S(S + 1) 3 χ0(T )

(42)

where χ0 denotes the bare Curie susceptibility. For non interacting spins the susceptibility χ (T ) follows a pure Curie law χ = χ0 ∝ 1/T as well and, hence, S+ S− is temperature independent. In the case of interacting spins the temperature dependence characterizes the correlations. The angular dependence of the second moment (polar angle ϑ and azimuthal angle ϕ ) with respect to the external magnetic ﬁeld reﬂects the characteristic anisotropy of the underlying perturbation. The exchange frequency is approximated by ωex ∼ J/¯h. Due to the fact that these results are obtained in the high-temperature approximation, when all states are equally populated, additional considerations are necessary concerning the temperature regime, where this approximation fails. In strongly exchange-coupled systems this approximation is only valid at temperatures exceeding the exchange-integral value J. For three dimensional magnets this restriction is not very severe, because at temperatures kB T ∼ J long-range order is established. Only on approaching magnetic order in the regime of critical ﬂuctuations the temperature dependence of second moment and exchange frequency have to be treated with care. However, in one-dimensional magnets long-range order, if it exists, is established only due to the weak inter-chain interactions at kB T J. Thus, there is a wide temperature range in the paramagnetic state where antiferromagnetic correlations have to be taken into account.

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Fig. 13 Temperature dependence of the linewidth taken from Oshikawa and Afﬂeck [2] for relaxation via (a) symmetric anisotropic exchange or (b) antisymmetric staggered DzyaloshinskyMoriya interaction

3.3 One-Dimensional Magnets An expansion of the ESR-linewidth theory to the case of one-dimensional systems is a very difﬁcult task up to now (cf. [39] and references therein). Besides the question, how to treat appropriately the temperature regime kB T < J, even in high-temperature approximation the general problem appears that the spin correlations do not decay fast enough that the correlation time can be approximated by the inverse exchange frequency. The corresponding line shape is intermediate between Lorentzian (fast exchange regime) and Gaussian (slow exchange regime). However this pronounced deviation from the Lorentzian shape is only observed, if the inter-chain exchange interaction J is negligible as for example in tetramethylammonium manganese trichloride TMMC [43, 44], where J is more than four orders of magnitude smaller than J. But in the quasi one-dimensional transitionmetal oxides treated in the present work, the inter-chain exchange J is usually only about one or two orders of magnitude smaller than the intra-chain exchange J, e.g. in CuGeO3 J /J ≈ 0.1. Already quite small inter-chain exchange (J /J ∼ 10−3) recovers the Lorentzian shape of the ESR line with renormalized exchange frequency h¯ ωex = J (J /J)1/3 [45]. Thus, for quasi-one dimensional systems with sufﬁciently strong inter-chain interaction J , the Kubo-Tomita approach works at least in hightemperature approximation kB T J, but the experimentally observed linewidth may exceed the expected value estimated from the exchange frequency ωex ∼ J/¯h. An expansion of the ESR-linewidth theory to the case of low-dimensional systems at temperatures kB T < J was performed e.g. by Soos [46] who included the temperature dependence of the spin-correlation functions within the traditional formalism. Recently, Oshikawa and Afﬂeck [2] used ﬁeld-theory methods to derive the temperature dependence of the ESR linewidth deeply in the one-dimensional regime (kB T J). One relevant result of these papers is that, if the ESR linewidth is due to a single anisotropic interaction, then the temperature dependence of the linewidth is an isotropic function. This means that for all temperatures the relative anisotropy of the ESR linewidth ∆ H should be the same as in the high-temperature approximation T → ∞, i.e.

Anisotropic Exchange in Spin Chains

∆ H(T, ϑ , ϕ ) = F(T )∆ H(ϑ , ϕ )|T →∞ .

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(43)

The isotropic function F(T ) approaches unity at high temperatures T J/kB . The behavior of the function F(T ) at ﬁnite temperatures depends on the origin of the anisotropic interaction which causes the line broadening (cf. Fig 13). For the case of symmetric anisotropic exchange interaction, F(T ) was predicted to increase linearly with increasing temperature at kB T J and to saturate at high temperatures kB T J, while for the antisymmetric Dzyaloshinsky-Moriya (DM) exchange interaction a divergence ∝ 1/T 2 is expected at low temperatures [2]. The exact temperature dependence of F(T ) is unknown. However, Eqs. (40) and (43) allow us to write down a simple expression for the normalized linewidth M2 (ϑ , ϕ )|T →∞ ∆ H(T, ϑ , ϕ ) ∆ H(ϑ , ϕ )|T →∞ = = . ∆ H(T, H l) ∆ H(H l)|T →∞ M2 (H l)|T →∞

(44)

Here l is some ﬁxed direction. As it follows from Eq. (44), the normalized linewidth should be temperature independent as long as only one type of interaction is responsible for it. It is also free from the uncertainty in the deﬁnition of the exchange frequency ωex . The price of this simplicity is that we can obtain only ratios of microscopic Hamiltonian parameters, once we express second moments in (44).

4 Anisotropic Exchange in Low-Dimensional Systems 4.1 Universal Relaxation Behavior The three quasi-one dimensional antiferromagnets LiCuVO4 , CuGeO3 , and NaV2 O5 exhibit a quite similar phenomenological behavior from the ESR point of view. In all three compounds the ESR spectrum consists of a single exchange-narrowed resonance line, which is well described by a Lorentzian line shape. Figures 14–16 summarize the temperature dependencies of spin-susceptibility χESR , resonanceﬁeld Hres and linewidth ∆ H in LiCuVO4 , CuGeO3 , and NaV2 O5 , respectively, as measured at X -band microwave frequency ν = ω /2π ≈ 9.4 GHz. The prototypical spin-chain compound LiCuVO4 crystallizes within an orthorhombically distorted inverse spinel structure, where the magnetic chains are built by equivalent edge-sharing CuO6 octahedra along the b direction [47, 48]. As depicted in the upper frame of Fig. 14, above 20 K the spin susceptibility is well described by a Bonner-Fisher law [49] with its characteristic maximum near 28 K and a corresponding Heisenberg-exchange constant J/kB = 45 K. To lower temperatures the susceptibility is enhanced with respect to the Bonner-Fisher ﬁt and indicates antiferromagnetic order at TN = 2.1 K [50, 51]. In CuGeO3 similar chains of edge-sharing CuO6 octahedra aligned along the c axis [52, 53] determine the quasi-one-dimensional magnetic properties (J/kB = 120 K). The two oxygen

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octahedra within the orthorhombic unit cell share the apical oxygen. Hence, there are two inequivalent chains which differ by the orientation of the octahedron axis in the ab plane and give rise to a sizable inter-chain exchange of about 0.1J. The susceptibility, which is shown in the upper frame of Fig. 15, exhibits a maximum near 50 K but fails to follow a Bonner-Fisher law, because of non-negligible next-nearest neighbor interactions [54]. Below the spin-Peierls transition TSP ≈ 14 K [55] the susceptibility vanishes exponentially due to the singlet formation on dimerization. From a mean-ﬁeld ﬁt, which is illustrated in the Inset of Fig. 15, an excitation gap of ∆ (T = 0)/kB = 22.4 K is obtained in good agreement with literature [56]. In the quarter-ﬁlled spin-ladder system NaV2 O5 the rungs consist of two VO5 pyramids along the a axis of the orthorhombic structure, which share one corner at their base [57, 58]. Each rung carrying one electron is connected with the neighboring rungs via additional corners at the pyramids’ bases forming the ladders along the b axis. As shown in the upper frame of Fig. 16, the susceptibility is approximately described by a Bonner-Fisher law with J/kB = 580 K but deviates to smaller values at temperatures lower than 200 K [59]. Below TCO ≈ 34 K (cf. Inset of Fig. 16) the susceptibility vanishes exponentially due to the opening of a spin gap of about ∆ (T = 0)/kB = 98 K driven by charge order [60]. The middle frames of Figs. 14–16 depict the temperature dependence of resonance ﬁelds in LiCuVO4 , CuGeO3 , and NaV2 O5 for the static magnetic ﬁeld applied along the three crystallographic axes. In all three one-dimensional systems under consideration the resonance ﬁelds reveal a distinct anisotropy and change only weakly with temperature, thus indicating stable orbital conﬁgurations. The corresponding g values obtained via the Larmor-resonance condition ω = γ Hres deviate only about 10% from the free-electron value ge = 2.0023, because the orbital moment is nearly quenched by the strong ligand ﬁeld of the oxygen ions surrounding the 3d transition-metal ion. The residual spin-orbit coupling gives rise to a positive g shift for a more than half ﬁlled d shell and to a negative g shift for a less than half ﬁlled d shell: In LiCuVO4 the nearly tetragonal g-tensor ga = 2.07, gb = 2.10 and gc = 2.31 [3] is typical for one hole occupying the 3dx2 −y2 orbital with the strongest g shift along the elongated c direction of the CuO6 octahedron. The local g-tensor in CuGeO3 is similar to that in LiCuVO4 , but the sum over the two equivalent Cu sites results in the orthorhombic tensor ga = 2.16, gb = 2.26, and gc = 2.07 [61]. In NaV2 O5 one observes again a nearly tetragonal g tensor ga = 1.979, gb = 1.977, and gc = 1.938 [59] identifying the ground state of the single electron as the 3dxy orbital in the basal plane of the pyramid which coincides with the crystallographic ab plane plane. The lower frames in Figs. 14–16 present the detailed temperature dependencies of the ESR linewidth in the three model compounds for the static magnetic ﬁeld applied along the three crystallographic axes. For all compounds the linewidth increases monotonously with increasing temperature in the paramagnetic regime and approaches saturation in the high-temperature limit. Differences appear only at low temperatures, where the linewidth increases again below the spin-Peierls

