Lecture Notes in Physics Edited by H. Araki, Kyoto, .I. Ehlers, Menchen, K. Hepp, Z~rich R. Kippenhahn, M(Jnchen, H. A. ...
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Lecture Notes in Physics Edited by H. Araki, Kyoto, .I. Ehlers, Menchen, K. Hepp, Z~rich R. Kippenhahn, M(Jnchen, H. A. WeidenmfJIler, Heidelberg and J. Zittartz, KSIn Managing Editor: W. Beiglb6ck
251 R. Liebmann
Statistical Mechanics of Periodic Frustrated Ising Systems
Springer-Verlag Berlin Heidelberg New York Tokyo
To Gudrun Behnke, for substantial help and encouragement
S T A T I S T I C A L M E C H A N I C S OF P E R I O D I C FRUSTRATED
ISING SYSTEMS
Rainer L i e b m a n n Max-Planck-Institut Heisenbergstr.
fur F e s t k ~ r p e r f o r s c h u n g ~) I, D-7OOO S t u t t g a r t 80
CONTENTS
I.
2.
I n t r o d u c t i o n and survey
I
1.1
C r i t i c a l p h e n o m e n a at second order phase t r a n s i t i o n s
4
1.2
Scope of this book
6
One-dimensional 2.1Periodic
3.
f r u s t r a t e d Ising systems
7
ANNNI-chain
7
2.1.1
G r o u n d s t a t e d e g e n e r a c y of the A N N N I - c h a i n
2.1.2
Periodic ANNNI-chain
for
T ~ O
9
11
2.2
D e c o r a t e d chains
2.3
P a r t i a l l y f r u s t r a t e d chains
2O
2.3.1
P e r i o d i c f r u s t r a t e d chains
21
2.3.2
R a n d o m f r u s t r a t e d chain
22
Two-dimensional 3.1
15
f r u s t r a t e d Ising systems
Transformations
26
of Ising systems
26
3.1.1
Duality transformation
29
3.1.2
Decimation transformation
34
3.1.3
a)
Decoration-iteration
b)
Star-triangle
transformation
transformation
C o n n e c t i o n between d i f f e r e n t lattices
present address: AEG Aktiengesellschaft,
Sedanstr.
10, D-7900 UIm/FRG
34 35 37
VI 3.2
T r i a n g u l a r lattice
38
3.2.1
39
3.4
Simple lower bound
41
Pauling m e t h o d
42
c)
Systematic cluster a p p r o x i m a t i o n
42
P a r t i t i o n function and exact isotropic system
3.2.3
Specific heat near the frustration points (J1 = J2)
48
3.2.4
Pair correlation function, (J1 = J2)
51
GS
entropy of the 46
d i s o r d e r lines
Mapping to the q u a n t u m xy-chain and to the kinetic nn Ising chain
54
Further frustrated s y s t e m s with n o n c r o s s i n g interactions
61
3.3.1
Union-Jack
3.3.2
V i l l a i n ' s odd model and its g e n e r a l i z a t i o n s
68
lattice
61
a)
G r o u n d s t a t e s and phase diagrams
68
b)
C o r r e l a t i o n functions
70
c)
P e r i o d i c a l layered models
71
d)
C h e s s b o a r d model
74
3.3.3
Hexagon lattice
75
3.3.4
Pentagon lattice
76
3.3.5
Kagom& lattice
77
3.3.6
C o n n e c t i o n between g r o u n d s t a t e d e g e n e r a c y and existence of a phase transition at Tc = O
81
F r u s t r a t e d Ising systems with crossing interactions ANNNI-model
82
3.4.1
2d
3.4.2
Brick model
3.4.3
F r u s t r a t e d triangular lattice with n n n - i n t e r a c tions and m a g n e t i c field
88
a)
Additional nnn-interactions
89
b)
A d d i t i o n a l m a g n e t i c field
c)
Corresponding
3.4.4
4.
a) b)
3.2.2
3.2.5
3.3
E s t i m a t i o n of the g r o u n d s t a t e d e g e n e r a c y
82 87
J2 H
lattice gas model
93 97
Square lattice with competing nn- and n n n - i n t e r actions: relation to vertex m o d e l s
99
a)
System w i t h o u t m a g n e t i c field
99
b)
Systems with m a g n e t i c field
101
c)
Connection to vertex models
105
Three-dimensional
f r u s t r a t e d Ising systems
4.1
fcc
antiferromagnet
4.2
Fully and p a r t i a l l y f r u s t r a t e d simple cubic lattice
109 109 112
VII
5.
4.3
AF
pyrochlore
4.4
ANNNI-model
4.5
fcc
four-spin
model
117 122
(quartet)
model
127
Conclusion
131
References
133
I.
Introduction
The
present
and Survey
b o o k ~) r e v i e w s
systems I with
competing
configuration
of
all
Ising
interactions
some
The tive
strength,
rate
phases.
havior.
are
spins
Even i.e.
to a m u l t i t u d e
critical
For
interactions
transitions of this
can m i n i m i z e
in I s i n g
competition the e n e r g y
in the g r o u n d s t a t e remain
no of
(T = O)
in the e n e r g e t i c a l l y
exponents
certain
ratios
becomes
at
T = O
The
present
review
now
in this
fast
may
tries
the
simplest
example
system
of t h r e e
Ising
occur,
but
with
including
to s u m m a r i z e
and
on t h e i r
also
un-
order,
nonuniversal
the d e g e n e r a c y
the p a i r
a power
rela-
incommensu-
is of s e c o n d
interactions
large,
decrease
developing
depending
transition
of the
especially may
leads,
of c o m m e n s u r a t e ,
If the c o r r e s p o n d i n g
function
with
Because
(s i = ±I)
'broken',
of the
groundstates
As
of p h a s e
configuration.
competition
different
theory
simultaneously.
interactions
favorable
the
interactions.
beof the
correlation
law or e x p o n e n t i a l l y .
the r e s u l t s
obtained
up to
field.
for c o m p e t i n g
spins
at the
interactions
corners
we
consider
of a t r i a n g l e
a
(Fig.
1.1)
the H a m i l t o n i a n H
=
-
(J1s2s3 + J 2 S 3 S l
+J3SlS2 )
(1.1)
Fig. i.i The triangle formed by the three spins si = ±i with the pair interactions Jj .
w
v
¢
In the sons
case
of a n t i f e r r o m a g n e t i c
at l e a s t
relative
one
strength
interaction of the
three
interactions
is a l w a y s
for t o p o l o g i c a l
'broken'.
interactions
J1"
Depending J2
and J3
rea-
on the a dif-
~) This book is based on the habilitation thesis of the author, which has been accepted in April 1985 by the Physics Department of the University Frankfurt, Fed. Rep. Germany.
ferent groundstate (a)
J1 ~ J2
< J3
degeneracy < 0
:
J3
Ng
occurs
is w e a k e s t ,
(Fig.
therefore
N g = 2 , (as in the configuration
(b)
J1
< J2 = J3
< O
:
either
1.2): broken
ferromagnetic
J1
= J2 = J3
< O
:
J2 or J3
either
J1'
broken
for
J2 or J3
broken
\\
,
In the
isotropic
In the
are g r o u n d s t a t e s .
from
disorder
most
case
of
sition pletely
T = O
(B) +
;
(Y)
\
(6)
ions
of t h e n e a r e s t
In the
the
resolved.
per
in the
the
magnetic
3.2 w e
shall
infinite
remains
competing
theoretical problem
lattice,
But
six o f
eight
field
triangular
finite
and
Tc = 0
(nn) even
which
interactions;
treatment
interaction
whether
the
susceptibility
spin glas
coined
glasses disorder
modelled
there
as a
between exist
al-
transition
is a r e a l p h a s e
effect,
.
we use
spin
strength
for s u c h m o d e l s
nonequilibrium
of
is the t o p o l o g i c a l is u s u a l l y
H
see t h a t
of s p i n g l a s s e s 2 T o u l o u s e 3 h a s
The question,
a dynamical
finite
in the
site
with
second
neigbor
as a c u s p
'only'
In a w e a k
also
systems
lattice.
results.
up e.g. or
H = 0
in the
is e x t r e m e ,
2 . In S e c t i o n
entropy
for
frustration
on a regular
showing
for
equivalently.
no e x a c t
of
investigations
frustration
of the m a g n e t i c
spins
(x)
the d e g e n e r a c y
by a factor
context
terms
apart
(c)
the g r o u n d s t a t e
term
both
(~)
case
isotropic
lattice
the
states
is r e d u c e d
g in the
for
The four possible configurations of broken interactions (dashed lines) in the triangle with antiferromagnetic interactions.
possible N
(~)
(s) +
I
;
"t
(c~)
Fig. 1.2
;
case),
T = 0
(a) +
Ng = 6 , configurations
/
T = 0
(s)
Ng = 4 , c o n f i g u r a t i o n s
(c)
for
is s t i l l
not
trancom-
This
complication
is one of the r e a s o n s
f r u s t r a t e d systems,
where
many exact results
are known.
stable states w i t h e x t r e m e l y glas-like behavior. teresting methods
In p e r i o d i c
this p e r i o d i c
but also w e l l
nificance.
systems with competing
For example,
systems are not only in-
interactions
in m a n y m a g n e t i c
alone
lead a l r e a d y to f r u s t r a t i o n ,
tions
in l a t t i c e s
(3d)
, additional
substances (nn)
additional
behavior with decreasing
(2d)
E v e n if they are
substances,
where
(spin)
in two o p p o s i t e d i r e c t i o n s
respectively.
can m o v e on a c i r c l e
we w i l l
(spin d i m e n s i o n
t r a n s i t i o n s e.g.
from d e s c r i b i n g m a g n e t i c
dimensional periodic
Ising system,
for c o n v e n degeneracy)
are also re-
The a b s o r p t i o n of
surface
(sub-)
can be m a p p e d on a two-
if a b s o r p t i o n takes place o n l y on a d i s c r e t e
l a t t i c e of a b s o r p t i o n sites d e t e r m i n e d by the substrate.
s i = +I
s i = -I
Ising systems
systems.
of atoms on a c r y s t a l l i n e
(n = 3)
of xy-type.
substances,
l e v a n t for v e r y d i f f e r e n t p h y s i c a l
systems,
or on a sphere
(often c o n n e c t e d to e n h a n c e d g r o u n d s t a t e
find also p h a s e
monolayers
(n = 2)
e.g. be-
of the single
A l t h o u g h we only r e g a r d Ising s y s t e m s here,
ient i n t e r a c t i o n s
Weak
from lower- to
In this r e s p e c t they d i f f e r f r o m xy- and H e i s e n b e r g
the spins
interac-
or t e t r a h e d r a
significantly.
fields the m a g n e t i c m o m e n t
ions can o n l y be o r i e n t e d
interactions
temperature.
are u s e d to d e s c r i b e m a g n e t i c
cause of local c r y s t a l
correspond
to a vacancy.
in these m o n o l a y e r s from
nn
interactions become very important.
higher-dimensional
n = I).
the i n t e r a c t i o n s
as for a n t i f e r r o m a g n e t i c
i n t e r a c t i o n s m a y also lead to a c r o s s o v e r
Ising s y s t e m s
are of g r e a t sig-
. If
c o n s i s t i n g of joint t r i a n g l e s
w e a k they can r e d u c e the g r o u n d s t a t e d e g e n e r a c y
Spins
too m e t a -
l e a d i n g to spin
s u i t e d to test a p p r o x i m a t i o n
are o f t e n not l i m i t e d to n e a r e s t n e i g h b o r s
Apart
systems
for t r e a t i n g spin glasses.
Experimentally,
where
(2d)
frustrated systems
long l i f e t i m e m a y o c c u r
Therefore,
in t h e m s e l v e s ,
for the i n t e r e s t in p e r i o d i c
at least for t w o - d i m e n s i o n a l
nn
(e.g.
interactions
to an a b s o r p t i o n
For
site o c c u p i e d by an atom,
the p r o p e r d e s c r i p t i o n of p h a s e t r a n s i t i o n s from c o m m e n s u r a t e
one needs c o m p e t i n g
to i n c o m m e n s u r a t e )
apart
interactions between more
d i s t a n t sites. Substitution
alloys
AxBI_ x
c o n s i s t i n g of two types of atoms
can also be d e s c r i b e d as Ising interstitial i , and
s
1
sites. = -I
Then
to a
systems,
si = I B
atom.
A, B
if the atoms c a n n o t go on
corresponds
to an
A
a t o m at site
As a l a s t e x a m p l e
for t h e m a n y
ments,
the
we mention
the p o s i t i o n described
racy
of the p r o t o n s
originally
temperature
of a f r u s t r a t e d
adjacent After very
to the
1.1
of t h e
vertex
system,
physical
fields,
critical
properties
defined
where
to e x p e r i -
For
e.g.
instance
the
the
spin orientation
of t h e p r o t o n
for
in f e r r o e l e c t r i c
on t h e g r o u n d s t a t e
systems
energy,
If it is p o s s i b l e
systems),
low-
degenes i = ±I
between
the
two
transitions
at
reduced
T c . Defining
terminology
M(t)
~
the
of m a g n e t i c
tp
is t h e
B
close
to the p h a s e
Tc
For
tion
length
the
for
r ~ ~
critical
properties
such phase
(n = I)
of s e c o n d behavior
quantities
further
t =
Ising
which
is f i n i t e
order
M
are
IT-Tcl/T c
for
in f e r -
character-
of t h e o r d e r
parameter
, in the
c , the
the asymptotic
systems
properties
asymptotic
critical
as t h e
finds
describing
For
. Other
heat
a n d the
at a
transitions
(1.2a)
exponent
transition.
specific ~
one
review.
the m a g n e t i s a t i o n
temperature
systems
short
transitions,
(T < T c)
Here
is a s c a l a r
of t h e i r
parameter,
(e.g.
but nonanalytic
a very
phase
to
susceptibility.
an order
ized by a continuous,
systems
Transitions
. Often
the
Ising
of t h e p r e s e n t
in t h e r m o d y n a m i c
T > Tc
phase
order
Phase
change Tc
heat and
for
survey
Order
show a sudden
to i n t r o d u c e
1.1 w e g i v e
at s e c o n d
temperature
specific
T < T c , but vanishes romagnetic
at S e c o n d
by singularities
the
of f r u s t r a t e d
in S e c t i o n
1.2 b y a d e t a i l e d
critical
are a c c o m p a n i e d
M
transitions
models.
positions
of the c o n n e c t i o n
Phenomena
Many physical
free
Ising
in S e c t i o n
Critical
well
or t h e
of ice c a n b e m a p p e d
two a l l o w e d
discussion
different
followed
in ice,
systems
transitions,
02--ions.
this
summary
of I s i n g
order-disorder
by different
degeneracy
corresponds
applications
structural
show analogous
susceptibility
parameter behavior
X , the
pair correlation
exponents
behavior
the o r d e r
correla-
function
G(r)
can be defined:
C (t)
~
t -a
,
X (t)
~
t -Y
m(H)
~
H I/6
,
G(r)
~
r 2-d-n
,
~ (t) :
t
=
~ O
t -~
at
: (I.