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∆

χ

Anisotropic Exchange in Spin Chains

ν

Fig. 14 Temperature dependence of spin susceptibility χESR (upper frame), resonance ﬁeld Hres (middle frame), and linewidth ∆ H (lower frame) for the external ﬁeld applied along the three orthorhombic crystal axes in LiCuVO4 . The solid line approximates χESR (T ) by a Bonner-Fisher law with J/kB = 45 K, the dashed (thin solid) lines ﬁt ∆ H(T ) by Eq. (45) with C1 = 60 ± 5 K and C2 = 15 ± 5 K without (with) a critical contribution (T − TN )−α , where α = 0.55 ± 0.05

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∆

χ

∆

ν

Fig. 15 Temperature dependence of spin susceptibility χESR (upper frame), resonance ﬁeld Hres (middle frame), and linewidth ∆ H (lower frame) for the external ﬁeld applied along the three orthorhombic crystal axes in CuGeO3 . The dashed lines ﬁt ∆ H(T ) by Eq. (45) with C1 = 235 ± 5 K and C2 = 40 ± 2 K. The inset shows a mean-ﬁeld approximation of the susceptibility below TSP with a temperature dependent energy gap ∆ (T )

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∆

χ

∆

ν

Fig. 16 Temperature dependence of spin susceptibility χESR (upper frame), resonance ﬁeld Hres (middle frame), and linewidth ∆ H (lower frame) for the external ﬁeld applied along the three orthorhombic crystal axes in NaV2 O5 . The solid line approximates χESR (T ) by a Bonner-Fisher law with J/kB = 580 K, the dashed lines ﬁt ∆ H(T ) by Eq. (45) with C1 = 420 ± 20 K and C2 = 80 ± 10 K. The inset shows a mean-ﬁeld approximation of the susceptibility below TCO with a temperature dependent energy gap ∆ (T )

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transition in CuGeO3 and below the charge-order transition in NaV2 O5 , because the singlet formation inhibits the exchange narrowing of the residual magnetic centers, but diverges already for T < 20 K in LiCuVO4 due to the critical spin ﬂuctuations on approaching magnetic order. In all three compounds it is possible to parametrize the temperature dependence of the linewidth above the transition into the ground state by the empirical expression [62]

C1 , (45) ∆ H(T ) = ∆ H(∞) exp − T + C2 where ∆ H(∞), C1 , and C2 are treated as ﬁtting parameters. For LiCuVO4 an additional critical divergence at TN has been added typical for the onset of magnetic order. Note that the parameter C1 corresponds to the order of magnitude of the isotropic exchange constant. This is reasonable, because the parameter C1 indicates the transition from the strongly correlated one-dimensional regime at low temperatures T J/kB to the purely paramagnetic regime T J/kB , where the high-temperature approximation is valid. The parameter C2 indicates the inﬂuence of the low-temperature phase-transition on the line broadening. But it is necessary to recall that this purely empirical parametrization has no underlying microscopic picture, yet. Nevertheless, the general feature of the observed temperature dependence, i.e. an approximately linear increase at temperatures small compared to the exchange constant T J/kB but large compared to any phase-transition temperature and a saturation behavior in the high-temperature regime T J/kB coincides with the theoretically expected temperature dependence for the case of dominant relaxation via symmetric anisotropic exchange.

4.2 Analysis of the Exchange Paths 4.2.1 LiCuVO4 In LiCuVO4 the basic exchange geometry corresponds to the situation depicted in Fig. 7. Within the spin chain neighboring copper sites are connected via two symmetrical oxygen bridges spanning the xy plane. For Cu2+ with 3d 9 electron conﬁguration there is a single hole with S = 1/2 in the 3dx2 −y2 ground state. As explained in Section 2.2.3, quantum interference plays an important role in this bond geometry. Due to symmetry reasons it is immediately clear that the antisymmetric DM interaction vanishes, because the contributions of both oxygen bridges exactly cancel each other. Hence, only the symmetric anisotropic exchange has to be considered: For this purpose we change from the usual coordinate system, where the Cartesian axes point from the center to the corners of the CuO6 octahedra into another Cartesian coordinate system, where the AE tensor becomes diagonal [39],

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i.e. we rotate the xy plane by 45◦ around the z axis that the y axis coincides with connection of both Cu ions, i.e. the chain direction along the crystallographic b axis. In this new system 3dxy denotes the ground state and 3dx2 −y2 the excited state. For the symmetric anisotropic exchange the leading contribution arises from the ring processes of Figs. 6(c) and (d) between the 3dx2 −y2 excited state of one Cu ion and 3dxy ground state of its neighboring Cu ion. The excited 3dx2 −y2 state is connected to the ground state 3dxy of the same Cu ion via spin-orbit coupling (17). There is only one matrix element < xy | lz | x2 − y2 >= −2i of the lz operator, which connects the 3dx2 −y2 and 3dxy states. Therefore this process contributes to the zz diagonal component of the symmetric anisotropic exchange tensor only, yielding the parameter Dzz = 2

λ

∆x2 −y2 ,xy

2 Jx2 −y2 ,xy =

1 (gzz − 2)2 Jx2 −y2 ,xy , 32

(46)

where ∆x2 −y2 ,xy denotes the energy difference between 3dx2 −y2 ground state and 3dxy excited state. Using the experimental value gzz = 2.31 and the exchange parameter derived for Cu dimers Jx2 −y2 ,xy /kB = −475 K [23] one obtains Dzz /kB ≈ −2 K. The other components Dxx and Dyy as well as the contributions of the dipolar interaction have been shown to be one order of magnitude smaller than Dzz [3]. The general calculation of the second moment M2 of symmetric anisotropic exchange can be found in literature [46, 61]. Neglecting the anisotropy of the g tensor and with only Dzz = 0 one obtains 4 M2 (ϑ ) = S(S + 1)(cos2 ϑ + 1)(Dzz )2 3

(47)

where ϑ denotes the polar angle between the magnetic ﬁeld H and the z direction which coincides with the crystallographic c axis in LiCuVO4 . Inserted in the saturation value of Eq. (40) with S = 1/2 and J/kB = 45 K, this yields a linewidth ∆ H∞ (ϑ = 0) ≈ 1 kOe which is twice as large for the magnetic ﬁeld applied parallel to the crystallographic c axis compared to the perpendicular orientation. This agrees well with the experimental data at high temperatures shown in Fig. 14. The residual anisotropy in the ab plane is due to the smaller matrix elements of anisotropic exchange and dipole-dipole interactions. Thus, in LiCuVO4 the spin relaxation is satisfactorily explained in terms of a dominant symmetric anisotropic exchange. The maximum linewidth appears for the ﬁeld applied along the direction of the leading matrix element of the diagonalized exchange tensor. The upper row in Fig. 17 shows the full angular dependence of the resonance linewidth for the three crystallographic planes ab, ac, and bc in the high-temperature limit at 200 K. The solid lines were obtained using the complete expression for the second moment with all matrix elements [3]. The ﬁtting parameters Dxx /kB = 0.16 K, Dyy /kB = −0.02 K, and Dzz /kB = −1.75 K nicely agree with the theoretical estimation. It turns out that the symmetric AE parameter Dzz is indeed the dominant contribution, whereas Dxx can be understood in terms of dipole-dipole interaction,

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Fig. 17 Angular dependence of the ESR-linewidth in LiCuVO4 (upper row), CuGeO3 (middle row), and NaV2 O5 (lower row) described in terms of symmetric anisotropic exchange. The relevant parameters are indicated in the ﬁgures

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alone. Concerning the parameter Dyy , which was difﬁcult to estimate, the experiment indicates that both dipolar interaction and anisotropic exchange nearly cancel each other. Remarkably, the largest anisotropic contribution to the superexchange amounts to about 5% of the leading isotropic coupling J/kB ≈ 45 K. Hence, it is three times larger than the conventional estimate of the anisotropy D ∼ (∆ g/g)2 J, where ∆ g denotes the shift of the g factor from the spin-only value [33]. In terms of the linewidth the latter estimate yields a value, which is an order of magnitude smaller compared to the experimentally observed one [63]. That comparison of the theoretical predictions with the experimental data underlines the importance of the particular bonding geometry for the magnetic anisotropy of low-dimensional transition-metal oxides Based on this fundamental analysis of the symmetric anisotropic exchange in the chains of corner sharing CuO6 octahedra of LiCuVO4 it is possible to consider the more involved situation in CuGeO3 which contains two such chains interconnected via the apical oxygen ions of the octahedra.