H = O 2b)
Experiments ferent
until
developed.
on the
o n the
of
and
1.1
Normal
d
a
20(log)
~
1.1
the
lattice
have
exponents
~
002
than
stars mark
in t h i s
one
and also
the
same
Ising
for
component,
not
systems
dif-
not be
theory 4 near-
depend , but not specific
on the
exponents.
are
5
:
d = 3
q I/4 ~ I ~
5.0±0.05
system,
as m e n t i o n e d
d
(!) on the
exponents
for
could
group
exponents
critical
15.04±0.O7
results
of v e r y
ferromagnetic
Ising
6
1.250±0.002
exact
This
dimension
d = 2 and
7/4~
O.31 2 _ +O. 0.005
with
critical
lattice
Y
If f r u s t r a t i o n o c c u r s in an I s i n g more
the
ferromagnetic
the n o r m a l
Ising
exponents.
systems
a n d the
interactions,
d = 3
I/8 ~
3 0.013±0.01
In T a b l e
the
n
classes'
and renormalization
interactions
example,
I
critical
in f e r r o m a g n e t i c
square
and
'universality
hypothesis
pair
For
d = 2
Table
identical
spin dimension
type.
triangular For
(nn)
strength
lattice
have
that whole
scaling Thus
est neighbor only
shown,
substances
understood were
have
.~^+0.002 O.bJ~_O.OO1
-
d = 2 .
the o r d e r
above.
parameter
General
may have
considerations
connection
and Alexander
h a v e b e e n w o r k e d o u t e.g. b y M u k a m e l a n d K r i n s k y 6 7 a n d P i n c u s . As the d i m e n s i o n of t h e o r d e r p a r a m e t e r
in f r u s t r a t e d
systems
not only havior Until
on
a magnetic
but
, one
now we have
yields this
d
as c o m p a r e d
field
depends
can expect
a much
richer
to the n o n f r u s t r a t e d regarded (one-spin
only
general
models
review
we will
consider
for
with
four-spin
variety
structure
and
of c r i t i c a l
interactions. and multi-spin
different
almost
lattice
be-
case.
two-spin
interaction)
more
a few results
on t h e d e t a i l l e d
universality
exclusively
interactions
are
Inclusion
of
interactions classes.
In
pair-interactions, added.
1.2
Scope
of This Book
The f o l l o w i n g ing to the systems
three
lattice
simple
interactions
e.g.
contrast
to this for
as high-
Id
to solve, transfer
for
3d
2 d
systems
cluster-approximations, Carlo
for
T = O
Chapter exact tems
(in 3.1)
generacy,
is
2.1 with
the specific
After
crossing
heat,
one,
del,
frustrated
In
and in
are available,
field-
(MF-)
group
(RG)
such
approximamethods
and
4 reviews
lattice
type differ
the
disorder
decorated
and Section
3d
a large number
Ising m o d e l
behavior
systems
function
(GS) de-
of the pair
and,
cor-
on the
In Section
without
for instance
of
sys-
on the triangu-
and the m a p p i n g
are discussed.
Ising
of Ising
the g r o u n d s t a t e
line,
differ
in
3.3
crossing GS
therefore,
entrobelong
classes. the properties
Especially,
very
(AF)
2d
to the exactly
Chapter
of the A N N N I - c h a i n
d = 2
Besides
which
3.4 summarizes
interactions.
is compared
crossing
solutions.
transformations
Ising chain
are considered,
universality
as for
the a s y m p t o t i c
the r e l a t e d
of further
Section
interaction~
interactions
methods
mean
frustrated
without
the p r o p e r t i e s
general
py and the decay of the c o r r e l a t i o n
Finally
crossing
2.2 describes
extensively.
and the kinetic
to d i f f e r e n t
systems
accord-
Ising chains.
the a n t i f e r r o m a g n e t i c
function,
interactions
with
periodic
short range
yield exact
renormalization
. Section
known.
is treated
quantum-XY-
2d
expansion,
3 is by far the longest
lar lattice
a couple
T % O
frustrated
results
relation
and for
are arranged
calculations.
in Section
and
2.3 p a r t i a l l y
systems with
systems
tion,
2 deals
of the various
only a p p r o x i m a t e
and l o w - t e m p e r a t u r e
(MC)
d Ising
m a t r i x methods
Monte
Chapter
(2 to 4) of this book
dimension
(d = I to 3)
are u s u a l l y
general
chapters
solvable
systems, much
of three
the first of them, brick m o d e l
which
and belong
depending
systems the
with
ANNNI-mo-
in Section
3.4.2.
on the specific
to d i f f e r e n t
universality
classes. Chapter
5 finally
frustrated
Ising
contains systems.
a short
summary
of this
review
on periodic
2.
One-Dimensional
To e x p l a i n lar
Ising
tioned
Frustrated
the e x p r e s s i o n cluster
as the
with
field
three
sider
one-dimensional
connecting stems they
because, show
trated
systems
can o c c u r only
for
systems.
T = 0
, as
d = I
2.1
several
2.1
Periodic
the the
2.1a
triangles
nn and n n n
~sing-model)
with
-
for
relative
J1
E
o
i
i ai+1
H
=
-
real
long
short
are
ways
more
con-
of systhem,
frus-
order
(LRO)
interactions
dimension
shown
a
we n o w
to solve
range
range
critical
to d i s c u s s
i
s
-
i
that
edges.
the
of such
to f o r m
closely
a chain
now.
By r e d r a w i n g
interactions
in a l i n e a r
-
will
chain.
J1
This
J2
field,
J
= si E
be d i s c u s s e d
it and J2
linear next
are
chain
is
nearest
versions
with
later.
is:
E
i
J2
~
(2
i ~i+2
< O
it o n l y
of one h a l f - c h a i n
oi ~i+I E
common
properties
9
B
lower
have
(AF)
a magnetic
orientation
Substituting
is the
seen,
d = I
antiferromangetic
ter w i t h o u t
with
v e r s i o n of the A N N N I - m o d e l (axial 8 ; its two- a n d t h r e e - d i m e n s i o n a l
different
Hamiltonian
=
Of c o u r s e
men-
with
one-dimensional
simplicity
possibilities
interactions
one-dimensional
H
chapter
to the h i g h e r - d i m e n s i o n a l
systems
are g o i n g
it is e a s i l y
characteristic The
we
was
six,
by d i f f e r e n t
these
triangu-
ANNNI-Chain the
2.]a')
~eighbor
later.
field
In this
of the m a t h e m a t i c a l
discussed
which
treat
the
interactions
obtained
We
similarities
In Fig.
triangles,
(Fig.
many
introduction
a magnetic
occur.
systems,
in o n e - d i m e n s i o n a l
from
In Fig.
Ising
triangles.
in s p i t e
already
Without
groundstates
frustrated
in the
antiferromagnetic
example.
degenerate
Systems
frustration,
equal
simplest
Ising
; the
,
of
with
J1
does
(see Fig.
to the o t h e r
the H a m i l t o n i a n
s i si+ I
sign
determines
(2.1)
B = J1
I)
not mat-
2.1a)
one. is m a p p e d
into
' J = J2
'
i (2.2)
the
J2
J2
J2
J2
J2
J2
J2.
J2
J2
J2
(1.) J~
Frustrated ways: (a)
i.e.
the
form tion.
Periodic
ANNNI
the
chains
(a') to (c')
with
nn-
of
which
Hamiltonian
•
(= Mock
~,/<-:-2-
J2
frustrated triangles
nn and nnn
interactions
'knot spins', o ANNNI
in different Jl and J2 ;
'decorated spins';
chain).
the chains are drawn strictly one-dimensional.
and
nnn-interactions
is
equivalent
nn-interactions
in
a homogeneous
antiferromagnetic sign
J2 J,
(n-l) J2
chain with
simple decorated chain;
chain
~
chains formed by connecting
n-decorated
the of
J2 (2.)
(c)
with
field,
" " "
(b)
In
chain
(n)
~ , ~
J2
Fig. 2.1
J2
of
course
later
to
is
irrelevant.
calculate
the
We
pair
will
to
use
correlation
the external this func-
2.1.1
Grounds~a~e_De~ene[a~z_£~_the__AN_NN~ichgin
L e t us
consider
on the r a t i o only
two
and
there
spins
simple
occur,
for
is c a l l e d states,
a = I/2
functions
nn No
for
spins,
ending
j+1
N2
there
is no
LRO
the p o i n t p o i n t 11.
spin
each
for
with
> 0
For
a > I/2
alternating (e.g.
with
long
J2
two
ref.
it d e p e n d s
and t h e r e
dominates,
spins
range
order
T = O)
in p h a s e
a large
number
is the
the
chain
requirement
is space
of g r o u n d of at l e a s t
over
Defining
orientations
(GS)
and
these
GS
the n u m b e r
++,
the r e c u r s i o n
+-,
-+,
the p a i r
correlation
can be c a l c u l a t e d of
--
GS
, with
of a c h a i n N~,
N
N
formulas:
N~ (2.3)
j+1 N3
=
N3
Combining
the
and
N3 + N3
NJ= m N j+1 p
=
GS
ending
with
respectively,
a
(+)- or from
(-)-spin
(2.3)
to
N j = N~ P
+ N3
follows:
N j + N j-1 p m (2.4)
N j+1 m
=
N j + N j-1 m
p
two
(LRO)
a = I/2
exists
up,
10).
. For
now
are
spin.
averaged
formulas.
J1
(a = I/2,
There
condition
immediately
=
groundstates
the o n l y
a = I/2
recursion
obtains
with
For
dominates
groundstates.
of the g r o u n d s t a t e s
simple
J1
<2>-groundstates
only
a frustration
of (2.1).
a < I/2
groundstates
frustrated",
for w h i c h
The n u m b e r
. For
we call
a ~ I/2
one p a r a l l e l
one
four
which
for
"completely
the g r o u n d s t a t e
ferromagnetic
are n o w
down,
Whereas
first
a = -J1/J2
from of
j N3
10
For
the
total
number
of
q , N~ = NJ + NJ
GS
•
o
N] pm
= Nj p N j+1
Nj m =
O
=
- N j-2 pm
(2.5a)
is
the
j ~ ~
N~)
Nj o The
for
the
difference
recursion
formulas:
(2.5a)
I
(2.5b)
well
one
and
m
known
obtains
definiton
of
(independent
a Fibonacci-series,
of
the
first
two
in
the
values
N2 o
:
~
qJ
resulting
S
the
O
N j+1 pm
limit and
gets
N j + N j-1
O
As
one
p
=
o
whereas
with
GS
entropy
l i m ~ in N j j~ 3 o
for
a #
I/2
a continuous
function
function
obtains
one
G(r)
=
~+1 2
q
per
=
in
the
GS
in
<~I~I+r>o
a
lattice
q
=
site
for
So
O.4812
entropy
N r+1 P Nr+1 p
~
(2.6)
for
I/2
_ N r+1 m + Nr+1 m
I/2
,
. For
Nr pm qr
~
is:
(2.7)
vanishes.
s =
s =
Thus the
S(T=O)
pair
not
correlation
-rlnq =
is
. Nr pm
e
(2.8)
From For
(2.5b) chains
tains
the
N~ pm
is
beginning simple
periodic with
1 0
(+-)-start:
1 I0
j
with
(++)-spins
periodic
(++)-start:
in
series
1 1 0
1
1 I0
or
1
...
j
r
for
6
(+-)-spins,
(period
...
period
6)
. for
one
Nr pm
ob-
:
r = 2,
3,
4,
5,
6,
7,
...
(2.9) In
the
middle
probable, requirement
the
of
a chain
correct
both
linear
=
start
configurations
combination
for
is
are
obtained
I << n - r
< n +
not
by
equally
the
r <<
symmetry j
:
]]
Nr pm
2 N (++)r pm
=
+ N (+-)r pm
=
(2),
I,
I,
2,
I,
I,
...
2,
(2.10) for
Finally
from
function
(2.6),
G(r)
lim G(r) r~
~
sional agree
2.1.2
decay
cos
limit
the
T ~ 0 systems
limit
Periodic
using
the p u r p o s e form
a = I/2
the
As
we o n l y
note,
for
that
averages
for
correlation
finite
6
the p a i r
transfer
is s h o w n
correlation
for h i g h e r - d i m e n do not
matrix
necessarily
for
finite
method.
to take
e.g.
function
the
Hamiltonian
by H o r n r e i c h ,
in the trans.
Liebmann,
function
(with p e r i o d i c
of a c h a i n
boundary
conditions)
can be w r i t t e n
Schuster
with
by p r o d u c t s
j
spin: of
matrices:
=
Tr
(2x2)
(~rT3-r) /Tr (T3)
transfer
at
temper-
T ~ 0
correlation
two
pair
of p e r i o d
again
the p a i r
with
...
.
it is c o n v e n i e n t
(2.2).
find
S e l k e 12'13
transfer
7,
is
by an o s c i l l a t i o n
the g r o u n d s t a t e
ANNNI-chain
temperature
and
5, 6,
asymptotic
for
we w i l l
. Here
T ~ O
to c a l c u l a t e
formed
3, 4,
<3 " r)
a = I/2
We n o w w a n t
For
the
chain
multiplied
point
frustrated with
(2.10)
2,
in (~+lh \---~'-----)
exponential
in the
(I),
infinite
e -r/~°
=
frustration
ature
and
=
(2.11)
6o
This the
in the
-I
with
(2.8)
r
matrices
#~ss
~ 2 i i+I
(2.12)
(s i = +I)
~(si,si+ I )
=
exp
m(si,si+ I )
=
s i T (si,si+ I) si+ I
:
+ KI ( s i + s i + 1 ) / 2
}
and ,
(2.13)
12
where
K I = B J1
Diagonalizing trices
' K2 = 8 J 2
w and ~
one o b t a i n s
and
B = I/kBT
and using
in the limit
the r e s p e c t i v e 18 :
transformation
ma-
j ~ ~
-rln (I I/I I ) a e
•
-rln (I I/I 2 ) +
(l-a)
e
for
~I ~ ~2
G (r) -rln (I i/~) e
(1+br/Y)
for
~I = ~2 = ~
' (2.14)
N
with
the e i g e n v a l u e s K2 11,2
----
e
=
e
of
w and • :
-2K2h 1/2 cosh K I ± (e 2K2 sinh 2 K 1 + e
K2 ~I ,2
It is i n t e r e s t i n g K11
= kBT/J I )
For
/ 2K2 sinh K I ± ~e
to note
according
cosh K I > e -2K2
simple
exponential
correlation
=
(2.14)
both
decay
-2K2h 1/2 cosh 2 K I - e
the two r e g i o n s to
/
in p h a s e
with different
~I and ~2
of the pair
/
are real, correlation
(2.15)
space behavior
(a = J2/J1 of
and one o b t a i n s function
with
in
(t 1/~'1) ]
(2.16)
,
cosh K 1 < e-2K2
~'1 ,2
are complex,
and the e x p o n e n t i a l
decay w i t h N
(In
is m u l t i p l i e d
q
=
by an o s c i l l a t i o n
-2K 2 e
with wavevectOr
(Im ~ I / R e ~i )
between =
(2.16')
I~I/I 1 I) -I
arc tan
The b o u n d a r y cos K I
a
the
length
whereas f o r
=
G(r)
these
(2.17)
two r e g i o n s
(2.18)
,
18
in ref.