4.2.2 CuGeO3 The intra-chain anisotropic exchange in CuGeO3 is comparable to that in LiCuVO4 . Hence, at ﬁrst glance one is tempted to describe the anisotropy of the linewidth in CuGeO3 just by the superposition of the single-chain contributions. However, it turns out that this is not enough to understand the experimental data, because the inter-chain contributions to the anisotropic exchange cannot be neglected. The schematic pathways of the possible spin-spin coupling between two neighboring chains are presented in Fig. 18. Again like introduced in LiCuVO4 , the local coordinate system of each chain is rotated by 45◦ with respect to the usual system. The y axis runs along the chain direction, i.e. the crystallographic c axis. The hole of the 3d 9 electron conﬁguration of Cu2+ is located in the ground state 3dxy (i). In the middle picture (ii) the excited 3dyz states of both copper ions couple via

Fig. 18 Schematic pathways of the exchange coupling in CuGeO3 (blue spheres: Cu, red spheres: O) y axis running along the chains, local z axis perpendicular to the corresponding CuO2 ribbons (i) In the ground state the hole occupies the 3dxy orbitals. Anisotropic spin-spin coupling between copper 3dxy states via (ii) ﬁlled 3dyz and oxygen 2pcy states – the 3dxy and 3dyz states are coupled via intra-atomic Coulomb interaction as well as due to SO coupling – (iii) occupied 3dz2 , 3dxy and 2p-oxygen states – 3dz2 and 3dxy orbitals are coupled via intra-atomic Coulomb repulsion, whereas 3dxy and 3dxz due to SO coupling

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the bridge-oxygen 2py orbital. In the right picture (iii) the excited 3dz2 state of one copper ion is coupled to the excited 3dxz state of the other copper ion via a linear combination of the bridge-oxygen 3px and 3pz orbitals. To determine the relevant components of the anisotropic exchange tensor, we have to consider the matrix elements of the spin-orbit coupling between the ground state and excited states of the copper ion. In the ﬁrst case (Fig. 18(ii)) the matrix elements xy|lx |yz = xy|lz |yz = 0 vanish and only the ly component of the orbital momentum yields a nonzero exchange parameter Dinter yy and spin projection sy . In the second case (Fig. 18(iii)) the only parameter, which does not vanish, is the parameter Dinter xx from the spinorbit coupling of the ground state with the 3dxz orbital. The states 3dz2 and 3dxy are not coupled via spin-orbit interaction but via intra-atomic Coulomb repulsion. Following this scheme, we obtain for the inter-chain anisotropic exchange parameters

2 2 tπ2 λ tπ inter Dyy = − 2 (3B + C) 2 ∆xy,yz ∆12 ∆π2

2 2 t2 λ tπ Dinter 2 (4B + C) σ2 sin2 (2γ ) (48) xx = − 2 ∆xy,xz ∆12 ∆π where the ﬁrst expression corresponds to Fig. 18(ii) and the second one is illustrated in Fig. 18(iii). Here ∆xy,yz ≈ ∆xy,xz denotes the crystal-ﬁeld splitting between the respective 3d states of Cu2+ . For σ - and π -bonds between copper and oxygen tσ and tπ denote the transfer integrals and ∆π ≈ ∆σ the charge-transfer energy. ∆12 is the charge-transfer energy from one Cu site to the other Cu site. The parameters B/kB ≈ 1374 K and C/kB ≈ 6092 K denote the Racah parameters of the Coulomb interaction between the copper electrons [29]. After transformation into crystalinter inter lographic coordinates and renormalization Dinter cc = Dyy − Dxx /2 the two other inter inter parameters |Dinter aa |, |Dbb | |Dcc | can be approximately neglected and we can write the Ising-like Hamiltonian for the symmetric anisotropic inter-chain exchange as inter = Dinter (49) HAE cc S1,c S2,c where S1,c and S2,c are spin projections of chain 1 and its neighboring chain 2 on the chain direction along the c crystallographic axis. The ratio between the spin-orbit coupling parameter λ and the energy of the crystal-ﬁeld splitting ∆xy,xz between the states 3dxy and 3dxz is approximately given λ by ( ∆xy,xz )2 ≈ (g⊥ − 2)2 ≈ 0.01, where g⊥ = (gx + gy )/2 in local coordinates. The ratio of the oxygen-copper transfer parameters to the charge transfer energies is known for oxides from studies of the transferred hyperﬁne interactions as (tπ /∆π )2 ≈ (tσ /∆π )2 ≈ 0.077 [64]. The oxygen-copper transfer integrals are approximately equal (tπ ≈ tσ ) and according to different estimations their value is about tσ 1.3 − 2.5 eV [65, 66]. The charge-transfer energy between the Cu ions is about ∆12 6 eV [65]. Therefore we can use (tπ /∆12 )2 ≈ 0.1. Substituting the structural angle 2γ = 68◦ between the chains, we get Dinter cc /kB ≈ −0.7 K, which shows that the inter-chain exchange coupling cannot be neglected compared with

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the intra-chain exchange. Note that in principal the inter-chain bond geometry via the apical oxygen allows the existence of the DM interaction. However, as there is no divergence ∆ H ∝ 1/T 2 observed in the experiment for T > TSP , the DM interaction seems to be of minor importance. Using the detailed algebra for the line width resulting from superposition of the second moments of the intra- and inter-chain symmetric AE [62] the data can be ﬁtted as shown in the middle row of Fig. 17. Each intra-chain contribution gives rise to a 2:1 anisotropy with the maximum linewidth perpendicular to the plane spanned by the CuO2 ribbons. The geometrical superposition of the intra-chain contributions (dashed line) results in a maximum linewidth for the magnetic ﬁeld applied along the b axis as observed in the experiment, but the corresponding width along the c axis is smaller than the experimental data. This is compensated by the inter-chain contribution (dotted line) which exhibits a 2:1 anisotropy with respect to the c axis. The intra relative strength of inter-chain to intra-chain symmetric AE Dinter cc /Dzz is in satisfactory agreement with the theoretical estimation. Note that the relative anisotropy intra shown at 120 K remains unchanged up to the and corresponding ratio Dinter cc /Dzz saturation of the linewidth at high temperatures. Only the experimentally determined absolute value | Dintra zz | /kB = 3.2 K of the high-temperature limit is about twice as large as the theoretical estimate. This may indicate a partially inhibited narrowing caused by the one-dimensionality.

4.2.3 NaV2 O5 In the spin-ladder system NaV2 O5 the exchange paths are somewhat different in comparison to the two spin-chain systems LiCuVO4 and CuGeO3 analyzed so far. One has to recall that NaV2 O5 contains quarter ﬁlled spin-ladders where each rung consists of two vanadium ions sharing one electron. Hence, the ground-state wave function is built by the superposition of two neighboring 3dxy orbitals hybridized with the intermediate oxygen 2py orbital on the same rung, as depicted in Fig. 19(i). The super exchange between two neighboring rungs is transferred via π bonds with the interconnecting oxygen 2px orbitals. Most relevant for the symmetric

Fig. 19 Schematic pathways of the exchange coupling in NaV2 O5 (blue big spheres: V, red small spheres: O) y axis running along the chains. (i) In the ground state the electron occupies the dxy orbital. (ii) Anisotropic spin-spin coupling via excited V dx2 −y2 states. (iii) Inter ladder coupling

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anisotropic exchange is the strong antiferromagnetic super exchange between the excited 3dx2 −y2 states of neighboring rungs transferred via σ bonds with the oxygen 2py orbitals, as it is illustrated in Fig. 19(ii). Thus, the symmetric AE in NaV2 O5 works via the ring process shown in Fig. 12(v,vi): The electron e.g. couples from the 3dxy ground state to the 3dx2 −y2 excited state via spin orbit coupling λ by lz , is transferred (tσ ) to the excited 3dx2 −y2 state of the neighboring rung, couples via λ to the corresponding 3dxy ground state and returns (tπ ) to the 3dxy starting orbital. Like in LiCuVO4 due to the spin-orbit coupling between 3dxy and 3dx2 −y2 states via the z component of the orbital momentum, the only non zero matrix element of the symmetric AE is given by 2 Dintra zz = 8λ

tπ tσ 2 ∆x2 −y2 ,xy ∆AA

[c∗l cl + c∗r cr ]2

(50)

where cl , cr and cl , cr denote the distribution coefﬁcients of the wave functions for the left (l) and right (r) vanadium site on rung A and neighboring rung A’ respectively. Following literature, the crystal ﬁeld splitting amounts about ∆x2 −y2 ,xy ≈ 0.36 eV [67, 68] and for the charge-transfer energy between neighboring rungs on ﬁnds ∆AA = 3 eV [69]. The hopping integral tσ between the excited states is difﬁcult to estimate, but as a lower boundary one can approximate it by the hopping integral between the ground states tσ ≈ tπ = 0.17 eV [58]. Using the free-ion value of the spin-orbit coupling λ = 31 meV, one obtains Dzz ≈ 0.6 meV at high tempera√ tures, where the electrons are equally distributed on each rung, i.e. cl = cr = 1/ 2. This yields a characteristic linewidth of about ∆ H = 300 Oe. The leading zz diagonal element of the symmetric AE tensor gives rise to a 2:1 anisotropy with respect to the corresponding crystallographic c axis. Both, the absolute value as well as the experimentally observed angular dependence of the linewidth is in good agreement with the theoretical prediction, as one can see in the lower row of Fig. 17 (dashed line). For the exact description of the data (solid line) it is necessary to take into account the corrections due to the inter-ladder interactions. Possible exchange paths are shown in Fig. 19(iii). These exchange paths involve a 90◦ -bond geometry which has been already discussed in LiCuVO4 and CuGeO3 , where it has been shown that it produces sizable symmetric anisotropic exchange contributions, as well. Considering all possible exchange paths, one obtains a maximum inter-chain xx component corresponding to an enhancement of the linewidth for the ﬁeld applied along the crystallographic a direction, which allows to ﬁt accurately the experimental data (dotted line in Fig. 17). So far it has been shown that the three quasi one-dimensional systems LiCuVO4 , CuGeO3 , and NaV2 O5 represent excellent model compounds where the spin relaxation is governed by the symmetric anisotropic exchange interaction. The microscopic analysis of the exchange paths correctly predicts the observed angular dependence of the linewidth and in the high-temperature limit the correct order of magnitude of its absolute value is estimated.