12 we c a l l e d
'Lifshitz
that S t e p h e n s o n 13 o b k a i n e d boundary
between
correlation two kinds a)
oscillating
function
of them:
a
condition'.
earlier.
and n o n o s c i l l a t i n g
'disorder
In d i s o r d e r
first kind
Subsequently
the same results
line'.
lines
the w a v e v e c t o r continuously
it t u r n e d
He c a l l e d
out,
the
d e c a y of the pair
In a d d i t i o n
he d i s t i n g u i s h e d
of the q
e.g.
of the o s c i l l a t i o n with
temperature,
varies
as in our
example; b)
second kind
q
is t e m p e r a t u r e
for example
Figure
2.214
shows
the t e m p e r a t u r e
line
q
values
varies of
q
for the chain
the d e p e n d e n c e
K~ I
line no o s c i l l a t i o n s
independent.
of
In the region occur
shown
, Eq.
on the
(q ~ O)
continuously;
q
left
is true
in Fig.
(2.17),
, whereas
the d a s h e d
This
on
2.1c.
s
side b e l o w
and the full
to the right of this
lines
correspond
to fixed
.
\
\
\
'\ \
\ ~
I
2
q -- 0u
0
Fig.
2.2
0.1
L i n e s of c o n s t a n t the (fully drawn)
~ ,\
0.2 0.3 04
~\
/I I/
/If
0.5
0.6
//
q:l.& ~_~
cX=- K2/K,
-1 w a v e v e c t o r in the KI - e - p l a n e ; disorder line q H O .
on the
left side below
14
The
frustration
point, from
because
which
Figure state
2.3 is
PF
limit
T = O) q
is a p p r o a c h e d .
shows
the behavior
line
monotonous
to
in
(~ = 0 . 5 , value
PF
ferromagnetic
disorder
minima
point the
where
for
the
of
~
s < I/2
decay
oscillating,
( , justifiing
of
of
Eq. the
the
at
s < I/2
, lim
6 -I = O
correlation one
13.
on
the
direction,
As
the
ground-
. Precisely function
finds
'disorder
!.5~-
is a s i n g u l a r
depends
for
(2.18),
name
obviously PF
maxima
at
the
changes in
(-i
from , e.g.
line'
/ t ~.jl
/
Ln(l+~2)
o
Fig. 2.3
The
of
=
In c o n t r a s t and
,.o
2.0 ksT/iJ,i
2.5
Temperature dependence of the inverse correlation length ~-1 (T) for fixed values of ~ . ~-I shows cusps along the disorder line (~ < 0.5 , compare with Fig. 2.2).
limit
~i
0.5
~-I
in
approaching
does
~S I
=
in
along
the
disorder
line
(V~+I)
to this
also
PF
not
(2.19)
for
a = I/2
vanish
\--~---)
is
for
,
~-I (T)
T ~ O
, but
does
not
approaches
show
any
spike,
a finite
value:
(2.20)
15
with
- 3 ~ ' in f u l l
q ~qo
from averaging Besides tities
the p a i r
lattice
U
=
U
=
in ~I - B
(~inlI/~)
c
B2
(~21nll/ZBa)
11
T ~ O
Figure
of t h e
entropy
(2.22)
(or
K I ~ ~)
singular
value
frustration
point
In a d d i t i o n
we note
S
as
2.2
the
are
of Fig.
2.1b,c.
the c
(each
spins
(2.23)
2 . 4 a , b 15 c l e a r l y the v a l u e
(~+I)/2) T = O)
the
completely
of
shows
monotonous
the
a = 0.5
sharp
~ O.481
in the v i c i n i t y
detailed
continue
decorated
=
e.g.
heat
quan-
(2.22)
increasingly
with
chain
asymmetric
, which
for
and reproduces
, Eq.
(2.7),
behavior
exactly
at t h e
of t h e
entropy
of the d i s o r d e r
line,
as these
of t h e p e r i o d i c
ANNNI-chain
~I,2
coupled
dependent
K2 + I/2
only
discussion a shorter
(Fig.
S t e p h e n s o n 13 • By d e d e c o r a t i o n
keff I
method,
specific
,
S o = in
relatively
temperature
matrix
thermodynamic
Chains
2.1a we
corated
other
and the
(s = 0.5,
independent
of Fig.
simple
S
near
becomes
U a n d c)
Decorated
After
also
,
(2.15).
quantities
obtained
(2.21)
from
(as w e l l
(2.11)
site):
S/k B
=
the r e s u l t
b y the t r a n s f e r
, the entropy
,
maximum
The
function
- in II/B
with
the
correlation
energy
with
the groundstates.
c a n be d e t e r m i n e d
internal per
over
agreement
2.]b)
has
at t h e d e c o r a t e d
also been
16 , t h a t is b y s u m m a t i o n
to n n - s p i n s
nn-interaction
in c o s h
look
2K I
one
obtains
treated over
chains
by
the d e -
an e f f e c t i v e
of t h e r e m a i n i n g
'knot spins'
(2.24)
16 S 0.5
0.4
0.3
0.2
0.1
0.0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Ct.---~
Fig.
2.4a
Entropy S ture K1 •
f
as function
of
~
for fixed values
of the inverse
tempera-
1.0
\ \ \ o.so
0.8 -'x. \ \ N " ~k 0.6
o.so/./3/
0.481 y / /
0.481
X" 0.46
0.46) " 1 /
,o.~\ \ X X , °.4° o.4o,,<,9,- /
0.4 0.2
-.
~
00
i
0.2
Fig.
2.4b
0.3
0.4
Lines of c o n s t a n t e n t r o p y point S is not analytic.
0.5
S
0.6
0.7
\i
0.8
in the T-e-plane.
A t the f r u s t r a t i o n
17
This
effective
interaction
vanishes
for
-2K 2 cosh The
=
condition
chain, to
2K I
Eq.
that
and
even As
value)
Stephenson Fig.
q
LRO for
The
been
the
it
T = Td from called
behavior
of
analogous
is
to
< a < I
along
the
The
frustration at
does
to
~
(the
a disorder ~eff(T)
(a =
1
. This
from
of ref.
ANNNI
opposed
disorder
line,
(q =
~/2)
point,
kind
the 13
of
disorder
independent second is
kind
shown
in
2.5.
I
1
I
.6 .5
0
Fig. 2.5
o.5
1.0
1.5
2.0
kBT~lasl
,
where
, T = O)
temperature line
the
. As
antiferromagnetic
occurs
, so
0
sign
decay.
0
line
interval
changes
exists,
switches
, has
13.
Td
exponential
no
vanishes
a disorder for
just
temperature
T = O
~/2
now
K eIf f
with
where
defines
but
here
the
K 7 ff
line,
(2.25)
cases
for
(2.25)
(2.18),
case
below
in b o t h
e
2.5
Temperature dependence of the correlation length ~ decorated chain for fixed values of ~ (indicated). order line ~ = O . (Ref. 13)
of the simple Along the dis-
by
18
After can
this
now
been
I
tration trices
the
ly.
As
has
the
For
the
spin
decorated
-
knot (o)
with
spins
, and
spins
part
of
K I , the
x
zeros
to
(a = 0 . 5 , summed
n
been
it n o w
(Fig.
2.1c.
JI/2 T = O)
out,
of , to
now
can
[(n+2)/2]
n the
knot
again
spins
the
frusma-
an e f f e c -
sign
repeated-
where
[x]
. are
given
explicitly
by
i n ( ½ ~(1+e-2K1) 2 + ( 1 _ e - 2 K 1 ) 2 t a n h 2 \/m ~
2I [i + ~I
ANNNI.
transfer
times,
hat
instead
shift
change
we
model
uniaxial
. Using
leaving
2.1b)
This
contains
nn-interaction
reduced
spins can be eff KI ( K I , K 2)
chain
of F i g .
higher-dimensional (.)
the
is
decorated
chain
eff decorated spins K1 eff s h o w n 17, K I vanishes
the
integer
given
simple
to t h e A N N N I - v a l u e
nn-interaction of
the
generalized
in c o n n e c t i o n
decorated
point
Because
is
nn
of
the
Inbetween
decorated
to their
tire
treat
investigated
m o d e l s 17. of
consideration
easily
.YI J]
,
(2.26)
with
m = O,
...,
[n/2]
, and
are
shown
as f u l l
1 q~:-~-~qs=-~
2
lines
in Fig.
2.618 .
n=9
K]I ,
\2 \
q~.
00
I
=~T~_ 3 ~\
I
0.2
i
q :T0
0.&
0.6 K2 KI
Fig. 2.6
0.8
Disorder lines of the second kind of the (n:9)-decorated chain (from Eq. (2.26)). In the regions between these lines the wavevector is constant: q = m -i-~ z , O < m < 5 .
19
For
T ~ K~ I ~ 0
frustration
c
As
a
chain the
asymptotically
point
with
-
=
2
{ in ~2 cos
n
second
(-) qm
straight
lines
through
the
slope
(2.27)
generalization
decoration
kind,
qm(-)
value
are
m ~ (n ~) ]
straightforward with
these
where
spins
thus
of the
simple
decorated
shows
In+2] IT]
disorder
lines
from
fixed
the w a v e v e c t o r (+) one qm :
q
switches
one
chain
the of
to the n e x t
m
=
n + I " ~ (2.28)
q(+) m
The
=
first,
m + I . n + I a
n-independent
the b o u n d a r y the
single
[n/2]
of the
disorder
ones
occur
occur,
The
chain
separated
one r e c o v e r s boundary
line
the
(2.18)
in a d d i t i o n
in the n - d e c o r a t e d qm
disorder
line
nonoscillating
in
(2.23))
(q ~ O)
of the p e r i o d i c
by d i s o r d e r
lines
variation
number lying of
forming
is i d e n t i c a l
~NNNI-chain.
a n d are n - d e p e n d e n t .
an i n c r e a s i n g
continuous
(m = 0
region
With
increasing
of r e g i o n s
with
dense
n ~ ~
q
(Eq.
to
The o t h e r
for
(2.17))
fixed . Thus
beyond
the
(2.18).
connection
between
can be d e s c r i b e d
the p e r i o d i c
even more
AN[NNI- and the n - d e c o r a t e d
precisely.
Inserting
(2.28)
in
chain
(2.26)
one
obtains
=
~m
in a n a l o g y structure varies The
to
(2.17),
continuously lines
of c o n s t a n t
qm-)
,
with
~
with
the
of the n - d e c o r a t e d
disorder
lines
.Im ~ \{ R ~ 1 1 ~ /
arc tan
qm
sole
chain
difference qm
in the p e r i o d i c of the n - d e c o r a t e d
wavevector
qperiod.
~
,
that
is d i s c r e t e ,
(2.29)
because
of the
whereas
is
ANNNI-chain. chain
in the p e r i o d i c
(+) qm
~ m~
=
are
at the
chain,
same
time
with
(2.30)
n
20
The continuous
strated
respective
stepwise dependence
for. the periodic and the
q = q(a)
in Fig.
2.7 for
K11
= 0.5
of the w a v e v e c t o r
(n=9)-decorated
i '
q
'x;f
1.0
7
0.5~
I
o
2,3
>c
f ~ I
!
0.2
I
l
O.Z,
larity
chains
and d i s t i n g u i s h
The motivation
in C h a p t e r
fully
pyrochlore-
4. The r a n d o m
of spin glasses,
analytically.
between
for c o n s i d e r i n g
to h i g h e r - d i m e n s i o n a l
the t h r e e - d i m e n s i o n a l
version
?
I
0.8
Chains
We now turn to the last o n e - d i m e n s i o n a l
chains.
,
0.6
Comparison of the ~ dependence of the wavevector q for the periodic ANNNI-chain (continuous) and the (n=9)-decorated chain for T/J i = 0.5 .
Partially Frustrated
trated
on
is demon-
I
1.5
Fig. 2.7
chain
case
examples,
periodic
the p e r i o d i c
frustrated
case
systems
(or B-spinell-)
is i n t e r e s t i n g
because
the p a r t i a l l y
and r a n d o m
frus-
frustrated is its simi-
as for example
lattice
discussed
as a o n e - d i m e n s i o n a l
here one can treat
the d i s o r d e r
still
21
2.3.1
Periodic
Doman
Frustrated
Chains
a n d W i l l i a m s 19 h a v e
geneous
magnetic
field
considered
B
, where
antiferromagnetic
(AF)
(here
chain):
called
H
=
I-3
- X Ji i
a periodic
one
nn-interactions
si si+1
Ising
ferromagnetic
- B X si i
are
chain (F)
repeated
in a homo-
and
three
periodically
;
(2.31a)
with
J4i
=
J
J4i+1
=
J4i+2
=
J4i+3
=
- J
;
J > O (2.31b)
Using
the
chain
is e q u i v a l e n t
transformation
nn-interactions
H
both
=
-
versions
to
J(2)i
X
= Ji
I ~i+2
the
I-3
- J(1)
chain
B=konst.; --
-J
--
J
with
(compare
Sec.
nn-interactions
2.1) J(1)
this = B
I-3 and
:
a.
i J(2)i
of
si = ai ~i+I a chain
(2.32)
Xi o.i ~i+I
are
shown
in Fig. 2 . 8 .
J•O A
-J
A
-J
A
-J
A
--
J
-J
-j
~ Transformation B
-J
B
J
-J
i-3 --
-J
B
-j
-j
J
Fig. 2.8
B
B
-j
B
J
-J
-J
chain of Doman and Williams. ferromagnetic interactions, - -
B
-J
-j
-j
J
-J
antiferromagnetic interactions.
22
With
the t r a n s f e r
have
determined
matrix
method
the energy
a n d the m a g n e t i s a t i o n
M
and the magnetisation
these
figures
For
= 2
(B/J)
apparently
are
Doman
, the
2.9 u n d
(B/J)cl
2.10
determine
of
heat
c
the enB/J
= I
configuration
J
a n d W i l l i a m s 19
specific
as f u n c t i o n s
ratios
the groundstate
the nn-interactions
S
in Figs.
reproduced
two c r i t i c a l
exist where < I
already mentioned,
, the entropy
. As examples
tropy
(B/J)c2
U
. From
and
changes.
the g r o u n d s t a t e ,
l
with
M° = O
interacting
and
SO = 0
spins
are o r i e n t e d
antiparallel
pairs
This yields
M ° = I/2
all
spins
critical cially ues
of s p i n s and
are p a r a l l e l values
large,
inbetween
. For
:
~ in 4
M CI o
-
g2 4
field, B/J
~
< 2
all p a i r s
to the magnetic
have
two allowed
in 2 ~ O . 1 7 3 3
ANNI-chain, 19 :
of
field,
the
(B/J)
> 2
. Just at the
entropy
and
F
configurations.
. For
M ° = I , So = 0
the g r o u n d s t a t e
in t h e p e r i o d i c
(V~+I)
--
parallel
S o = I/4
to the
the n e i g h b o r i n g
S CI o
(B/J)
inbetween
of the r a t i o
like
I <
M
is e s p e -
exhibits
val-
o
'phases'
0.2203
0.3536 (2.33)
For
sC2 o
_
I in 3 4
MC2 o
=
2
The
0.2747
3
finite
temperature
o u t to a s y m m e t r i c
2.3.2
~
this
continuous
discontinuous
Ising
chain
in a h o m o g e n e o u s
magnetic
nn-interactions
has b e e n
by a number
investigated
essential
chains
is a g a i n w a s h e d
R andom Frustrated Chain . . . . . . . . . . . . . . . . . . . . . . .
and antiferromagnetic
The
behavior
curves.
already
configurations
additional discussed for g i v e n
problem
field with
random
of c o n c e n t r a t i o n
ferro-
(l-x)
and x
of a u t h o r s 19-25. compared
is the n e c e s s i t y concentration
x
to the p e r i o d i c to a v e r a g e . Whereas
frustrated
over all bond
the p u r e
antifer-
23
03
S/kB 02
01
05
10
l'S
20
215
3~)
B/J
Fig.