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4.3 Inﬂuence of the Phase Transition On the base of the universal relaxation pattern worked out above it is possible to extract important microscopic information on the critical ﬂuctuations on approaching the phase transition into the ground state, i.e. antiferromagnetic order in LiCuVO4 , the dimerized spin-Peierls phase in CuGeO3 , and the charge-order accompanied by spin-dimerization in NaV2 O5 . As it was discussed from Eq. (44), if the linewidth is determined by only one interaction – in our case the symmetric anisotropic exchange interaction – then the normalized linewidth should be temperature independent. To check this one has to plot the linewidth ratios dependent on temperature, as it is done in Fig. 20. Indeed the linewidth ratios are approximately constant above 40 K, 100 K, and 200 K for LiCuVO4 , CuGeO3 , and NaV2 O5 respectively, but below these temperatures distinct systematic changes show up. In LiCuVO4 this temperature regime coincides with the critical line broadening ∆ Hcrit ∝ (T − TN )−0.55 , which becomes sizable already at about 10 TN . Concomitantly on decreasing temperature an enhancement of both longitudinal 1/T1 and transversal nuclear magnetic resonance relaxation rates 1/T2 is observed on 7 Li and 51 V [70]. This is the usual effect, when magnetic order sets in accompanied by critical spin ﬂuctuations. The critical temperature regime is quite broad in comparison with conventional three dimensional systems. The reason may be both due to the one dimensionality, which favors short-range order, and due to some degree of frustration on the triangular lattice formed in the ac plane by the Cu chains. CuGeO3 and NaV2 O5 do not exhibit any critical line broadening on approaching the phase transition (only below the transition the ESR line broadens because

Fig. 20 Temperature dependence of the linewidth ratios in LiCuVO4 (left), CuGeO3 (middle), and NaV2 O5 (right). The insets show the ratio of the inter-chain/ladder to the intra-chain/ladder anisotropic exchange parameters for CuGeO3 /NaV2 O5

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the exchange narrowing is slowing down with decreasing number of residual non dimerised spins). Nevertheless the anisotropy of the linewidth changes already far above the phase transition. In general these changes may arise on different grounds. One reason can be that another interaction becomes important at low temperatures: e.g. the Dzyaloshinsky-Moriya interaction may give rise to a divergent contribution at low temperatures, or the soliton dynamics can inﬂuence the spin-spin relaxation, or simply some contribution due to impurities may affect the linewidth. But the observed deviations may also result from a change of the exchange integrals due to lattice ﬂuctuations on approaching the spin-Peierls transition in CuGeO3 or charge ﬂuctuations due to the onset of charge order in NaV2 O5 . This last possibility is supported by several independent experiments in both compounds. In CuGeO3 inelastic neutron scattering reveals two low-lying phonon modes associated with the structural distortion in the dimerized phase [71]. The eigenfrequencies of these modes remain approximately constant down to 150 K at 3.1 THz and 6.5 THz. On further cooling they exhibit an increasing shift to higher frequencies and a pronounced line broadening already far above the spin-Peierls transition. The frequency shift saturates below 30 K at values about 10% higher compared to room temperature. Electron-diffraction experiments exhibit a diffuse scattering with the shape of an inﬁnite ribbon parallel to the chain direction also clearly above the spin-Peierls transition [72]. Its intensity rises rapidly below 100 K and again saturates below 30 K. The diffuse scattering is proposed to result from low-frequency twisting vibrations of the Cu-O2 ribbons along the chains. In addition Ramanscattering experiments revealed an anomaly with a quite similar behavior. Finally ultrasonic experiments found a broad anomaly in the temperature dependence of the sound velocity along the c-axis centered around 100 K [73]. This anomaly is attributed to magnetoelastic couplings. In NaV2 O5 a strong frequency dependence of the ESR linewidth in the temperature regime 34 ≤ T ≤ 100 K [74], and the anomalous features observed by optical spectroscopy measurements [75, 76, 77] provide evidence for the onset of charge disproportions already far above TCO . Note that the uniform susceptibility has significant deviations from the Bonner-Fisher law already at T < 200 K, too (cf. Fig. 16). The coupling of these charge ﬂuctuations to the lattice has been revealed by a softening of the elastic constants below 100 K detected by ultrasound experiments [78, 79] and by the shift of the phonon energy found in light-scattering measurements around 80 K [80]. All these experiments in CuGeO3 and NaV2 O5 reveal a systematic change of the lattice couplings already far above the phase transition, which should affect the anisotropic exchange as well and, therefore, justiﬁes an evaluation of the resonance linewidth in terms of symmetric anisotropic exchange only. This is in agreement with the monotonous increase of the linewidth with increasing temperature above TSP or TCO , approaching saturation for T → ∞. The deviation from a purely linear behavior at T J/kB is not in contradiction to Oshikawa and Afﬂeck, because their theory for ideal 1D systems does not take into account structural ﬂuctuations, which exist already above the spin-Peierls transition, and therefore holds only in the case T TSP or TCO [2].

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Finally it is useful to consider the quantitative changes in the exchange parameintra ters: In CuGeO3 the ratio |Dinter cc /Dzz | deduced from the anisotropy of the linewidth intra gradually increases below 120 K up to approximately |Dinter cc /Dzz | ≈ 1 ± 0.05 below 40 K, i.e. the inter- and intra-chain AE become of equal strength [62]. One can estimate the inﬂuence of the lattice changes on the inter-chain exchange, using structural data [53]. The Cu-O1 distance, which is relevant for the inter-chain exchange, ˚ at 295 K down to R = 2.7295 A ˚ at 20 K. If one takes decreases from R = 2.7549 A (tπ /∆π )2 ∝ R−6 , then on lowering the temperature the value of Dinter cc becomes about 10% larger, which is the correct order of magnitude. In addition, the strong sensitivity of the exchange interaction to lattice deformations can be illustrated e.g. for the super exchange in MnO, which was established as J(R) ∝ exp (−15R/R0) for the 90◦ Mn-O-Mn bond [81]. For small deviations δ R from the equilibrium Mn-Mn distance R0 the resulting change in J can be expanded as δ J/J(R0 ) ≈ −15δ R/R0, which is about 15% for δ R/R0 ≈ 0.01, in agreement with the present results for the anisotropic exchange in CuGeO3 . In NaV2 O5 the ratio |Dinter /Dintra | even increases from 0.4 at high temperatures up to 1.6 near T = 34 K, i.e. the intra-ladder AE becomes even stronger than the inter-ladder AE at low temperatures. This can be understood taking into account the strong dependence of the AE parameters from the coefﬁcients cl and cr , which describe the electronic occupation on the vanadium sites. That means, e.g., for Dintra = Dzz (cf. Eq. (50)) the coefﬁcient [c∗l cl + c∗r cr ] is equal to 1 for the case V4.5+ –V4.5+, but it vanishes for the zigzag charge order (V5+ –V4+ ) realized below TCO [82]. The observed increase of the |Dinter /Dintra | ratio already far above TCO indicates that precursors of the developing charge order set in at about 150 K, weakening considerably the intra-ladder anisotropic exchange interaction.

5 Conclusions and Perspectives We presented an up-to-date survey of theoretical concepts in the ﬁeld of exchange interactions and of their relevance to experiment on low-dimensional materials. The main emphasis of the theoretical part of the work is on quantum phenomena which can drastically change the magnitude, the sign as well as the anisotropy of spin exchange. It was shown that instead of conventional estimations an explicit microscopic analysis of the involved orbitals and exchange paths is necessary to obtain a reliable estimate of exchange parameters. These theoretical conclusions are supported by the analysis of the electron spin resonance data in several low-dimensional systems which came into focus of interest during the last decade: LiCuVO4 , CuGeO3 , and NaV2 O5 . All the systems exhibit a similar relaxation behavior which is identiﬁed as due to the symmetric part of anisotropic exchange interaction. On the basis of the presented microscopical model the magnitude of the exchange parameters as well as their changes on approaching the phase transition have been estimated. Moreover, each of the systems considered allows to study quantitatively quantum effects peculiar to low-dimensional

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materials: effects of quantum interference in the ribbons of LiCuVO4 , inter-chain exchange in CuGeO3 and the anisotropic ring-exchange processes in NaV2 O5 . Having established the universal relaxation behavior due to symmetric anisotropic exchange in prototypical spin-chains presented above enables to study more complex systems. An advanced example of application of this knowledge is given by the vanadate compound β -Na1/3 V2 O5 , where an investigation of ESR in single crystals [83] allowed to address the questions of electronic occupation and charge order pattern. A further example is represented by a recent study of TiOCl [84], where the contribution of Dzyaloshinsky-Moriya interaction was estimated from the analysis of the temperature dependence of the ESR linewidth. These examples and the content of this work clearly show, that due to its efﬁciency in determining the anisotropic parts of the exchange interaction, ESR provides a very powerful experimental technique to investigate a large variety of related physical phenomena. An outstanding challenge for further ESR studies may be given by KCuF3 – one of the best realizations of an ideal 1D antiferromagnetic Heisenberg chain. The correct determination of the spin relaxation source takes on special signiﬁcance for resolving the up to now puzzling inconsistencies between the magnetic and crystallographic structures [85] and, consequently, for the whole physics related to this prototypical system. Acknowledgements We kindly acknowledge V. N. Glazkov, L. E. Svistov for useful discussions and A. Prokoﬁev, L. I. Leonyuk and G. Obermeier for the preparation of samples. This work was supported by the German BMBF under Contract No. VDI/EKM 13N6917, by the DFG within SFB 484 (Augsburg) and by the RFBR (Grant No. 06-02-17401-a). One of us (J.D.) was partly supported by the Swiss National Science Foundation through the NCCR ’Materials with Novel Electronic Properties’. The work of D. V. Z. was supported by DAAD.