2.9
Entropy S of the i-3 the inverse t e m p e < a t u r e OCCUr.
chain as function of B/J for fixed values of J/T (ref. 19). Two critical m a g n e t i c fields
Z
5
O&
0
0.5
I0
~.s
2,o
is
lO
B/J
Fig.
2.10
Magnetisation M temperature J/T
of the i-3 (ref. 19).
chain for f i x e d values of the inverse
24
romagnetic only
one
cussed
chain
in a m a g n e t i c
critical
above
magnetic
exhibits
shows
steps,
state
magnetisation
can
be
traced
e.g.
of
tion
than
After and
spin
the
back
(B c = 2J) where
random bond chain for
the
ANNNI-chain)
, and
the
I-3
groundstate
there
are
steps
in
all
with
opposite
r'
more
exhibits chain
Ising spins oriented
the ...,
ground~
. They 20 superspins , in o n e
direc-
one.
et
w o r k , for i n s t a n c e b y M a t s u b a r a 21 , L a n d a u a n d B l u m e 22 25 24 al. , D e r r i d a et al. in 1978 o b t a i n e d e x p l i c i t a n a l y -
expressions
Figure x = 0.5
2.11
shows
. The
is e v i d e n t ,
for
the
their
series
inbetween
of S
groundstate
result
for
spikes
at t h e
o
properties.
the
groundstate critical
entropy
fields
B
SO
for
= 2J/r
is c o n s t a n t .
~ 00~
|
1
2
H/J
Fig. 2.11
The
Entropy S of the random frustrated chain as function of H/J for x = 0.5 and T = O ; the value of S for H/J = 2 is O.143 , beyond the drawing (ref. 24).
analytical
fields
dis-
magnetisation
B (r) = 2J/r, r = I, 2, 3, c to f l i p p i n g s p i n b l o c k s , also called
clusters
in t h e
field ones,
(~ p e r i o d i c
previous
Puma
tical
in
two
field
are
expressions
given
in D e r r i d a
for et
So
, at
al. 24.
and
between
the
critical
25
In
comparison
to Fig.
finite
temperature
Figure
2.12
0.5
and
with For B
0.8
24.
x ~ I = 2J
Cl chain
are
shows
increasing
2.9
the Note
here
not
(e.g.
in
steps
of
(B ~ J1'
x
pure
agreeing
J ~
iJ21
the
continuous
the magnetisation
the monotonous
the
2.11
curves
for
given.
concentration
survives,
in Fig.
chain)
0
only
with
the
the
B = 2J ~ J1
r/
x = 0.2,
of M for o antiferromagnetic
the
course
' thus
for
decrease
of
AF of
M°
first
results
= - 2 J2
B
step for
or
at
the
ANNNI-
s = 0.5)
o [
1
2 H/J
Fig. 2.12
Magnetisation as function of H/J for three concentrations x of antiferromagnetic interactions: (a) x = 0.2 , (b) x = 0.5, (c) x = 0.8 (ref. 24).
At
P u m a a n d F e r n a n d e z 25
first
tropy
of
atUre
the
could
calculated energy, the
see
the
critical only
As
shape
bond
the
shape
entropy,
depend the
random
1978 for
numerically T % 0
few minima.
the
specific
fields on
first of
in
chain
continuous
heat
, and
Then
Doman
finite
curves
for
low and
in
the
enough Williams
temperature
and magnetisation
B(r)c . T h e s e
determined at
the
IB-2J/rl
en-
temper19
curves
for
vicinity << k B T
of
=
B -I
T ~ O
is
B(B-2J/r)
of
these
'washed-out'
qualitatively
similar
corresponding
r e s u l t s 19.
continuous
to t h e A N N N I - c a s e ,
here
curves we
only
for refer
to
the
26
3.
Two-Dimensional
Frustrated
In the last chapter we have with
different
behavior
tion.
These
phase
transition
with
We add,
discussed
continuous
du = 2
In this
chapter we now c o n s i d e r
systems
special
which
case exhibit
ratios
Transformations
structure,
square This
nection
the
individual
(LRO)
(n > I)
of t w o - d i m e n s i o n a l
types
of phase
interactions
systems
behavior
di-
frustrated
in the non-
transitions;
some may
depend
to w h i c h m o s t
for
show no or sev-
(a)
duality
(b)
decoration-iteration
(c)
star-triangle
(c)
by Kramers
will be b r i e f l y Ising
is devoted. by the con-
by the c o m b i n a t i o n
transformations
transformation, transformation,
transformation
and W a n n i e r 27
. A good r e v i e w gives
frustrated
given
These
lat-
3.2 the
chapter
but s u g g e s t e d
tables
transformations.
on the d e t a i l l e d
and in Table
of this
is not arbitrary, in both
(a)
Syozi 30
discussed,
systems
,
because
the lower c r i t i c a l
to the u n i v e r s a l
the h e x a g o n a l
lattices
three
introduced
transformations
3.1
are shown,
between
and O n s a g e r 29
d U =I
Systems
of f r u s t r a t e d
of lattices
of the f o l l o w i n g
have been
of Ising
in Table
lattices
choice
spins
is
order
transitions.
As the p r o p e r t i e s tice
func-
1966) 26 .
a number
different
of the c o m p e t i n g
eral s u c c e s s i v e
3.1
in c o n t r a s t
chains
systems
dimension
long range
symmetry
(Mermin and W a g n e r
for Ising
critical any
Ising
correlation
could not yet show a
because
destroy
rotation
is
frustrated
of course
for systems with m u l t i c o m p o n e n t
of the o c c u r i n g
frustrated
and the pair
the lower
fluctuations
mension
Ising
several
temperature,
interactions
thermal
that
systems
at finite
short range
Systems
of the e n t r o p y
one-dimensional
below w h i c h
Ising
28 , Syoz±
(1972).
b e f o r e we consider
in detail.
(b)
These the
,
27
Table
3. I
v" (a)
Cb}
\/k
/-X {XX
/
/ /
(c)
(d)
(e)
(a) (b) (c) (d) (e)
T r i a n g u l a r lattice Hexagon lattice K a g o m @ lattice D i c e d lattice T r i a n g u l a r lattice
with
nnn-interactions
(only p a r t i a l l y
drawn)
28
Table
3.2
~/IWIW
/~
T/~ T/?\ (a)
(b) J J
J
"';,; ,,x "
1
!/Z//~
•
j'
J
J
I i X
X
(e)
(d)
(c)
>¢<XX XXXX XX>C_X
x X'
x×x~ (g)
(f)
Ii!
Y iY ~Y ~
J i? I~? i71 (i)
(h)
(a) (b) (c) (d) (e)
Union-Jack lattice 4-8 lattice PUD model ZZD model Chessboard model
(f)
and
(g) (h) (i)
Square lattices interactions Brick model ANNNI-model
with
crossing
29
P~!i~-~[~i~
3.1.1
The d u a l i t y
transformation
functions
of pairs
tems have
no c r o s s i n g
geometrical
a)
gives
of Ising systems
Geometrical
part:
For a given
lattice
ses)
the centres
them
(by d a s h e d
is c a l l e d
dual
between
the p a r t i t i o n
to each other,
if these
The t r a n s f o r m a t i o n
consists
sysof a
part.
lattices
its dual
is c o n s t r u c t e d
of the e l e m e n t a r y lines);
that always
are i d e n t i c a l
dual
interactions.
and an a l g e b r a i c a l
is e v i d e n t
a connection
see Fig. pairs
as the square
by m a r k i n g
polygons
3.1a,b.
of dual lattices
(by cros-
and then c o n n e c t i n g
From these
lattices in Fig.
examples
occur. 3.1a,
it
When both
the
lattice
self-dual.
I
I
---~
.....
4
I I
-j
I I
I
' tJ
---~- .... I
.....
I
1
.,.
I
----'YI" ---"
" . . .i.. / " "k.
.-~-.. ~
i
r
!
i
./ "--.
X.-'-i"-.
.;
..!-..Z--im, Fig. 3.1a,b
The first tices: diced
four
lattices
triangular lattice
lattice ly
Construction of the dual lattice (for
lattice,
Sec.
In the t r i a n g u l a r each o t h e r dual
3.1
lattice
form the o t h e r one.
is of special
interest
(as t w o - d i m e n s i o n a l
chlore
in Table
and h e x a g o n
analogue
form
of dual
one pair,
Of the second pair,
Kagom~
latand
the Kagom~
of the t h r e e - d i m e n s i o n a l
pyro-
4.3).
lattice
to it.
are two pairs
e x p e r i m e n t a l l y 31 and t h e o r e t i c a l -
with nnn-interaction
and the n n - i n t e r a c t i o n s
lattice
d = 2). (from ref. 30).
and,
these
therefore,
are c r o s s i n g
there
is no
30
The first two lattices in Table 3.2, the Union Jack and the
4-8
lattice also form a dual pair. The three square lattices w i t h differing d i s t r i b u t i o n of nn-interactions
F and AF
can also be treated exactly and show e s s e n t i a l l y
d i f f e r e n t behavior. The Ising square lattice with crossing n n n - i n t e r a c t i o n s second square can be m a p p e d on the 16-vertex model,
in every
solvable
exactly for special cases. This is no longer possible,
when as in
III.2g n n n - i n t e r a c t i o n s exist in every square. Finally the brick and the uniaxial A N N N I - m o d e l s
are shown. The
first one can be solved exactly and is interesting as c o m p a r i s o n to t h e A N N N I - m o d e l
B)
for w h i c h an extensive
literature exists.
A l g e b r a i c part: dual p a r t i t i o n functions The p a r t i t i o n function of a nn-interactions
K(b)
(K(b)
2d
Ising system w i t h n o n c r o s s i n g
= J(b)/kBT
, b
is an index for the
interaction>
Z(k)
=
E
{o=+_1 }
(3.1)
exp (b K(b ) o.i oj)
for n o n f r u s t r a t e d systems
(for instance w h e n all
J(b)
> O) is
p r o p o r t i o n a l to the p a r t i t i o n function of an Ising system on the dual lattice with new n n - i n t e r a c t i o n s
e
-2K ~ (b)
Equation
:
(3.2a)
-2K(b)
= J~(b)/kBT
tanh K(b)
,
(3.2a)
can be rewritten in several ways:
=
tanh K~(b)
sinh 2K(b)
sinh 2K~(b)
e
K~(b)
I n t e r e s t i n g l y Eq. systems locally,
(3.2)
(3.2b)
=
I
(3.2c)
connects the interactions of the dual
i.e. this t r a n s f o r m a t i o n is a p p l i c a b l e for ar-
bitrary inhomogeneous d i s t r i b u t i o n s of f e r r o m a g n e t i c interactions,
not only in the h o m o g e n e o u s case.
The p a r t i t i o n functions of homogeneous dual systems for large 3O lattice size are linked by
31 Z(k) 2N / 2
where and
_
(cosh
N and N ~
its d u a l
are
Z*(k)
In s u c h
_
either
in p a i r s .
Ising
square
in the o r i g i n a l
lattice, same
interactions T h e n Eq.
in
(3.3)
o n e has
type
N = N*
as the o r i g i -
2d
systems),
simplifies
in
to
(3.4)
systems
connected
c a s e Eq.
2K c
sites
.
(cos 2K*) N
K = K*
sinh
the
holds.
(3.3)
Z (k*)
2K) N
self-dual
In t h i s
lattice
are of the
nn-pair
= Z(k)
Z (k) (cosh
as
interactions
(for i n s t a n c e
addition,
of
S = N + N*
is s e l f - d u a l ,
the dual
nal ones
and
,
(cosh 2K*) S/2
the n u m b e r s
system,
If a l a t t i c e If a l s o
Z* (k*)
2N * / 2
2K) S/2
=
(3.2)
singularities
b y Eq.
(3.2)
in
Z(k)
can occur
o r as a s p e c i a l
case
for
yields
I,
or transformed
Kc
This
=
(I/2)
is t h e
magnetic easily
sinh
nn
Ising
sinh
first
between
case
local
leaving
the
gauge
complex
of f r u s t r a t e d
=
of t h e
lattice.
The
result
function this
J1
are present,
or w i t h
external
one
has
frustration.
field)
to a p u r e l y
to d i s -
In the
can be transformed
ferromagnetic
system,
i n v a r i a n t 3.
is n o t p o s s i b l e ;
(3.2a,b)).
systems
can
(3.6)
without
(without
ferro-
case with different 8O a n d J2
I
interactions
(see Eq. Ising
square
anisotropic
transformations
systems
(3.5)
temperature
interactions
lattices
system
o n the
to t h e
2K2 c
the partition
In.frustrated become
transition
system
and vertical
2Klc
--~ 0 . 4 4 0 7
inverse
antiferromagnetic
tinguish
by
exact
(I+~)
be generalized
horizontal
When
in
with
As
real
the
dual
a consequence interactions
interactions the p r o p e r t i e s
on d u a l
lat-
32
t i c e s are n o t
linked
is a f u r t h e r
hint
systems
of t h e
The duality
as c l o s e l y
at the d i f f e r e n t
same
spacial
transformation
to h i g h e r - d i m e n s i o n a l where
e.g.
Refering on the
products
to t h e
as in the n o n f r u s t r a t e d
ready
that Wegner's
state
degeneracy
has
spins Ising
discussed paper
caused
been generalized
systems
2n
self-dual
fcc-lattice
of f r u s t r a t e d
This
Ising
dimension.
Ising of
behavior
case.
with multispin
occur
Wegner
32
interactions,
in the H a m i l t o n i a n .
system with
in C h a p t e r
contains
b y F.J.
four-spin
4, w e p o i n t
also
the
not by frustration,
case
interactions
out here
al-
of h i g h g r o u n d -
but by
local
gauge
symmetry. To explain two
this
systems
four-spin
local
symmetry,
(M22 a n d M32
interactions
are
in Fig.
3.2a,b
in the n o m e n c l a t u r e
v
of ref.
cells
of
32) w i t h
shown.
p •
f,
the u n i t
A
k/
A
kJ f
.
b--
,,
(a)
(b) (c)
32 (a), (b) : Two-, three-dimensional lattice gauge model of Wegner The spins (.) are in the middle of the edges of the squarerespecitve cubic lattice, whose sites are marked by crosses (x) . The four-spin interactions are drawn as hatched squares. (c) : Local symmetry: Flipping spins i to 4 for d = 2 leaves the Hamiltonian invariant, as two spins change sign in each fourspin interaction.
Fig. 3.2
F r o m Fig.
3.2c
of c o n v e n i e n t twofold
the
degeneracy
per marked
invariance
clusters
lattice
of t h e H a m i l t o n i a n
of s p i n s
of e a c h site.~
is e v i d e n t .
state,
Thus
under one
the
flipping
obtains
a
n o t o n l Y of t h e g r o u n d s t a t e ,
33
As there are two r e s p e c t i v e in the two- r e s p e c t i v e generacy
for the m o d e l s
boundary
effects)
state
entropy
is
three
spin
sites
three-dimensional M22
2 N/2
per spin
and M32
and 2 N/3
per m a r k e d
case,
with
N
and the
lattice
site
the g r o u n d s t a t e spins
de-
(without
corresponding
ground-
is
(22) So
=
I ~ in 2
(d = 2)
S(32) o
=
I ~ in 2
(d = 3)
(3.7)
and
For c o m p a r i s o n
the e n t r o p y
for
.