Appendix The Hamiltonian of the electronic subsystem can be expressed as H = ∑iα |h| jβ c+ iα c j β +

1 + iα1 jβ2 |g12 |kα1 l β2 c+ iα c j β ckα cl β , 2∑

(51)

where i, j, k, l represent the lattice sites, and greek letters indicate the orbital substates of a shell. c+ iα is a fermion operator in the second quantization and creates an electron at the site i in the state α , ciα is the annihilation operator for the same electron. The localized orbitals φiα are assumed to form a complete orthogonal set. We divide H = H0 +V into the unperturbed Hamiltonian H0 and the perturbation V with V = ∑iα |δ h| jα c+ iα c j α +

1 + iα1 jβ2 |δ g12 |kα1 l β2 c+ iα c j β cl β ckα , 2∑

(52)

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231

Fig. 21 Sketchy energy schema supposed in our analysis. Possible on-site excitations (denoted by m, m , ..) have a much smaller energy compared to the excitation with the transfer to the neighboring site (denoted by l, l , ..)

where

δh = h −' h,

iβ |h| jβ iα |' h| jα = δiα , jα ∑ δαβ , 2[β ]

(53)

and

δ g12 = g12 − g'12,

(54) αβ

iα1 jβ2 |' g12 |kα1 l β2 = δiα ,kα δ jβ ,l β ∑ δατ δβ τ iτ1 jτ2 |g12 |iτ1 jτ2 , αβ

g12 =

g12 (1 − P12) . 2[β ] 2[α ] − δα ,β

(55) (56)

Here [α ], [β ] are the orbital degeneracy or the number of states in the subshell (so that the subshell α is complete when the number of electrons is equal to 2[α ]). The operator P12 exchanges two electrons, and the eigenvalues of H0 are the average energy of electrons in the Hartree-Fock approximation. The perturbation V can be rewritten as a sum of two terms V = V (1) +V (2) . V (2) is the part of the Hamiltonian with nonzero matrix elements between the excited (with electron transfer) and ground (with some electron excitations on the same site) state (2) (2) (2) conﬁgurations. In other words Vmm ≡ Vll ≡ 0,Vml = 0, the deﬁnitions introduced in Fig. 21 were used. Note that the energy scheme pointed out in Fig. 21 corresponds to the large energy distance between the excited and ground states ∆ab with respect to the energy splitting of each of them ∆ E ∆ab , what is usually very well fulﬁlled in real systems. The case ∆ E ∼ ∆ab is brieﬂy discussed in Section 2.2.4. Hoppings of electrons to the other site can be described by V (2) = ∑ tiα , jβ c+ iα c j β ,

(57)

where tiα , jβ are the transfer integrals determined by tiα , jβ = iα |h| jβ + ∑

nτ iα1 l τ2 |g12 (1 − P12)| jβ1 l τ2 . 2[τ ]

(58)

Here nτ is the number of electrons in the shell τ . The operator V (1) contains all interactions which do not change the number of electrons of the site. Therefore, it acts within the quasi-degenerate states of the

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(1)

(1)

excited and ground state conﬁgurations: Vml ≡ 0,Vmm = 0,Vll = 0. The most important of them are the Coulomb interaction and the crystal ﬁeld: V (1) =

1 + + iα1 iβ2 |δ g12 |iα1 iβ2 c+ iα ciβ ciβ ciα + iα |δ h|iα ciα ciα . 2∑

(59)

Here we would like to emphasize, that in the excited state (with changed number of electrons on the site) the crystal ﬁeld can be much larger than in the ground state. In order to take into account the electron transfers we employ the canonical Schrieffer-Wolff transformations: i.e. we transform the initial Hamiltonian H by 1 n using an unitary operator e−S = ∑∞ n=0 n! S into the effective exchange Hamiltonian 1 1 Heff = e−S H eS = H + [H , S] + [[H , S], S] + [[[H , S], S], S] + ... 2 6 ∞ 1 (n) = ∑ [H , S] . (60) n=0 n! The Hamiltonian Heff describes all effects of spin exchange between the different sites. But by a peculiar choice of the S matrix we can cancel all hopping terms of this Hamiltonian (Heff )lm ≡ 0, thereby describing the effects of the electron transfer as a perturbation to the ground-state energy of the localized spin. This approximation assumes naturally that V (2) ,V (1) ∆ab , what is usually well fulﬁlled. For determination of the S-matrix we perform the following iteration procedure S = S(1) + S(2) + S(3) + S(4) + S(5),

(61)

where the indices 1, 2, .. correspond to the order of perturbation theory. By this (2) transformation the hopping terms Vml vanish in the new basis if: [H0 , S(1)] = −V (2) , [H0 , S(2) ] = −[V (1) , S(1) ],

(62)

1 [H0 , S(3)] = −[V (1) , S(2) ] − [[V (2) , S(1)], S(1) ], 3 1 1 (4) (1) (3) [H0 , S ] = −[V , S ] − [[V (2) , S(1)], S(2) ] − [[V (2) , S(2) ], S(1) ], 3 3 1 1 [H0 , S(5)] = −[V (1) , S(4) ] − [[V (2) , S(1)], S(3) ] − [[V (2) , S(3) ], S(1) ] − 3 3 1 (2) (2) (2) 1 (2) (1) − [[V , S ], S ] + [[[[V , S ], S(1) ], S(1)], S(1) ]. 3 45 These equations give us the following expressions for the matrix S: (1)

Sml = − (2)

Sml = −

(2)

Vml

(0)

(2)

(0)

Em − El

Vml , ∆ml

(1) (2) (2) (1) Vmm Vm l Vml Vl l 1 , = −∑ ∆ml ∑ ∆m l ∆ml m l

≡−

[V (1) , S(1) ]ml ∆ml

(63)

Anisotropic Exchange in Spin Chains (3) Sml

233

(1) (1) (2) (2) (1) (1) Vmm Vm m Vm l Vml Vl l Vl l 1 − ∑ = −∑ + ∆ml ∆m l ∆m l ∆ml ∆ml m m l l

1 1 1 (1) (2) (1) +∑ Vmm Vm l Vl l + + l l ∆ ∆ ∆ m ml m lm

1 1 1 2 (2) (2) (2) + ∑ V V V , .. − − 3 nl ∆ml ∆nl ∆l n ∆nl ∆ml ∆l n ml l n nl

Putting the relations (61–62) into the Eq. (60), one gets for the effective exchange operator 1 1 1 [[[[[V (2) , S], S], S], S], S]. Heff = H0 + V (1) + [V (2) , S] − [[[V (2) , S], S], S] + 2 24 144 (64) This is the most general form of the interaction Hamiltonian including all terms up to the sixth order of the perturbation theory. Equation (64) is very convenient for estimations of exchange integrals. In order to demonstrate it, we will give the corresponding expression for the integral of isotropic exchange Hiso = J (Sa · Sb )

1 J= J + J ( µ ) + J ( ν ) (65) ∑ ηζ 2Sa Sb ∑ ηζ ∑ ηζ between two magnetic ions in the ground states with the maximal spin values. Jηζ are the individual channel contributions for pairs of half-ﬁlled orbitals η and ζ of the ions a and b, respectively. The exchange integrals Jηζ (µ ) and Jηζ (ν ) may be called correlation corrections, since they depend on the spatial distribution of the electrons of vacant (µ ) and ﬁlled shells (ν ) of interacting ions. After substituting necessary terms instead of V (1) (Eq. (59)) and V (2) (Eq. (57)) into (Eq. (64)), the quantities Jηζ = Jηζ (a → b) + Jζ η (b → a) read [86] 1 Jζ η (b → a) = − η1 ζ2 |g12 |ζ1 η2 + |Bζ η |2 ∆ζ η − ∑ Mηκ ζ1 κ2 |g12 |η1 ρ2 Mρζ − 2 κρ

− ∑ ζ1 η2 |g12 |κ1 η2 Mκζ − Mζ κ κ1 η2 |g12 |η1 ζ2 κ

− ∑ Mζ κ Mρζ η1 κ2 |g12 |ρ1 η2 + ∑ Mηκ Mκζ Bζ η (∆ζ η + ∆κζ + ∆κη ) + κρ

+ ∑ Mηκ Mκζ Mζ ρ Mρη κρ

− ∑ Mηκ Mκζ Mζ ρ Mρη κρ

κ

∆κζ ∆ρζ (∆ρη + ∆κζ )(∆ρζ + ∆κη ) + − ∆ζ η 2∆ηκρζ

(∆ρη + ∆κζ )(∆ρζ + ∆κη ) κ1 ρ2 |δ g12 |ρ1 κ2 . 2|∆ηκρζ |2 (66)

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This expression includes all types of contributions: direct exchange, indirect Yamashita-Kondo’s exchange, Kramers’s exchange, Anderson’s superexchange (see Section 2.1.2). The introduced dimensionless quantities Mηκ = tηκ /∆ηκ and Bζ η can be expressed in terms of the overlap integrals Sηκ and covalence parameters γηκ by the following relations:

Sηκ , Mηκ = − γηκ + 2

Sζ η |∆κζ | 1 3 + ∑ Sκη + Bζ η = − γζ η + γ + S (67) 2 2 κ |∆ζ η | ζ κ 4 ζ κ

|∆κη | 3 1 . + ∑ Sζ κ γκη + Sκη 2 κ 4 |∆ζ η | All parameters entering these expressions can be determined from the experimental data obtained by electron-nuclear double resonance and optic measurements. Formulas for estimates of Jηζ (µ ) and Jηζ (ν ) are given in [87]. The term corresponding to Anderson’s model of exchange interaction (Section 2.2.1) can be written in accordance with (66) as Hiso (b → a) = ∑

+ + + tκη c+ κ aη · tζ κ bζ cκ · tρζ cρ bζ · tηρ aη cρ

∆κη ∆κκ ∆κζ

.

(68)

The shell of the intervening diamagnetic ions (e.g. O2− , F− ) is assumed to be fully ﬁlled, i.e. in the initial ground state we always have two electrons with opposite spins on the p (or s) orbital of the ligand ion. Therefore, the following relations can be used to simplify the Hamiltonian (68): c+ κ cκ = δκκ ,

+ c+ κ cρ cρ cκ = δκκ δρρ − δκρ δρκ .

(69)

1 + t t t t · aη b+ ζ bζ aη , 2 ∆ ∑ κη ζ κ ρζ ηρ ∆ca ab

(70)

The result is given by Hiso (b → a) =

where ∆ab (∆ca ) denote the energy of the electron transfer between the magnetic ions (between the diamagnetic and magnetic ions). That energy can be factored out, because it is large with respect to the distance between the orbital levels. Obviously, the electron does not change its spin during the hopping that results in the additional conditions: sη = sζ and sη = sζ . Then, using the usual relations between the spin operators s(z) , s± ≡ s(x) ± is(y) and the Fermi operators a+ , a (z) : s(z) s s(z) =

1 + a↑ a↑ − a+ ↓ a↓ , 2

s+ = a+ ↑ a↓ ,

s− = a+ ↓ a↑ ,

(71)

Anisotropic Exchange in Spin Chains

235

+ and a+ ↑ a↑ + a↓ a↓ = 1, we obtain ﬁnally

1 1 Hiso (b → a) = 2 tηρ tρζ tζ κ tκη − + 2(sa · sb ) , ∆ca ∆ab ∑ 2

(72)

in accordance with the classical Anderson’s result [27]. The values of hopping integrals entering this expression can be estimated, e.g., using the LDA+U approach [88] or from magnetic susceptibility data. Using perturbation theory allows easily to describe the anisotropy of exchange interaction, too. The spin-orbit (SO) interaction (17) which provides the coupling to the lattice is going into the effective exchange Hamiltonian (64) as an additional on-site perturbation V (1) = HLS beside four hopping terms V (2) . The resulting expressions corresponding to the ﬁfth Heff(5)

(1)

∝∑ m

(4,hop) (Heff )mm

(2) (2) (2) (2)

(1)

V V V V V V · m m = ∑ ml ln nl l m · m m , ∆mm lnl m ∆ml ∆ln ∆nl ∆mm

(73)

and the sixth order of perturbation theory Heff(6) ∝

mm

(1)

(2) (2) (2) (2)

(1)

Vmm Vm l Vln Vnl Vl m Vm m · · . ∆ ∆m l ∆ln ∆nl ∆mm ll n mm

∑

(74)

give the Hamiltonians of the antisymmetric and symmetric parts of anisotropic exchange, respectively (see Section 2.3).

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A Review of Bose-Einstein Condensation in Certain Quantum Magnets Containing Cu and Ni Vivien S. Zapf

Abstract We summarize a set of recent experimental and theoretical work on the phenomenon of Bose-Einstein Condensation (BEC) of magnetic degrees of freedom in quantum magnets. We will focus on two examples: BaCuSi2 O6 , which contains planes of Cu S = 1/2 dimers, and NiCl2 -4SC(NH2 )2 (DTN), an organic magnet with Ni S = 1 weakly-coupled chains. In both compounds, the BEC corresponds to a ﬁeld-induced antiferromagnetic ordered state that persists over a magnetic ﬁeld range Hc1 to Hc2 . At the quantum critical point at Hc1 , the predicted power-law behavior of the critical ﬁelds and the magnetization for 3-D BEC is observed at low temperatures. In addition, BaCuSi2 O6 shows a dimensional crossover from a 3-D BEC to 2-D behavior below 1 K.

1 Introduction In the past decade there has been a surge of interest in the topic of quantum magnetism in condensed-matter systems. Quantum spin systems come in a wide variety of structures including reduced-dimensional ladders, chains or planes, dimers, frustrated spins, and single-molecule magnets. These materials all share the common trait that quantum effects, such as spin ﬂuctuations and quantized spin levels, play a signiﬁcant role in shaping the ground state and the physical properties of the system. The resultant behavior can be complex and challenge our understanding. In particular, there have been a number of attempts recently to observe BoseEinstein Condensation (BEC) in quantum magnets. BEC was ﬁrst observed in dilute gases of 87 Rb atoms [1], leading to a nobel prize being awarded in 2001. It turns out that a form of BEC can also be observed in quantum magnets, e.g. crystalline lattices containing spins. BEC in quantum magnets was ﬁrst predicted to occur in 1991 by V.S. Zapf National High Magnetic Field Laboratory, Los Alamos National Laboratory, e-mail: [email protected]

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Ian Afﬂeck [2]. In the past few years, several reports of BEC in real spin systems have been published including TlCuCl3 [3, 4], BaCu2 SiO6 [5, 6], CsCuCl4 [7, 8], and NiCl2 -4SC(NH2 )2 [9, 10]. In these compounds, the BEC phase transition occurs at a critical magnetic ﬁeld at which XY antiferromagnetic order is induced. The quantum phase transition into the magnetically ordered state belongs to the same universality class as Bose-Einstein Condensation, and the bosons can be identiﬁed via a mapping from the spin degrees of freedom. All quantum magnets investigated for BEC to date contain either Cu2+ (S = 1/2) or Ni2+ (S = 1). In this paper we give a review of some recent experimental data on a material containing Cu, BaCuSi2 O6 and an organic crystal containing Ni, NiCl2 -4SC(NH2 )2 (DTN). We describe the details of these systems and how BEC can be derived from the Hamiltonian and observed via thermodynamic measurements. Both BaCuSi2 O6 and DTN have body-centered tetragonal crystal structures (see Fig. 1) with antiferromagnetic exchange coupling. In BaCuSi2 O6 , the strongest antiferromagnetic coupling (J = 51 K) occurs within the Cu dimers, such that the two S = 1/2 spins in each dimer form a S = 0 singlet groundstate separated by a gap J from the S = 1 excited triplet (see Figs. 2 and 6). The dimers are arranged in square planes with an intraplane coupling between dimers of J = 6.7 K. The planes in turn are staggered along the c-axis, which results in frustrated antiferromagnetic coupling between planes of J f = 1.3 K. Thus J >> J >> J f [12, 5]. DTN, by contrast, is a chain compound, consisting of Ni S = 1 spins strongly coupled along Ni-Cl-Cl-Ni bonds in the c-axis with Jc = 2.2 K. The interchain coupling Ja = 0.18 K is an order of magnitude weaker than the intrachain coupling and no diagonal couplings analogous to J f in BaCuSi2 O6 have been observed in experiment. The exchange couplings were determined from inelastic neutron diffraction

Cu

Ja

J' Jf

Ni Cl

J Jc

BaCuSi2O6

c

NiCl2-4SC(NH2)2 (DTN)

b a

Fig. 1 Tetragonal crystal structures of BaCuSi2 O6 (left) showing the Cu S = 1/2 dimers arranged in staggered planes, and DTN (right) showing the Ni and Cl atoms. The lines indicate the antiferromagnetic coupling strengths J (intradimer), J (intraplane), and J f (interplane) for BaCuSi2 O6 and Jc and Ja for DTN [9, 11]