T ~ ~
(3.8)
for a r b i t r a r y
Ising
systems
is
S
=
in
The m o d e l tions dual
2
M32
is dual
on the simple a phase
transition
is a r e m a r k a b l e finite The
first
groundstate
Ising
systems
to the usual
cubic
of second order
example
entropy
The d u a l i t y several don't
Tc
with multispin
interactions
introduced
interest,
in q u a n t u m
e.g.
Tc ~ O
.
by Weg-
similar m o d e l s
play an i m p o r t a n t
latwith
role.
can also be g e n e r a l i z e d to spins w i t h 33 (n = 2) , but here we
for X Y - m o d e l s
this further.
of a
as they are the s i m p l e s t
chromodynamics
of f r e e d o m
transformation
components;
its
and it
in spite
with
spin degrees
discuss
like
a transition
tice g a u g e models; general
at a finite
for a system w h i c h
nn-interac-
it has
shows
ner are of such p h y s i c a l
more
Ising m o d e l w i t h
lattice 32. Therefore,
34
3.1.2
Decimation
.
.
Already chains can
.
.
.
.
.
.
.
in the
is n o t
first
.
over
.
.
.
.
.
.
.
.
.
single
.
.
.
.
.
.
of the
spins.
between
spins
the r e m a i n i n g
of
(ref.
and multi-decorated
the partition
One obtains
or g r o u p s
system
of s i m p l e
systems,
of the r e m a i n i n g
.
.
.
.
the
.
.
1 in
.
.
.
but
spins
34).
system,
function
one
a temperature
de-
spins.
This method
can be applied interact
When
they
with
only
interact
on obtains
in
with
also new
three-
Transformation .
.
.
.
simplest
with
K I and K 2
.
.
.
.
case
.
.
.
.
.
.
one obtains
(KI+K2) h
\cosh
(KI-K2) /
.
shown
two n e i g h b o r i n g
/C ° s h
in Fig. spins.
3.3a,
where
For different
the e f f e c t i v e
a cenoriginal
i n t e r a c t i o n 34
(3 9)
'
symmetrical
=
tanh K I tanh K 2
KI = K2 = K
this
direct
contribution
chain,
Eq. the
(2.25). inversion
yields
This
not
a c t on the
discuss
also
K'
transformation
frustration
= I/2
transformation
of t h i s
star-triangle
in c o s h
of the
can always because
cannot
to a m a g n e t i c
intermediate
applications
on t h e
again
is n o t u n a m b i g u o u s
decoration-iteration spins,
(3.10)
to the n n n - i n t e r a c t i o n
nnn-interactions
intermediate
section
talked
for c o m p u t i n g
interactions.
2
t a n h K'
direct
.
we have
where
part
.
interaction
interacts
-
or m o r e
We
.
to o n e - d i m e n s i o n a l
consider
spin
K'
does
.
chapter
spins
interactions
The
.
Decoration-Iteration .
ever,
.
2.1),
where
or f o u r
3.1.2a
.
of the r e m a i n i n g
four-spin
For
.
effective
two spins
tral
Transformations
restricted
all c a s e s ,
three
.
last
sum exactly
and
.
(see Fig.
pendent
We
.
2K
simple
, the
in-
decorated
be i n v e r t e d .
How-
in a c h a i n w i t h o u t
occur.
can b e g e n e r a l i z e d field
(ref.
34),
as
to s e v e r a l long
as it
spins. transformation
transformation.
in the n o w f o l l o w i n g
36
KI (a)
"--
C
K'
K2 ."
_
K~ K~ K3
(b}
(c)
y K23
Fig. 3.3
Decimation transformations: (a) Decoration-iteration transformation, (b) star-triangle transformation, (c) star-square transformation.
3.1.2b
S~a~zTEian~le Transformation
This t r a n s f o r m a t i o n is completely analog to the d e c o r a t i o n - i t e r a t i o n transformation,
only the i n t e r m e d i a t e spin or group of spins is now
i n t e r a c t i n g w i t h three other spins. Figure 3.3b again shows the simplest case. W i t h o u t a m a g n e t i c field the t r a n s f o r m a t i o n from the star i n t e r a c t i o n s (no primes) is30,34:
to the triangle interactions
(primes)
36 ~o
=
cosh
(KI+K2+K3)
~I
=
cosh
(-KI+K2+K3)
~2
=
cosh
(KI-K2+K 3)
~3
=
cosh
(KI+K2-K 3)
(3.11)
4K~ e
When
~o ~I ~2 ~3
-
all
star
4K~ '
e
interactions
~o ~2 #3 ~I
-
are
4K~ '
identical
e
-
(K i ~ K)
~o ~3 ~I ~2
, Eq.
(3.11)
be-
comes
4K ~ e
Independent romagnetic (3.12) the
cosh cosh
-
of the
sign
realizes,
corresponding
when
In a d d i t i o n the
of The
star
3.3c
spin
generates K i . Two
inverted
As
the
are n o t
only
for
the
the
triangle
K
becomes
is p o s s i b l e 30,
from
triangle
(K' < O)
Similar
transformation
on the real
Ising
to maps
Hamilton-
is p r e s e n t .
seven
with
new
transformation
four
cases
are
Therefore, and
neighboring
interactions
interactions
independent. special
fer-
imaginary.
star-triangle
star-square
new
are
for a f r u s t r a t e d
of the n e w ones
seven
interactions
transformation
interactions
interacts
original
they
that
also
real
transformation
ones,
inverse
interaction
with
in Fig.
interaction.
the n e w
no f r u s t r a t i o n
central
ones
K
however,
transformation
Hamiltonians
ians only,
where
(3.12)
(K' > O)
one
the d u a l i t y Ising
3K K
depend
is of m i n o r
This
K~ f r o m the four i , one is a f o u r - s p i n
nnn
this
is shown, spins.
on the f o u r
transformation importance.
original can be
37
3.1.3
Connection
The d e c i m a t i o n connections
Between
the D i f f e r e n t
transformations
between
of the inverse
triangles
of the t r i a n g u l a r
lattice,
in a d d i t i o n
the lattices
plication
to d u a l i t y
d e p i c t e d in Tables
star-triangle
to all triangles
Lattices
lattice
transformation
(Fig.
to the diced
3.4)
yield
3.1
leads
further
and 3.2.
Ap-
to half of the to the h e x a g o n
lattice.
,\\ 1,,"I",, ,,'?',, ,,"1",,\\ [,,
IIII
Fig. 3.4
After
Transformation between triangular and hexagon lattice.
decoration
mediate
\\\\
of all bonds
spin the s t a r - t r i a n g l e
tice.
The c o n n e c t i o n s
derived
tions
b e t w e e n the v a r i o u s 3O 35
of the h e x a g o ~
lattice w i t h
transformation from d u a l i t y
hexagon
lattices
leads
an inter-
to the K a g o m ~
and d e c i m a t i o n of Table
3.1
lat-
transforma-
are summed up
in Fig.
As to the square formation
lattices
and the Union
Jack
lattice
m o d e l 35'36 w h i c h w i l l In the f o l l o w i n g mentioned fects.
of Table
3.2,
one can show the t r i a n g u l a r
in m o r e
to be special
be further
sections detail,
using
lattice,
we discuss with
cases
discussed
the s t a r - s q u a r e Villain's
of Baxter's
in Section
regard
8-vertex
3.3.4.
the t w o - d i m e n s i o n a l
special
trans-
odd m o d e l
Ising
system~
of the f r u s t r a t i o n
ef-
38
Fig. 3.5
3.2
Connection between the hexagon lattices of Table 3.1, derived from duality (d) , star - triangle (Y-A) and decoration-iteration (I) transformations (ref. 30).
Triangular
Lattice
The a n t i f e r r o m a g n e t i c first f r u s t r a t e d appe138
using
exact
mined
the p a r t i t i o n
only appel
solution
and also
considered treated
system on the t r i a n g u l a r
has been
transfer-matrix
ger's
tities,
Ising
system,
investigated
methods
already
of the f e r r o m a g n e t i c function
and several
f o u n d the t r a n s i t i o n the case of i s o t r o p i c
the a n i s o t r o p i c
e nted n n - i n t e r a c t i o n s
case,
have d i f f e r e n t
lattice,
a few years
square related
lattice.
Tc
(h O)
nn-interactions, the three
values
(Fig.
after OnsaThey deter-
thermodynamic
temperature
where
the
by W a n n i e r 37 and Hout-
whereas
differently
3.6).
quan-
. Wannier Houtori-
39
31
A
W
Fig. 3.6
Unit cell of the anisotropic triangular lattice.
In the p a r a m e t e r ficient
space
to c o n s i d e r
simultanuous
change
ic q u a n t i t i e s only every fixed
interaction
of a c o m m o n
is f l i p p e d ;
always
triangle
ishes mark
is f e r r o m a g n e t i c Along
(here
the e d g e s
these
e.g.
o v e r the o t h e r
GS's without
chains
square
because
of the third,
(JIJ2J3),
Its c o n t o u r s
F
the
(J1,-J2,-J3),
to c o n s i d e r AF I, AF 2 and are d e f i n e d
. I n s i d e this t r i a n g l e marks
lattices.
the i s o t r o p i c interactions The p o i n t s
van-
Qi
lattices.
(full lines),
J2 = -J1 ) . The
. The p o i n t
. Also
field),
the c o r n e r s
J3 + J1 = 0
sphere,
the t h e r m o d y n a m -
it is s u f f i c i e n t
lines one of the t h r e e
to a n i s o t r o p i c square
J3 ) d o m i n a t e s
J3-direction),
(F)
the d a s h e d
corresponding the i s o t r o p i c
Along
four p o i n t s
3.7 w i t h
and
(> O)
leaves
in the d i r e c t i o n
one f o u r t h of the surface.
J1 + J2 = O , J2 + J3 = 0
ferromagnet.
factor
it is suf-
on a u n i t
of a m a g n e t i c
Therefore,
in Fig.
interactions
properties
scaling
(in the a b s e n c e
are e q u i v a l e n t .
AF 3 , c o n t a i n i n g
GS
(GS)
of s i g n of two i n t e r a c t i o n s
invariant
the s p h e r i c a l
the
of the t h r e e
s e c o n d r o w of spins o r i e n t e d
(-J1,-J2,J3) only
J1' J2' J3
the g r o u n d s t a t e
as they are i n d e p e n d e n t
Ki
W
J1
~!~!2~_2~_~h~_Q[2~5~_~£~[~2Z
3.2.1
by
A
AF I - K 3 - AF 2
c o n s i s t of i d e a l l y order between
are e x a c t l y
one i n t e r a c t i o n
two of e q u a l a b s o l u t e ordered
different
decoupled
a l so for
value
Id-chains
chains. T # O
(here
(here in
At the p o i n t s . Though
the
40
J!
A%
AF,
Fig. 3.7
Parameter space of the anisotropic triangular lattice. Special points are: F
:
(I/N)
(i,i,i)
Qi
:
e.g. ( I / H )
Ki
:
e.g.
(i,O,O)
AF i :
e.g.
(i/H)
isotropic ferromagnet
(O,i,i) id
is d e g e n e r a t e ,
per
site
vanishes
proportional
we
are mainly
interested
spherical All
three
triangle
triangle,
Before turn
fully GS
system
entropy
SO
entropy the per
systems,
to in
are
have
the
GS site,
ways
for which
the
the
same
the
degeneracy
exact
no e x a c t
absolute for
eight
the
GS
is h i g h
results
which are
of
value,
and
a single
be d i s c u s s e d
So
corners
points
states
determination
to e s t i m a t e
N ~ ~
(equivalent)
frustration
Already
as w i l l
limit
entropy
N-I/2
the
(six of
on square lattice,
frustration points, T c = O , corresponding to isotropic AF on triangular lattice.
thermodynamic
frustrated.
considering
to t h r e e
d = 3)
which
interactions is
to a h i g h finite
in t h e
F
on triangular lattice,
chain, T c = O ,
(I,I,T)
GS
Here
isotropic
F
(F)
of
SO
leads inGS
below.
by Wannier,
we
first
also
for
other
(especially
for
applicable
available
the
a finite
detail
the
elementary
this In
to y i e l d
in m o r e
are
each
GS).
of
system.
triangle
are
enough
AF i
the
41
~!~H!£_b22£~_~2H~
3.2.1a
In h i s ent
paper
tropies N
Wannier
statistical
is
per the
weight, the
a
lower
SO
which
SO
taking
lattice
number
and
of
I ]
can
be
>
~
notes
for
in
S
2
that
5
in
at
least
2
--~
+
-
third
free.
are
has
shown,
the
spin
This
differ-
with
enwhere
highest
(all
spins
immediately
0.2888
nn-corrections
÷~----÷-----+
-
every
c
with
GS
on
one
yields
0.2310
/\/\/\/\/\ +----~+-----+---/\/\/\/\/\/\ +~---+------+-----+ /\/\/\/\/\/\/\ +----- +----- +--_--+--_ \ /- -\-/+\~ -/-\-/+ -\-/-\- -/+\-/- \ /+ ~\-/- \- +/-\- /- -\-/- \+ -/- \/\/\/\/\/ -
GS
, const .
configuration
completely
of
3.8a,b,c
to
account
--
classes Fig.
N -I , N - 1 / 2
The
is
in
:
o
-~
raised
different
examples So
sites.
sublattices)
bound
>
As
site
of
one
three
into
discusses
weight.
-
_
-
-
+
-
-
(a)
(3.13)
37
.
---+-----+---
/\/\/\/\/\ +------+-----+--/\/\/\/\/\/\ ---+------+------÷~_ / \ / \ / \ / \ / \ / \ / \ ---__+ ---.+------+-----+ \ /- -\-/+\- -/-\ / \ -/-\+ /- -\ / - - - + ~ \ /+ -\-/- \- -/+\- /- -\-/- \+ -/- _ \/\/\/\/\/ -
-
-
-
+
-
-
-
-
-
+
-
-
-
(b)
x
Fig. 3.8
Three types of groundstates in the AF triangular lattice: (a) AF chains periodically stacked, (b) AF chains random stacked, (c) ~3x~3 structure, two of three sublattices are F ordered, the spins on the third (o) are completely free.
42 3.2.1b
~!!D~
A second bound,
approximation
is the m e t h o d
estimated states,
2N . T h e s e of
the
six of e i g h t
u s u a l l y is n e i t h e r an u p p e r nor a lower 39 Here the n u m b e r of GS N is g r e d u c t i o n f a c t o r s to the t o t a l n u m b e r of
factors lattice
states
are the p r o b a b i l i t i e s in
are
GS
GS,
to find
configurations.
Wo(3)
= 6/8
As
. There
the e l e m e n t a r y
in an
AF
are two
triangle
triangles
per
hence
=
SO
In this
lim ~I in N N-~ g
=
N
drQps
in 2 + 2 in Wo(3)
This
is by far too
tory
for
yields
--~ lim ~I in N~
approximation
So(3)
lattices
very
good
low, with
out,
-~
in this high
results
< 2N Wo(31
2Nh/
and one
(3.141
obtains
O.118
(3.151
simple
form
coordination
for the K a g o m 6
the m e t h o d
number. lattice
is u n s a t i s f a c -
However, and
for
this m e t h o d
ice m o d e l s .
§ZS~g£!2_~!~[_5~{2~!~!2~
3.2.1c
One
which
of P a u l i n g
by m u l t i p l y i n g
triangles
site,
Method
obtains
a much
better
approximation
for S
by c o v e r i n g
the
lattice
O
not
only with
size
M
exactly M ~ ~
elementary
, calculating and
finally
. This
triangles,
the
GS
extrapolating
is a s i m p l e
but with
probability
f o r m of
the
clusters
for t h e s e
corresponding
'finite
size
of d i f f e r e n t clusters S
scaling'
Wo(M)
(M) to o40-42 . As the
deviations
ASo(M)
are
=
an e f f e c t
proportional
AS ° (M)
Therefore, So(M)
of the
- So(M~)
cluster
to the r e l a t i v e
~
V•
~--
=
M
M-I/a
(3.16)
boundary, number
one
can e x p e c t
of b o u n d a r y
ASo(M)
, as one
to be
atoms
--1/2
for e x t r a p o l a t i o n
versus
behavior.
So(M)
(3.1 7)
to
M ~ ~
should
it w i l l
find
then
be u s e f u l
to p l o t
an a p p r o x i m a t e l y
linear
43
We have shown
considered
in Fig.
different For
case
(a)
case
to c o v e r
for
the
lattice
triangular
(M = 6)
by clusters
clusters
which
and which
are
yield
So
one obtains
=
I in 2 + ~ in w(M)
=
2 in 2 + ~-- in w(M) P
I ~ in Ng(M)
=
,
(3.18)
(b)
s(b) (M) o
where
for
approximations
s(a) (M)o
for
two ways
3.9
is t h e n u m b e r
M
(3.19)
of e l e m e n t a r y
triangles
in t h e M - c l u s t e r .
P
v\/vV
A/X/k/X/ .*~"...."°....""...""....""...."",..""...."
t xXY
(a) Fig. 3.9
As
Two ways to cover a lattice with clusters. (a)
Every lattice site belongs to one and only one cluster corresponding to Eq. (3.18);
(b)
no space between clusters corresponding to Eq. (3.19).
in t h e
limit
expressions Eq.
[b)
(3.15)
for '
M ~ ~ So
the
ratio
coincide.
corresponds
to
2M/Mp ~
We
S (b) o
note
that
I , in t h i s the
simple
limit
the two
Pauling
method,
(M = 3)
is an u p p e r b o u n d , as o n e o v e r e s t i m a t e s the n u m b e r of g r o u n d S O(a) (M) states when only the triangles inside the different clusters are required S o(b) (M) densely
to b e in t r i a n g l e is a l w a y s packed,
but
lower
GS than
. s(a) (M)o
it n e e d n o t b e
, as the c l u s t e r s
a lower bound.
are more
44
We have
determined
for t r i a n g u l a r and h e x a g o n a l shape
the
GS
(M = 3, 6, (M = 7, 21)
are shown
in Fig.
degeneracy 10,
15,
Ng(M)
21)
clusters,
and thus also
, rhombic the first
Wo(M)
(M = 4, 9, 16, two clusters
25)
of each
3.10.
@ A
Zg V\!VV Vx/\I
(a) Fig. 3. i0
We note
that
Ng(M)
is the true
along
(c)
The first two clusters of each shape used for extrapolation.
is d e t e r m i n e d
one down or the reverse This
(b)
GS
of a finite
the edge of the
here by r e q u i r i n g
for each e l e m e n t a r y
cluster
cluster
triangle
only,
are r e d u c e d
when
two spins
up and
of the cluster. the i n t e r a c t i o n s
by a factor
of two.
However,
this way the d e p e n d e n c e
of N on the b o u n d a r y is reduced s i g n i f i c a n t g for the e x t r a p o l a t e d So(~) are obtained. Also
ly and b e t t e r values only using upper
this way
to c a l c u l a t e
-(a) (M) S O
Ng(M),
is n e c e s s a r i l y
an
bound.
In Fig.
3.11
the e n t r o p i e s
s_(a) O
and S o(b)
are p l o t t e d versus
M -I/2
s(a) (M) is very close to linear in M-I/2 w i t h i n the three series o except for the very s m a l l e s t clusters (M ~ 7) . The e x t r a p o l a t e d value
5o-(a) (~)
agrees
with
the exact result
(full circle)
to w i t h i n
3 percent,
s(b)(M) is m o r e s e n s i t i v e to cluster shape than s_ o( a ) ( M ) , o but e x t r a p o l a t i o n w i t h i n each series also agrees with the exact result within
similar
We add that
error bars.
AS °(i) (M)
lar and smallest effect, gonal
the t r i a n g u l a r
ones
(especially
for h e x a g o n a l
the smallest
for
clusters.
clusters relative
i = b) This
possessing number
is largest
for triangu-
is also a cluster
the
of edge
largest, sites.
edge
and the hexa-
45
I 0.6 9
So{M)
J
0.~
10 9
25 1916
7
4
II
21 15
0.2
I0
I 6
0
I
t
I
0
!
0.2
I
M-~
Fig.
3.11
The
shape
CE
=
dependent
aE
aE =
and
hexagonal
for
method small
clusters.
I
06
--~
SO(M) for M > 7 is to good approximation linear in M -1/2 . Extrapolation to M ÷ ~ agrees to within 3 percent with the exact value (.) .
with
This
I
0.4
part
of
edge
• M -I/2
,
, 4
~
3 ~
, 6/~
sites
for
M ~ ~
is
(3.20)
4.24,
40,
3.46
for
triangular
rhombic,
clusters. to
calculate
cluster
sizes
the and
GS can
entropy be
yields
augmented
quite
precise
systematically
results
using
larger
46
As already
mentioned,
the p a r t i t i o n frustrated function
W a n n i e r 37 a n d H o u t a p p e 1 3 8
function
a n d thus
triangular
per
site
l
lattice. (Z = 1N,
also
internal
Houtappel
have
calculated
energy
obtained
F = - kT in ~)
exactly
and entropy
of the
for t h e p a r t i t i o n
:
/ =
in
8~ 2 f o
f o
in |OkI C 2 C 3 + $ I S 2 S 3 - S 1 c ° s ~ 1
-
S2cos~ 2 -S3cos(~i+~2) ) d~ I d~ 2
,
(3.21) with C. l From
this
specific
=
cosh
result heat
2K. 1
the
c(T)
,
c a n be d e t e r m i n e d .
spherical
triangle
of
( 2 J i / k T c)
Fig.
3.7
be set
J1
for
corresponding
Tc
to the
(with
I
(3.22)
=
the w e l l
lattice;
the
2K c y
spherical
=
known
only one
case
of the n o n f r u s t r a t e d
of the t h r e e
interactions
has
I
has
to
shown
The vanishing crossing
on the
3.12a.
long r a n g e the
GS
is n o t y e t a f i n i t e of
these
(3.22)
in Fig.
two-dimensional
there
of Fig.
, Eq.
AF i - Ki+ 2 - AFi+ I , where although
(3.23)
triangle
+ J2 + J3 = const.
right
one obtains
, the
):
contains
2K c s i n h x
Tc/(JI+J2+J3 ) GS
range
S(T)
to zero:
sinh
When
1
, the e n t r o p y
for the p a r a m e t e r
c c c c c o S I S2 + S2 S3 + S3 S I
square
U(T)
2K
energy
Tc
S~l = s i n h
(3.22)
sinh
internal
system
anisotropic
=
1
and
In the a n i s o t r o p i c
Equation
S
Tc
lines
lines.
along the
the
3.7 is m a p p e d yields One
GS
lines
later we
entropy AF
per
- K - AF
shall
whenever
. Along
Id L R O
causing
of c o n s t a n t
Tc % 0
(LRO)
has o n l y
on the plane
lines
notes
order
GS c h a n g e s ,
However,
the
lattice
lines vanishes site.
is p l a u s i b l e ,
a higher
see
the
, Tc
the
as
degeneracy
(for the s q u a r e
lat-
47
A
w
(a)
Fig. 3.12
tice)
(a) :
Triangular lattice. Lines of constant transition temperature T c /(Jl+J2+J 3) (after ref. 38).
(b) :
Hexagon lattice (not frustrated in this parameter space). Lines of constant Tc/(Jl+J2+J 3) . Here T e vanishes only when at least one interaction Ji vanishes, as the lattice then falls apart into seperate id chains (after ref. 38).
that
even
for
finite
the
properties
S(T=O)
For
comparison
in Fig.
isotropic the
the
cases
points
Whereas
to
is
KB
the
where
nonfrustrated
case
. Because IJI
for
_
square , the of
from
the
GS
3
n16 f
~
o
the
lattice
and
frustration
-1.5
IJl
(2cos~)
. From
is
are
Eq.
GS
Ising
at
the
de
~
0.323066
no
energy
(3.21)
So
U(T) equal
be
finite;
simultaneously for
the
two
corresponding
no e s s e n t i a l
exhibits
the
may
occur
to
3.12.
a normal
antiferromagnet
entropy
in
of Fig.
Tc
not
energy
IJil
there
site
need
internal all
in t h e
Tc
per
Tc = O
AF i
-0.5
So
3.13
shown,
entropy
and
respectively
ferromagnetic
value
GS
% O
Fi
at a f i n i t e T # 0
(b)
one
frustration
difference
transition
singularity U(O)
to
occurs for
is r a i s e d
obtains
the
exact
points37:
(3.24)
48
-o.2.
•
~cu~----~'7-~,
t
u -o.~
I
-0.8
--II.~62~.-I'O I,-cu.POI qlEiNT.1 4 I0 0
Fig. 3.13
This
6
12
8
!=
2~'r
ILl
IJI
14
16
20
18
Internal energy U(T) of the isotropic ferromagnet and antiferromagnet on the triangular lattice (ref. 37).
value
cluster
2
was
used
already
in F i g .
3.11
for
comparison
with
the
finite
results.
3.2.3
Specific
Heat
Near
the
Frustration
Points
(J1
= J2)
the
frustrated
....................................................
Vaks
and
gular J3
Geilikman
Ising
= J +
6
43
system , J
have
near
< O
,
r61
studied
the <<
the
frustration IJi
. Then
behavior
of
points two
for
cases
the
have
case to
be
J1
trian= J2
= J '
distinguish-
ed: I.)
6 > O
:
the
GS
shows
2d L R O
2.)
6 < O
:
the
GS
shows
Id L R O
In Fig. and
as
3.14 dashed
the
specific
line
for
heat 6 < O
c(T) .
is
; T
c
# 0
only;
shown
as
;
Tc = 0
full
line
for
6 > O
49
T
Fig. 3.14
As
Specific heat of the anisotropic AF triangular lattice (J1 = J 2 = J3 - ~ , I~I << IJil) For ~ > O (full curve) there is a logarithmic divergence, not for ~ < 0 (dashed curve). (after ref. 43).
expected
T >>
161
c
the
=
is
Ising
transition
t -3/2
6 < O the
maximum
at
occurs
at
only
of
occurs
and
are
broken
with
the
weaker
ones.
For
6
only (see
the
comparison
tremely
than
the
Vaks
J1
heat
into
Sec.
and
T
of
6
in
the
range
range
t-I/2
an
26/ln
near
2
Tc
, .
respectively
uncommon. maximum
at
The
for
16f , b e c a u s e
T
of
probability,
even
T ~
low
interactions
equal this
IJf
Tc =
maximum
strength
for
T
<
J 16r
T = O
only
leads
the
specific
heat
of
square
lattice,
Fig.
3.15.
to
3.2.4).
ferromagnet
ones
the
6 <<
critical finds
temperature.
Geilikman
independent
the one
quite
]61
But
positive
temperature
analytic
any >
in
is
almost
in vertical
horizontal
decomposes cific
next
anisotropic
interactions
. For
. For
lower
Tc
an
at
to
J +
much
which
2d LRO
SRO
IJl
m
below
behavior,
there
absence
Id L R O
independent
divergence
respectively law
T the
corresponds
almost
is
(3.25)
logarithmic
above power
For of
a broad
typical
Further
heat
IKI 3 e - 4 1 K
There
with
specific
:
J2
discuss on
the
direction ' J1/J2 horizontal
are
= Y
<<
supposed I
chains;
. For the
to
be much
y = O
the
corresponding
the
exThe
weaker system spe-
50
cO':
J
T
Fig. 3.15
Specific heat of the extremely anisotropic ferromagnet on the square lattice (ref. 43).
c(T)
has
a broad maximum
(y << to
K 2 c o s h _ m K2
=
J1
I)
the
= 0
for
y <<
c(T)
Near
c
logarithmic
2d
ferromagnet, 6 > 0
Tc
one o b t a i n s
divergence and
begins
I , a n d the
. Above
. For
in the r a n g e
order
9__y (iny) a in --~ 4~
the
case
T
T m ~ J2
heat
. Interchain
temperature
behavior.
around
specific
T c ~ 2 J2/in(I/y)
(3.26)
Ising
is m u c h
up only
transition
at m u c h
compared lower
from
2d
to
Id
:
(3.27)
weaker
compared
dependence
lattice
J1
is at
;
the t e m p e r a t u r e
of the t r i a n g u l a r
additional is u n c h a n g e d
is a c43 rossover c(T)
(I/Itl)
Tm
to b u i l d
there for
a small of
(Fig.
to the
isotropic
is c o m p a r a b l e
3.14).
to the
51
3.2.4
Pair
Leaving
Correlation
the
thermodynamic
tion
function
der.
S t e p h e n s o n 44 h a s
G(r)
general
anisotropic
Here
discuss
J3
we
= J1
6 > O For
Function,
+ 6
as
or
, 6 = O ~ < I
functions
containing
we
Lines
now
detailled
calculated
the
pair
(J1 = J2 )
consider
the
pair
information
on
correlation
function
the
correlaspin
or-
for
the
case. in Section
J3/J1
, 6 < O
there
Disorder
= ~
3.2.3
and
only
like
there
respectively
is a t r a n s i t i o n
the
case
J1
distinguish
= J2 = J < 0 the
three
,
cases
I < I , X = I , X > I at
a finite
TD
(dashed
T
(see Fig.
3.14).
C
Above
Tc
along
decays
exactly
a disorder
line
exponentially;
along
the
line
(I)- a n d
in Fig.
3.16)
G(r)
(2)-directions
- r / ( i / 2 (T) G1/2(r)
along
the
=
(tanhK1)r
=
(-I) r e
,
(3.28)
(3)-direction
G3(r)
=
(-tanhK3)r
=
e-r/(3(T)
(3.29)
\
1/K
I
o
0.5
1.o
A
Fig. 3.16
Phase diagramm of the anisotropic AF triangular lattice. Full line: T c , dashed line: disorder line T D (after ref. 36).
52 with -1 ~I/2(T)
=
in
Itanh KII
This p u r e e x p o n e n t i a l a
Id
ferromagnetic
lar s y s t e m
Below
d e c a y of
TD
of
(-I) r
. Contrary
and
factor
r-I/2
wavevectors
to this
G3(r)
qi/2
GI/2 (r)
for
~
q3
r ~ ~
(tanhK I ) r
to
G(r)
in
line the t r i a n g u -
The d i s o r d e r
line ends
are m o n o t o n o u s l y r ~ ~
an a d d i t i o n a l
dependent
respective range
(3.30)
exactly
the d i s o r d e r
l i m i t for
T > TD
and t e m p e r a t u r e
In the a s y m p t o t i c
2~;~ 2
at
(I = I, T = O)
r , which have a finite
(~ LRO)
=
corresponds
Along
one-dimensional.
frustration point
G1/2(r)
functions
~31(T)
G(r)
I s i n g chain.
is e f f e c t i v e l y
the i s o t r o p i c
;
T < T
C
characteristic
oscillations
vanishing
decreasing when
occur,
continuously
with
for
T ~ TD
one has:
r - 1 / a cos
lq1/2 r +
~i/2> (3.31)
G3(r )
In Fig.