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measurements [9], reﬁned via Quantum Monte Carlo calculations and Electron Spin Resonance experiments [10]. The important feature of both DTN and BaCuSi2 O6 is that the zero ﬁeld ground state is nonmagnetic and is separated by a gap from a magnetic excited state (see Fig. 2). In BaCuSi2 O6 the gap between the singlet ground state (S = 0) and the excited triplet states (S = 1) is created by the intradimer coupling J. In DTN, there is no S = 0 singlet – only a S = 1 triplet. A uniaxial anisotropy D splits this S = 1 triplet into a Sz = 0 ground state and a Sz = ±1 excited doublet. In both compounds, the excited Sz = +1 state can be suppressed with magnetic ﬁeld until it becomes degenerate with the S = 0 or Sz = 0 ground state. The critical region where these two states overlap is where the antiferromagnetic order/Bose-Einstein Condensation occurs. The Sz = ±1 spin levels are dispersed by the antiferromagnetic coupling, e.g. the spins can raise or lower their energy depending on their orientation with respect to their neighbors. In Fig. 2, the Sz = ±1 energy levels are shown schematically as broad bands where the upper and lower edges of the bands correspond to the k = 0 FM wave vector and the k = (π , π , π ) AFM wave vector, respectively. Thus, the region of overlap between the ground state and the ﬁeld-suppressed Sz = +1 excited state occurs over a broad range of ﬁelds between Hc1 and Hc2 . In both systems, 3-D XY antiferromagnetic order is observed between Hc1 and Hc2 , (23.5 T to 49 T for BaCuSi2 O6 and 2.1 T to 12.6 T for DTN) [5, 13]. The magnetic order occurs in a dome-shaped region of the phase diagram with a maximum N´eel temperature TN of 3.8 K for BaCuSi2 O6 and 1.2 K for DTN. The temperature-ﬁeld phase diagrams are shown in Fig. 3. In BaCuSi2 O6 the groundstate is nonmagnetic for H < Hc1 (S = 0). In DTN, the spins form a disordered Sz = 0 spin liquid for H < Hc1 . When the ﬁeld is increased to Hc1 , the spins order antiferromagnetically in the plane perpendicular to the applied ﬁeld and then cant along the applied ﬁeld direction as the ﬁeld is increased from Hc1 to Hc2 . Finally, the spins polarize along the ﬁeld direction above Hc2 . The longitudinal magnetization therefore increases monotonically between Hc1 and Hc2 , as shown in Fig. 3. It should be noted that for DTN, this phase diagram is only valid for ﬁelds along the tetragonal c-axis. For ﬁelds perpendicular to c, the spin levels within the triplet mix with one another leading to paramagnetic behavior for all ﬁelds. For BaCuSi2 O6 , there is no uniaxial anisotropy created by the crystal structure. Thus, BEC should occur for all ﬁeld directions with a small anisotropy due to an anisotropic g factor. The data discussed in this review was taken for H||c.

2 Boson Mapping The bosons are created via a mapping from the spin levels. In the spin language, the Hamiltonians HDT N for DTN and HB for BaCuSi2 O6 are given by: HDTN = ∑ D(Sjz )2 + ∑ Ja Sj · Sj+eα + ∑ Jc Sj · Sj+eγ j

j,α

j

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DTN Sz = -1

E

Sz = 1 Sz = 0

Hc1 = 2.2 T

BaCuSi2O6

Hc2 = 12.6 T

H

Sz = -1

E Sz = 0 Sz = 1 Hc1 = 23.5 T

Hc2 = 49 T

S=0

H

Fig. 2 Energy level diagrams of BaCuSi2 O6 and DTN as a function of magnetic ﬁeld parallel to the crystallographic c-axis. The broad colored bands for the Sz = ±1 levels schematically indicate the width antiferromagnetic dispersion due to the coupling Ja and Jc for DTN, and J and J f for BaCuSi2 O6

HB = ∑ JSj,1 · Sj,2 + ∑ J Sj · Sj+eα + ∑ Jf Sj · Sj+eγ j

j,α

j,γ

Here eα is the unit vector along a and b. eγ is the vector for coupling along the c-axis. For DTN, eγ = ec . However, for BaCuSi2 O6 , the coupling occurs between dimers on staggered planes (see Fig. 1) such that eγ = ec ± 1/2ea ± 1/2eb. In both Hamiltonians, the ﬁrst term creates the zero-ﬁeld splitting between the ground state and the excited state. The indices 1, 2 in this term refer to the two Cu spins within the dimer of BaCuSi2 O6 . The second term of the Hamiltonians expresses the AFM coupling within the plane, and the third term couples along the c-axis. For BaCuSi2 O6 it is implied that the couplings J and J f are summed over both spins within each dimer. When a magnetic ﬁeld H is turned on, both Hamiltonians acquire an extra Zeeman term −gµB H ∑j Szj . These Hamiltonians can be transformed into boson language via the identiﬁcation S+ = b† , where b† is the boson creation operator. Then

Bose-Einstein Condensation in Quantum Magnets 1.2

0.8

AFM/BEC

0.6 0.4 MCE

0.3

0.2

Specific Heat

0

QMC

0

5

Hc1

H (T)

10

Hc2

4 2

0.8

data QMC

0.6

2

0.4 1 0

Torque Specific Heat MCE QMC

20

Hc1

30

40

sat

N

1

6

H || c

M/M

T (K)

0 15 1.2

BaCuSi O 3

sat

data QMC

0.6

M/M

N

1

DTN H || c

0.9

T (K)

243

0.2 0 50 Hc2

H (T)

Fig. 3 Temperature T - Magnetic ﬁeld H phase diagram of DTN (top) and BaCuSi2 O6 (bottom). Antiferromagnetic order/Bose-Einstein Condensation occurs under the dome-shaped region with the spin conﬁguration indicated by arrows in the bottom ﬁgure. The N´eel temperatures were determined from magnetocaloric effect (MCE) and speciﬁc heat data for DTN (top) [9] and torque, speciﬁc heat and MCE data for BaCuSi2 O6 (bottom) [6]. Quantum Monte Carlo (QMC) calculations of the phase diagrams are shown for comparison [5, 10]. The magnetization vs ﬁeld at 16 mK (DTN) [13] and 0.5 K (BaCuSi2 O6 ) [5] is overlayed onto the phase diagram together with the predicted magnetization determined from QMC calculations [5, 10]

He f f = t ∑(b†j+eα bj + b†j bj+eα ) + t ∑(b†j+eγ bj + b†j bj+eγ ) j,α

j

+ V ∑ nj nj+eα + V j,α

∑ njnj+eγ + µ ∑ nj j

j

The deﬁnitions of t, t , V , V , and µ in terms of the parameters of the spin-language Hamiltonian are summarized in Table 1. The Hamiltonian consists of kinetic energy terms (t and t ), potential energy terms (V and V ), and a chemical potential µ . The kinetic energy terms allow the bosons to hop on the lattice in the square a-b plane (t), and along the c-axis t , subject to the hard-core constraint of one boson per lattice site. The bosons repel each other proportional to V and V in and out of the plane, respectively. The repulsion is also proportional to the number of bosons, and therefore to the Sz = +1 component of the spins. Finally, there is a chemical potential µ that is linear in the magnetic ﬁeld and controls the number of bosons.

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In this two-level boson-mapping model, [2, 14, 3] the mapping from spins to bosons treats the S = 0 (or Sz = 0 for DTN) state as an unoccupied bosonic state, and the Sz = +1 state as occupied bosonic state. The magnetic ﬁeld acts as the chemical potential in this system, tuning the number of bosons, e.g. the weight of the Sz = +1 component of the ground state. Condensation occurs as the number of bosons is tuned from zero to nonzero at Hc1 . As the magnetic ﬁeld is further increased, the bosons become less dilute, and eventually near Hc2 , where the ground state is mostly Sz = +1, a reverse mapping is necessary to create a dilute Bose gas, with Sz = +1 being the unoccupied state and S = 0 the occupied state. A second BEC transition then occurs across Hc2 . This two-level model described above, while didactic, is not exactly accurate in DTN because the third spin level Sz = −1 is low enough in energy that it needs to be taken into account. In comparing the phase diagrams of BaCuSi2 O6 and DTN in Fig. 3, it is clear that the region of AFM order in BaCuSi2 O6 is more symmetric in ﬁeld about the midpoint between Hc1 and Hc2 . This reﬂects a particle-hole symmetry of the bosons in this compound. In DTN, by contrast, the upper Sz = −1 level is low enough in energy that it breaks particle-hole symmetry of the bosons, and distorts the phase diagram in the T − H plane. Therefore in DTN, the idea of two spin levels corresponding to occupied and unoccupied bosons is not a complete picture. Two theoretical papers have presented models of Bose-Einstein Condensation for DTN that take all three spin levels (Sz = 0, Sz = +1, Sz = −1) into account. Wang and Wang [15] treat each spin level as a different type of boson and Ng et al. [16] interpret the Sz = −1 level as an energetically unfavorable double occupancy state. In any case, the concept of bosons that condense is still valid in a three-level model and the universality class of Bose-Einstein Condensation is still applicable at the quantum phase transition. A key condition that separates bosonic systems that condense from those that don’t is boson number conservation. The boson number must be set by some external constraint or else the bosons will merely be excitations of the system and vanish as the temperature is lowered to zero, as is the case e.g. for phonons. In DTN and BaCuSi2 O6 , the number conservation is created by the tetragonal crystal structures, which provides an approximate uniaxial symmetry of the spin environment about the direction of the applied ﬁeld. In the effective Hamiltonian it can be seen that every creation operator b† is multiplied by a destruction operator b. This is an indication that the Hamiltonian obeys the uniaxial symmetry. If the Hamiltonian were Table 1 Relation between the parameters of the Hamiltonian in the boson-picture and the Hamiltonian in the spin picture