~
(-tanhK3)r
r -I/2
3 . 1 7 a , b 44 the t e m p e r a t u r e
83 = ~ - q3
qi(T)
disorder
larity between
and this
when comparing sus
T
more
convenient
For
~ = I
similar
(Fig.
the
2d
Fig.
3.17 to Fig.
3.18 c o n t a i n s
of
el = qi/2
~ = J3/J1
above
T D , like for the
line is of the f i r s t kind. Id
system becomes
3.1813,
and
where
8 = q
the same i n f o r m a t i o n
The simi-
especially
clear
is p l o t t e d v e r -
as Fig.
2.2, b u t is
Tc = TD = O
. For
T > O
S t e p h e n s o n 45 es-
finds asymptotically (tanhK) r
as for f i n i t e
GO(r)
of
for c o m p a r i s o n ) .
(isotropic)
~
8 ~ ~/3
dependence
are c o n t i n u o u s
periodic ANNNI-chainthe
G(r)
63 >
is s h o w n for s e v e r a l v a l u e s
As the w a v e v e c t o r s
sentially
cos
(see Fig. _~
corresponding
r-I/2
TD
3.17).
O.63222"
cos
(Eq. For
r-I/2
to the a b s e n c e
(er)
(3.32)
(3.31)),
but for
T ~ O
one has
T = O /2 cos L~ ~ r) of
LRO
for
, T > O , b u t for
(3.33) T = O
53 I
I
I
7
Fig. 3.17
5
Anisotropic triangular lattice. Continuous temperature dependence of the wavevectors
I
0
2
3
q3 = ~ - 83
(b)
ql/2 = 81
and for fixed
values of J3/Jl (with J2 = Jl ) in the disordered phase (ref. 44).
(a)
0
(a)
5
4
kBT//IJ21
8
I
i
I
I
2
"it" T 3
(b)
2
3 4 ksT//I J21
5
1.5
Fig. 3.18
8
For comparison: id ANNNIchain. Wavevector q = e versus temperature for fixed values of Jnnn/IJnnl (ref. 13).
1.0 0.5
o.5
the
correlation
G°(r) second found ty
~
r-I/2 order.
for
class.
all This
t.o
1.5
length , which However,
~ is
2.5
diverges usual
the
for
because
of
the
T = Tc
at
a phase
exponent
is
Ising
systems
nonfrustrated r-I/2
2.0
kaT/I JLI
power
law
at
q =
T = 0
I/2
power
law
transition
, contrary
to
forming
a single
we
find
will
q =
of I/4
universali-
also
for
54
several, complex For
b u t not
I > I
disorder
q3(T)
For
T = O
that
J3
(r)
G(r)
behave
=
(-I)
is in the
one
G °(r) I
GS
After
first
(3)-chains
down
to
lar b e h a v i o r lattice. again to
we
For
AF
r
in the
• r -~/2
short
there
shall
last 44
cos
of Eq.
hint
; there
(3.31),
I ~
order
to the
but
is no ql/2(T)
I
Along
in
(3)-direction,
the o t h e r
the o r i e n t a t i o n s
even
however,
for the
and a p p r o a c h
of the o r d e r e d this
distance
the f a c t o r
the w a v e v e c t o r s
two di-
(3.35)
<2 r)
with
again
occurs
section.
independent;
again
find
3.17)
range
Tc = O
case
Id
expect
. Chains
T > 0
(see Fig.
and
form the
perfect
one m i g h t
and
another
(3.34)
to be c o m p l e t e l y
T = 0
the
from
asymptotically
0.58835
for odd d i s t a n c e
systems,
,
already
sight
J1 = J2
has
indeed
finds
~-
again
2d
systems.
over
different
r
as was m e n t i o n e d rections
frustrated
of f r u s t r a t e d dominates
line.
and
G 03
all o t h e r
behavior
r
r-I/2
qi/2
and q3 for
only
correlated
occurs.
anisotropic
qi = ~
is true
are
Very
simi-
frustrated
vary
T ~ ~
square
continuously
, corresponding
order.
~!~_~_~h~_9~_~[=~h~!~_~_~e_~h~_~!~£~!£__~a__!~!~
3.2.5
Chain In this
section
correlation tice
and
several
The m a p p i n g on the short
we w a n t
functions,
fact, range
of
to m e n t i o n used
other 2d
that
2d
thermodynamic
transfer
matrix
of P a u l i
matrices,
noninterchangeable
Ising
classical
interactions V
. This
an i n t e r e s t i n g
by P e s c h e 1 3 6
AF
to d e t e r m i n e
triangular
lat-
systems.
systems
to
Id
quantities
can be d e r i v e d operator
method
for the
can
but usually
has
exponential
factors.
quantum
of
from always
2d a
systems
Ising
Id
systems
operator,
be e x p r e s s e d
a complicated
structure
The m a t r i x
V
is b a s e d with
the
in terms because
becomes
of
espe-
55
cially
simple
I.)
One
for t w o
succeeds
commuting 2.)
with
it s u f f i c e s
Pesche136
has
lattice
and
shown,
special
ratios
First
the
tice,
satisfying
weights and Green
w I w2
When
this
V
in the
(Hamiltonian
limit).
system
the
of
the
H
Id
form (3.36b)
respective quantum
AF
odd model
lattice,
triangular
interactions.
'free f e r m i o n
=
condition
possible
the
only
model
complete
the
is a n a l o g .
o n the
condition'
Jack
XY-chain
consider
square
for
lat-
the v e r t e x
solution
w5 w6 + w7 w8
is s a t i s f i e d ,
are
the U n i o n
the procedure
to an 8 - v e r t e x
w I .... , w 8 , making 46 1964) :
often
to t h e q u a n t u m
Here we
cases
the
so-called
which
lattice,
can be mapped
f o r the o t h e r
H
systems.
c a n be m a p p e d
+ w3 w4
, (3.36a)
to w r i t e
how
H
;
describe
Villain's
triangular
a new operator
to i n v e s t i g a t e
and which
AF
V
(-H)
hermitian
with
in f i n d i n g
it is p o s s i b l e V = exp
Then
cases:
(Hurst
(3.37)
the Hamiltonlan
] =
- nE
commutes
with
lattice,
if 47
J
Jx the
x On - J y transfer
=
I
=
tanh 2 K 3
z On-1
x z On ~n+1
matrix
V
z z ? + B On °n+1 j
(3.38)
of t h e o r i g i n a l
AF
triangular
x
Jy
B
=
,
(sinh2K1sinh2K2cosh2X 3 +cosh2K1cesh2K2sinh2K3)/cosh2
K3
(3.39) Finally
the H a m i l t o n i a n
(3.38)
c a n be r e w r i t t e n
by
the d u a l
transfor-
mation Z Z °n (~n+1
=
Z Tn
X ,
no
=
X Tn-1
X ~n
'
(3 40)
56
as the H a m i l t o n i a n
HXy
where
=
- n E
only
of the q u a n t u m x x ~n Tn+1
Jx
nn-interactions
d i r e c t i o n of t h e o r i g i n a l 36 iables :
G(r)
=
<~
=
z z
Fermi
in a
(transverse)
~y ~y z} n n + 1 + B Tn
+ Jy
occur.
Spin
lattice
,
correlations
can be expressed
field,
(3.41)
in the h o r i z o n t a l with
the d u a l v a r -
z ~Z+r >
z ~£+r-I
"'"
operators
(3.42) >
i ~ £+r-I E i=I
(-1)n < e x p
where
XY-model
have
been
c.+ c i ) > 1
,
introduced
using
the W i g n e r - J o r d a n
transformation. A t the t r a n s i t i o n corresponding other
of the o r i g i n a l
XY-chain
as a f u n c t i o n
changes,
(which c a n b e p r o v e n )
yields
largest
the
In the F e r m i o n tains gap.
eigenvalue
the t h e r m o d y n a m i c s
But
for the
(a)
J
(b)
J
(c)
J
x
x
x
+
J
+ J
= J
with
the g a p v a n i s h e s .
y
y
y
HXy ~(q)
the g r o u n d s t a t e
statement
GS
o f the t r a n s f e r
spectrum
three
This
t h a t the
of the o r i g i n a l
representation
the e i g e n v a l u e
system
is t w o e i g e n s t a t e s
of t h e p a r a m e t e r s .
assumption
mines
Ising
that
wave
cross
contains
function
matrix
of the each the
of
and thus
Hxy deter-
system.
c a n be d i a g o n a l i z e d usually
having
a n d o n e ob-
a finite
energy
Cases:
=
- B
;
~ (O)
=
O
=
+ B
;
~(~)
=
O
~(e)
=
O
; IBI < 2 J -x 8 = a r c cos
;
(-B/2J x)
,
(3.43)
57
For
J1
line
= J2
I <
I
~(q)
is
shown
linear q =
in
=
X J1
< O
(b)
corresponds
, T > 0 in F i g . 3 . 1 6 , w h e r e a s c point X = I , T = 0 . For I <
frustration
around
' J3
around
n/3
Fig.
q =
for
T = 0
~
for
T = Tc
. Examples
to
the
(c) I
~(q)
transition
corresponds
the
; for
of
phase
to
excitation I =
I
from
it
ref.
the
spectrum is
36
linear are
3.19.
_~=K-K¢-0.745 1"0.5
I(..~
I
/ .s
/
• 1.0
Fig.
3.19
Fermion excitation spectrum AF triangular lattice with
Because
of
the
continuous relation law
whereas
X =
G°(x)
=
a
is
the
the
oscillating
m o d e l s 48,
the I
cos
behavior
For
with
for
reproduces the
Luttinger function.
behavior
where
linear
i < I well
T = O
(2~ xh
• ~/
lattice
value
~ =
prefactor
of
~(q)
which and
known
and
k-~-- ~ /
£(q) of the XY-chain corresponding I = 0.5 and I = i.O (ref. 36).
reproduce T = Tc
normal he
constant.
first
and
cases
the
can
asymptotic
Peschel
Ising
one
obtains
exponent
~ =
construct pair
cor-
the
power
I/4
,
gets
(xk-~/~
I/2
in b o t h
to the
,
At
(3.44)
the
also
obtained
frustration
point
continuum
version
in by
Stephenson
(Eq.
this
method
(x/a
= r)
(3.33)).
58
The
advantage
Jack
and
tion
point
G
that
one
(x)
=
is,
tween
A closely
~
an
~
too;
is its a p p l i c a b i l i t y
too.
\a)J
For b o t h
\a)
because
also
and E m e r y 49 h a v e This
is the
Id
just
result
the
considered
be-
lattice
re-
later.
be m a p p e d
kinetic
frustra-
correlation
as for
at this
can
to the U n i o n
at the
(3.45)
of the p r e f a c t o r
look
which
systems
'
vanishes,
a closer
problem,
Peschel
on a
to o b t a i n
Ising m o d e l
Id
quantum
exact
corre-
introduced
by
the e n e r g y
- J E a n ~n+1 n are
by a m a s t e r
8t
have
functions.
:
cos
odd d i s t a n c e
related
H(~)
+
method
odd m o d e l 36
here
with
G l a u b e r 50 w i t h
where
I
shall
chain
lation
obtains
~ = I/2
spins
sult 13. We
spin
of the p r e s e n t
to V i l l a i n ' s
classical
equation
-..oco,t~
=
-
Ising
for
(3.46)
spins.
Its
the p r o b a b i l i t y
time
development
distribution
is d e s c r i b e d
p(a,t)
49,50.
T(o,o') ~(a',t)
z a
'
I
with
p(~,t)
=
po(~)
po(~) T
po(~)-I/2
=
const,
is the
=
exp
;
distribution.
by P a u l i
matrices
N,xzxz,
- n=IZ < n n A a + B o
(3.47)
(-BH(~))
equilibrium
can be e x p r e s s e d
T
• p(~,t)
The and
time
development
is h e r m i t i a n 5 1 :
zz -I
n
operator
z
n+1 + C ~ n o n + 1
- D~zn ~n+2 - E
) (3.48)
where spin from and
the c o n s t a n t s flip the
rates last
in g e n e r a l
a system
of
A,
...,
E
are
functions
a n d of the n n - i n t e r a c t i o n
two terms, (for
identical
D ~ O)
interacting
after
fermions.
J
of
the t h r e e
. This
to the H a m i l t o n i a n a Wigner-Jordan
independent
operator H
is,
, Eq.
transformation
apart
(3.38), yields
59
The
essential
space
of a
idea now
2d
Hamiltonian
H
its
matrix
transfer
distribution
this
po(~)
of t h e
exactly
and have
chain with
exponentially
Id
for all
Peschel
line of the
AF
advantage
triangular
only
in t h e H a m i l t o n i a n
(refs. The
52,
summed
point
(a)
(which
AF °
points
F
, and
A normal phase.
•
n
(3.46))
kinetic G(r)
reproduced already
limit,
2d
Eq.
de-
(3.36b)),
description
by StephenPeschel
ANNNI-model
which
questions
of t h e A N N N I - m o d e l
of d i s o r d e r
c a n be
the d i s -
thus
uniaxial
discussion
Ising
At
behavior: q = ~)
f r o m Fig. (1,1,1)
found
and order
in R u j a n
(a)
' J1% , which
are
point
AF °
the pair
whereas
for
central
occurs TD > Tc
correlation T < TD
T > TD
this
in the p a r a m e t e r
have
no singularity.
correspond
law decay,
r -~
are in
isotropic
AF
and the
For
, the
space
T = Tc
ferro-
T
K~
LRO
is f i x e d
correlation
(q = O
According
surface
system.
of
its a s y m p t o t i c
The
transition;
2d
special from
temperature
changes
of a d i s o r d e r
the usual
into a
c
continuously.
of t h e
two
lines
(c)
disorder
function
the
are
itself
at a f i n i t e
to s o m e p h a s e
, with
there
the d o u b l e
the wavevector
is c h a r a c t e r i s t i c
does
is s h o w n
the
J2 # J3
it c h a n g e s
first face
lattice
in the c e n t e r
(b)
the
range
because
case
a temperature
not
parameter 3.12
is n o w
Stephenson kind
triangular
corners.
transition
, for
frustrated
> Jn+2
finally
Stephenson,
power
in t h e
detailled
corresponding
at the
O > J n = Jn+1
or
found
interactions
of t h e
differs
from the general
cases,
The
A careful
properties
A F o = -(3) -I/2
Apart
simple
is its f l e x i b i l i t y ,
line
competing
up n o w a n d the
3.20
magnetic
to
3.4.1.
with
in
c a n b e ob-
53).
essential
Fig.
a disorder
diagramm.
in S e c t i o n
in s y s t e m s
and
Therefore,
system
(Eq.
with
is a l s o
system,
(for the
exactly
lattice,
of t h i s m e t h o d
(however,
lines
chain
2d
2d
character
a n d E m e r y 49 h a v e
found
follows
Ising
eigenvalue.
of t h e
nn-interactions
also
phase
commuting
the equilibrium
. Then
T > O).
and Emery
the o r i g i n a l
the
(3.36a))
of t h e
largest
the p a r a m e t e r
T
kinetic V
functions
ferromagnetic
this method
The
to the
strict
to
Id
matrix
of
and temperature)
(3.36b),
identical
transfer
subspace
constants
(Eqs.
correlation
Ising
order 13 son
H
that one belonging the
in a s p e c i a l
becomes
of t h e
cayes With
V
function
subspace
tained
that
(interaction
respective
an eigenfunction especially
is,
system
to
of the
disorder
sur-
S, U a n d c functions
Ising value
v show
~ = I/4
60
F I
Fig. 3.20
(b)
AF triangular lattice. Lines of Tc/(Jl+J2+J3 ) = const. . In the hatched area the system is frustrated. Along the double lines from K~n to AF o T c vanishes.