t t V V µ

DTN

BaCuSi2 O6

Ja Jc Ja /2 Jc /2 D − g µB H

J Jf J /2 J f /2 J − g µB H

Bose-Einstein Condensation in Quantum Magnets

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rotated by an angle φ in the plane perpendicular to the ﬁeld, then b† → b† eiφ and b → be−iφ , such that (b† eiφ )(be−iφ ) = b† b and the Hamiltonian is independent of φ . The fact that every creation operator is paired with a destruction operator also ensures that the boson number is conserved. Since b† b is the number operator, a Hamiltonian containing only b† b terms will commute with the number operator. Thus the uniaxial symmetry of the Hamiltonian creates a number conservation law for the bosons. We should mention that there are several caveats to the idea of BEC in quantum spin systems. First of all, the uniaxial spin symmetry of the Hamiltonian is approximate. The square lattice of the crystal does introduce a small anisotropy in the a-b plane and dipole-dipole interactions and Dzyaloshinskii-Moriya interactions could also create anisotropies. However, these effects have been shown to occur at lower energy than the mK temperatures measured scales (possibly) and can thus be neglected at the temperatures of tens to hundreds of mK at which these quantum magnets are studied [8, 10]. Another caveat is that the uniaxial symmetry is only obeyed in equilibrium and the conservation of the boson number is therefore only obeyed on average. On short time scales, thermal and quantum ﬂuctuations can distort the symmetry of the lattice thereby produce ﬂuctuations in the boson number. Thus, the effects of Bose-Einstein Condensation are studied through thermodynamic measurements in equilibrium. Nonequilibrium effects such as supercurrents are not robust. The boson number ﬂuctuations create relaxation mechanisms for supercurrent excitations and thus supercurrents have ﬁnite lifetimes in quantum magnets. Nevertheless, the Bose-Einstein condensation picture in quantum magnets is valid for the temperatures at which these compounds are studied, and more importantly it provides a way of understanding the observed thermodynamic behavior near the quantum phase transition. The thermal phase transitions in these system belong to the d = 3, z = 1 universality class of an XY antiferromagnet, where d is the spatial dimension and z is the dynamical exponent. However, the ﬁeld-induced quantum phase transition belongs to the d = 3, z = 2 universality class, and the challenge, as with all quantum phase transitions, is to ﬁnd a classical phase transition to map it onto, allowing us to create a physical picture of what is happening. Bose-Einstein Condensation provides an intuitive way to describe the quantum phase transition in this system.

3 Experimental Investigation into Bose-Einstein Condensation One experimental approach to identifying Bose-Einstein Condensation is to measure the power-law temperature dependencies of the critical ﬁelds and the magnetization near T = 0. The mean-ﬁeld theory for Bose-Einstein condensation has a critical dimension d + z ≥ 4. Near the quantum critical point (QCP), z = 2 for these systems, thus the condition d ≥ 2 must be satisﬁed for the theory to hold. The theory then predicts the following:

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Hc − Hc (T = 0) ∝ T α Mz (Hc ) ∝ ρ ∝ T α

(1) (2)

where Hc is Hc1 or Hc2 , α = d/2 and the longitudinal magnetization Mz is proportional to the boson density ρ . Thus, α = 3/2 for a 3-D BEC and α = 1 for a 2-D BEC [2, 3, 14]. In contrast, the prediction for an Ising magnet is α = 2. These power laws are valid in the dilute boson limit, which is satisﬁed near Hc1 and Hc2 . Experimentally, these power laws have proven difﬁcult to measure since they are a low-temperature approximation to the boson distribution function and only valid as T → 0. In addition, the power-law exponent α is very sensitive to the ﬁtting range used and to the extrapolated value of the critical ﬁeld Hc [8]. To solve these problems, a method has been developed to determine the critical ﬁeld Hc1 independent of the exponent α , and to then extrapolate the exponent α in the limit as T → 0 [6]. Using this approach, an exponent within experimental error of α = 3/2 was found [6, 9] for BaCuSi2 O6 down to 1 K, and DTN down to 100 mK. For BaCuSi2 O6 , Hc1 (T ) was determined using torque, magnetocaloric effect and speciﬁc heat data [6]. The magnetization was extracted from torque measurements. For DTN, magnetocaloric effect and speciﬁc heat data was used to measure Hc1 (T ) and the magnetization was measured using a VSM [9].

4 Frustration and Dimensional Reduction Recent experiments have found that the exponent α = 3/2 for BaCuSi2 O6 is only valid down to 1 K. Magnetic torque data taken to down to 35 mK showed that the critical ﬁeld vs temperature Hc1 (T ) and the magnetization Mz become linear with temperature (α = 1) as shown in Fig. 4 [11]. This data suggests a dimensional reduction at the quantum critical point with 2-D behavior occurring below T ∼ 1 K in the vicinity of Hc1 . 1.2

BaCuSi O 2

1

6

H || c 0.6

c

T (K)

0.8

0.4 0.2 0

23.2

23.6

24

24.4

H (T) Fig. 4 Critical temperature vs ﬁeld for BaCuSi2 O6 determined from magnetic torque data, showing linear behavior between 35 mK and 1 K [11]

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Fig. 5 Body-centered tetragonal structure of NiCl2 4SC(NH2 )2 (DTN) showing the two interlocking Ni sublattices (red and blue atoms). The two sublattices are decoupled due to frustration, but this doesn’t lead to a dimensional reduction. The signiﬁcant AFM couplings along the c and a axes are shown as thick lines (Jc = 2.2 K, Ja = 0.18 K). The thin lines along (1,1,1) indicate the frustrated couplings, which are either very small or absent [9]

This dimensional reduction at the quantum critical point can be understood by considering the body-centered tetragonal structure of the BaCuSi2 O6 (see Fig. 1). The square a-b planes are arranged in a staggered structure such that each spin has an even number of nearest neighbors on the next plane with coupling J f along (1, 1, 1). Since the AFM wave vector is commensurate with the lattice, this results in frustration that suppresses the coupling along the c-axis. The c-axis boson hopping term, which is derived from the c-axis AFM coupling J f , therefore goes to zero, restricting boson motion to the 2-D planes. One might expect phase ﬂuctuations at the quantum critical point to restore 3-D behavior. However, the magnitude of these ﬂuctuations in quadratic in the boson number ρ , and thus goes to zero as the critical ﬁeld Hc1 is approached [17]. This conﬂuence of effects leading to 2-D behavior at the QCP in BaCuSi2 O6 make it a unique system. 2-D quantum critical behavior has been predicted to occur in many systems near quantum critical points but has never unambiguously observed. DTN also has body-centered structure with a commensurate AFM wave vector. However, frustration does not lead to dimensional reduction in this system. The reason is that the dominant AFM coupling occurs along the c-axis, Jc with a weaker coupling Ja in the tetragonal a-b plane. As shown in Fig. 5, the frustration occurs between spins at the edge of the unit cell, and those in the center. This decouples the two interlocking lattices (shown as red and blue in Fig. 5), giving rise to a two-fold degenerate ground state. However, each lattice is still three-dimensional so no dimensional reduction would be expected near the quantum critical points. Inelastic neutron scattering measurements at zero ﬁeld [9] do not show any diagonal couplings to within experimental error, conﬁrming that the frustrated couplings are either suppressed to a small value, or entirely absent. Due to the prevalence of quantum ﬂuctuations in these systems, particularly in DTN, Quantum Monte Carlo (QMC) simulations are necessary to calculate the phase diagram and all ﬁnite-temperature behavior. QMC calculations have been performed for both BaCuSi2 O6 and DTN and the resulting predictions for the phase

248

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S = -1

60

z

S =0

50

z

S =1

T (K)

40

z

30

BaCuSi O

DTN

20

2

6

S = -1 z

10 0

S =1 z

0

H

c1

10

H

c2

20

H

30

c1

H (T)

40

50 H

c2

Fig. 6 Energy levels (lines) and N´eel temperatures TN (points) for DTN and BaCuSi2 O6 , shown on the same scale for comparison. AFM/BEC occurs under the dome-shaped regions [9, 6]

diagrams are shown in Fig. 3, together with the experimental data [5, 6, 9, 10]. For both systems, the QMC calculations ﬁt the experimental phase diagrams very well. In addition, the longitudinal magnetization is also shown together with the QMC prediction. Much further work is needed in both compounds. For example, the 2-D behavior in BaCuSi2 O6 should be observable as a broad Shastry-Sutherland crossover in the speciﬁc heat at Hc1 . Neutron diffraction data examining the low-energy spin excitations in these systems will also shed light on the symmetry of the spin conﬁgurations. Finally, in DTN, a study is underway to investigate magnetostriction in the soft organic lattice, and its effects on the phase diagram and energy levels. In summary, the quantum magnets DTN and BaCuSi2 O6 have different spin level conﬁgurations, antiferromagnetic couplings, and energy scales (see Fig. 6). However, the underlying physics of ﬁeld-induced antiferromagnetism corresponding to Bose-Einstein Condensation is largely the same and both systems show the critical exponent α = 3/2 at the quantum critical point. The most signiﬁcant difference between the two compounds is that in BaCuSi2 O6 a frustration-induced decoupling along the c-axis results in a dimensional reduction at the quantum critical point Hc1 , leading to 2-D behavior with α = 1 below T = 1 K.

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