For
T = 0
is S
only
Id L R 0 = 0
, as
. In
o decays
as
to
the
belongs
to
G(r)
r-l/2
for
one
calls
For
isotropic
enough comes
to
G(r) but
are
~
exp
~
r-l/2
now
the
any
(S o ~ 0 . 3 2 3 ) independent.
; thus system
again
at
~ =
isotropic.
the in
all
and
function
distance
that
the
GS
however,
Tc = O
frusAs and
.
the
three
. This
diverges
frustration
system
are
I/2
class.
length
the
there
entropy
D =
shows
, and
general,
I/2
odd
Tc = O
A F o)
T > 0 T = O
GS
universality
T = 0
. In
For
. For
is
value
with
(point
LRO
the
distance
correlation
transition
interactions
(-r/~(T))
Ising
interaction
correlation
with
even
a different , the
the
chains
with
usual
T = O
a phase
inhibit
finite
perties G(r)
this
ones
strong
Therefore,
. Thus
the
system
the
nn-directions
(~r/2)
for
of
chains.
two
trated ~
q
direction
single
cos
and
in
the
other
r-l/2
difference
(c)
in
the
uncorrelated,
in
these is
is
strong
entropy two
bepro-
paramagnetic,
nn-directions as
in
case
(b)
,
61
3.3
Further
After
Frustrated
the d e t a i l l e d
last
section
more
briefly.
tice
Here we
and
variant
The
Then
the
an e x a c t
only
models
stacked
AF
of the
Id
difficult
lattice
lattices
those models
with
of the
for t h e the
same
are
translational
perpendicular
repeated
lattice,
after
and also cases
to w h i c h
we
~
lines
one
s o m e of the
of
turn
the
lattice
these
pair
transfer
triangular
Along
in the
3.1,2
noncrossing
operator
than
in the
of T a b l e
solved which
as s p e c i a l
systems,
Interactions
triangular
the o t h e r
is p o s s i b l e .
triangular
can b e c o n s i d e r e d
diagonally
only
can be
are periodically
anisotropic
systems
more
in o n e d i r e c t i o n ;
constants
of the
structure
solution
stacked
With Noncrossing
describe
consider
is n o t e s s e n t i a l l y
riodically
tion
discussion
(3.2), w e m a y
interactions. matrix
S~stems
lat-
also pein-
interac-
c o n s t a n t s 54.
following
horizontally
at the e n d of t h i s
or sec-
tion.
3.3.1 As
Union
Jack
Lattice
the triangular
lattice
c a n be r e g a r d e d J1
there
as s h o w n
as a s q u a r e
is o n e d i a g o n a l in Fig.
with
nn-interactions
lattice
where
interaction
J2
the U n i o n
apart
Jack
lattice
from nn-interactions
in e a c h
elementary
square
3.21.
////
/s// //// J1 Jl Fig. 3.21
(a)
Comparison of the triangular lattice
OI (a)
(b)
and the Union Jack lattice
(b)
62
In this spins 2
lattice
($2)
interact
independent elementary by Yaks, the
there
of s u b l a t t i c e via
ofd t h e
J1
interact
with
s i g n of
triangle
Larkin
are two nonequivalent I ($I)
the spins
J1
J2 of
is f r u s t r a t e d .
sublattices.
, the $I
(here a s s u m e d
and Ovchinnikov
free energy
square
via
spins
only.
The
on s u b l a t t i c e
For
J2
to be p o s i t i v e )
< O
(AF)
each
This model was first investigated 55 , who obtained exact results for
1966
a n d f o r the p a i r c o r r e l a t i o n
function
of s u b l a t t i c e
Sl L e t us f i r s t
consider
actions
[J21/J I . L i k e
i =
different
i < I :
I = I :
cases
the
GS
as a f u n c t i o n
I < 1 , I = I
The
interactions
LRO
, is o n l y
and
J2
(I = I, T = O) lattice.
Tab.
three
GS
ferromagnetic
and
has So = O
This
elementary leads
.
point
triangle
to a
of t h e
any one
GS d e g e n e r a c y
inter~
Ng
CN
> O . The degeneracy N is e q u a l to the o g of d i f f e r e n t l y c o v e r i n g its d u a l 4 - 8 - 1 a t t i c e (see
number
to
are
inter-
there
is the f r u s t r a t i o n
In e a c h
a c t i o n m a y be b r o k e n . and thus
the
degenerate,
Union
of t h e
lattice
I > I .
are weak,
twofold
The point Jack
of the r a t i o
in the t r i a n g u l a r
3.2b)
S
with
nn-dimers,
for which
2N / 2
(1 + I / 1 6 ) N / 2
a simple
lower bound
can
be found: N
From
>
g
this
So
I > I :
Now plete free,
I 17 ~ in --~
We add that only
AF LRO
for the
So(i=I)
Figure
3.22
disorder
of
spins
$I
for
spins
on o n e
on
than
$2
have
com-
completely
the
GS
situation
entrowe
shall
of V i l l a i n .
So(I>I)
whereas
are
similar
'odd m o d e l '
larger
are b r o k e n ,
T = 0
sublattice
in 2 . A v e r y
anisotropic
be
on
the
LRO
S o = I/2
must
J1
the
. Therefore,
a n d in s p i t e
interactions
interactions
(3.49)
dominates,
p y is f i n i t e , find
(17/8) N / 2
follows
>
J2
=
for
because
~ = I
both
for
I > I
J1 o r J2
c a n be b r o k e n . shows
line
TD
the p h a s e
diagramm
(Stephenson13),
of V a k s
where
et al. 55 t o g e t h e r
as a b s c i s s a
with
and ordinate
the
83
i
i
I
~
I
i
TI
|
l
I
!
I
|
[
l
1
T
f
it
%, P
;T
/ ,% I/
,
-0.5
-I.0
Fig.
the
3.22
F
0.0
-0.5
0.5
1.0
Phase diagramm of the Union Jack lattice. The full lines are the transition temperatures TCF and TcA F of the F and AF phases. The dashed line is the disorder line T D .
scaled
quantities
~ = -
X/g4
+
l ~
and
T'
= K 121 / () 2 9 -4 + l
are
used. At
the
frustration
< aFp
the
(T > O ) - L R O
phases.
paramagnetic agreement tion
point
F and
(P)
with
along
(I =
the
T = O)
aFp
are
down decay
to of
= -
I/~
connected
transition
extends
exponential
disorder
I,
groundstates
Between phase
the
the
AF
lines the
the
to TCF
. For s > sFP and corresponding and
frustration diagonal
TCA F
the
point,
in
correlation
l i n e 13
-4K~ cosh
for
T
4K~
> O
=
e
: ( ~ r h - I /2 \2 )
G(r)
(3.50)
r (tanh
K 2)
for
even
r (3.51)
=
0
for
odd
r
func-
64
For
TD << J1
TD/J I
As
for
~
(K~) -1
T << J1
ponentially,
0.907
the
_
4 in 2
F and AF
reentrent
transitions
2.)
antiferromagnetic
3.)
paramagnetic
three
point, phase
all
which
we
as an e f f e c t
system
is shown,
to an e f f e c t i v e
Kef f (compare in Fig.
= Eq. 3.24
where
temperature
three
salt 56.
Such
are
of the n o n frustration
an u n u s u a l
The m u l t i p l e
interactions $I
which
to the U n i o n
In Fig.
Jack
the one of the
lattice interior
spins
3.23
also
to the
sign
to an at
investigated Jack
lattice
right
simplest
can be s u m m e d
found
can be
leading
changes
and E g g a r t e r 57 h a v e similar
multiple but
transition
transitions.
the
class for the
theoretically,
competing
Fradkin
which
true
of the
decorated
out
leading
interaction
I K 2 + ~ in (3.9)) for
is not
detail.
of s u b l a t t i c e
systems
of the U n i o n
this
interesting
of the
t e m p e r a t u r e 30.
show multiple
cell
decreasing
to the u n i v e r s a l i t y
in m o r e only
for r o c h e l l e
Ising
With
- paramagnetic,
belong
is not
nn-interaction
finite
behavior.
However,
consider
transition
also
ex-
(3.54)
model.
effective
unit
line
,
understood
and
the d i s o r d e r
asymptotically
transitions
Ising
experimentally
some
(3.52)
- ferromagnetic.
r-I/~
decorated
approach
I
- antiferromagnetic,
transitions
G(r)
is,
phase
< --
relation
occur:
paramagnetic
frustrated
1
'
linear
(3.53)
I.)
that
(I-I)
the
X < I
shows
consecutive
At all
one o b t a i n s
in a range
N<
the s y s t e m
(3.50)
f r o m Eq.
(cosh2K I)
with
fixed
a schematical ~ = _ K 2 / K I 30
(3.55) temperature
dependence
shown
65
j~
-
j
-
2
Fig. 3.23
Comparison of the unit cell of the Union Jack lattice with a model of Fradkin and Eggarter.
K ¸
Kc 0
- Kc
Fig. 3.24
Effective temperature dependent interaction (ref. 30).
For
convenient
Kc
is t h e
tice, at
inverse
and the
LIC
ordered
the
transitions of
transition
system
becomes
Finally
when
summation
given
paramagnetic, IKJ
F
L2C
phase.
Ising
Kc This
Jack
lat-
temperature
it b e c o m e s
below
the U n i o n
square
increasing
at
L3C
sequence
lattice
AF , there of
in the
(3.53).
the d e c o r a t e d
the
at
L = JI/T
Kef f > + K c , where
of t h e With
decreases
to t h a t of
b y Eq.
between over
phase.
to t h e p a r a m a g n e t i c
is i d e n t i c a l l
F
as function of
(L > LIC)
temperature
is in the
transition
The difference where
at low temperatures
system
again.
is a t h i r d
interval
l
K
central
model
spins
and the Union
of t h e u n i t
cell
Jack
lattice
(Star-square
66
transformation, between
the
currence
Fig.
3.3c)
remaining
of
a multiple
temperature
dependence
of
of
the
found
Vaks
Geilikman
more
effective
specific 43
complicated
obviously
transition
the
behavior and
is
phase
perature by
generates
spins,
heat
is
shown
only can
interactions
quantitative.
be
understood
nn-interaction. near
the
in F i g .
The from
The
low
frustration
oc-
the tem-
point
as
3.25.
c(r)
r:,
T
Fig. 3.25
Specific heat at the Union Jack lattice for I = i - @ (with l~I << i) For 6 > O (full line) two very close transitions occur at Tc2,3 corresponding to LIC and L2C of Fig. 3.24, in addition to a transition at higher temperature Tcl , existing also for @ < O (dashed line) (after ref. 43).
For
I
I :
(6 = O)
Cv(T)
For
0
almost
<
~
low
lytical
also
13
from
TD
jumps
two
Tcl
temperature
= KDI
similar
the
the
TD •
diagonal line
G(r)
(-I) r from
the
(4/in2)
disorder
below
TD
For
to
T/J I = -
G(r) at
low
is v i s i b l e .
determined
phase
above
wavevector
has
below
,
(3.56)
(TD/JI)-I
and
near
away
paramagnetic whereas
the
divergence
temperature,
Stephenson G(r)
I
with
maximum
is m o n o t o n o u s
K I e -4KI
I - I <<
coincide
logarithmic at
6.4
Cv(T)
one
and I >
fixed
I
Fig.
(I-i)
is
no
lattice
correlation (Eq.
(3.50)).
monotonously like
value
LIC, only
L2C
one
transition only
an
ana.
occurs.
pair TD
3.25
there
triangular
decays
behaves
transitions
in
this to
function In
(q = ~)
another
the
(q = O) . The
one;
the
,
67
Union
Jack
Finally
lattice
we
exhibits
consider
t i o n of Eq.
(3.51)
G(r) along
right
thus
If
~
~ = I/2
F o r g a c s 25 u s e d detail
G(r)
directions. order
line
Both lines is,
and
Eq.
When
of
one
frustration
line
down
to
k i n d 13
point.
Extrapola-
T = O
yields:
,
point T > 0
for
T ~ 0
line
is v a l i d
for
point
are
to are
to
the p h a s e
exponentially
along
r-I/2
T = 0
as the
close
the dis-
decay.
boundaries
for
idential,
in m o r e
from different
approaching
r-I/2
all three
frustration
is a p p r o a c h e d
(3.51),
along
lattice.
e t al. 55 to d i s c u s s
from exponential
frustration for
triangular
of V a k s
r-I/~
the
AF
f r o m Eq.
is a c r o s s o v e r
G(r)
finds
=
odd
frustration seen
the d i s o r d e r
(3.51)
G(r)
the
can be
approaching
angle,
r
results
the
second
(3.57)
the exact
from
the
,
isotropic
there
limits
even
as in t h e
when As
Approaching crossover
r
l i n e of
at t h e
the d i s o r d e r
- 1/ 2 G(r)
a disorder
a similar
is found.
two transition
for
T = O
; that
lines.
point
in Fig.
3.22
from a different
asymptotically:
const.
(from t h e
F
phase)
,
and
(3.58) G(r)
with
=
(-I) r • c o n s t .
const.
the
frustration
havior
G(r)
depends
The mapping
AF
phase)
< I
Therefore, of
(from t h e
on the quantum
tice
is p o s s i b l e
power
law when
here
point
is a s i n g u l a r
on the d i r e c t i o n
too;
approaching
XY-chain Pesche136
discussed this way
the frustration
point,
where
the be-
of a p p r o a c h . for the
triangular
reproduced
point
along
the TD
.
lat-
r-I/2
08
3.3.2
Villain's .
.
.
.
.
.
.
.
.
.
.
.
Odd Model .
.
.
.
.
.
.
V i l l a i n 58 w a s
the f i r s t
on t h e
lattice
lue,
square
where
be odd Andr~
.
.
a n d its G e n e r a l i z a t i o n s .
.
.
.
.
.
.
.
.
.
.
to c o n s i d e r
of
.
.
.
.
the
.
.
.
.
fully
frustrated
Ising
of e q u a l
absolute
all nn-interactions AF
bonds
around
each
elementary
system va-
square must
( ' o d d - m o d e l ' ) . In T a b l e 3 . 2 c , d t w o d o m i n o m o d e l s i n t r o d u c e d b y 59 are shown, which have one AF interaction J' and
F
to V i l l a i n ' s
interactions
the a r r a n g e m e n t in the
center
3.3.2a
first
Here
the
For
X < I
and
S
per
of t h e u n i t (dominos)
o
consider
AF
elementary for
square
X ~ - J'/J
cells
consisting
and behave
(X
a high
coverings
= 0
of
and which
= I . They of t w o
differently
for
are
equi-
differ
squares
only by
with
J'
I ~ I
G
J'
is
F
total
form
(Piled-~p domino)
contiguous
, the weak
J'
of T a b l e
3.2c.
chains.
interactions
are broken,
.
the d u a l
is the
identical
frustration to the n u m b e r
square
lattice.
=
G --~ --~ O . 2 9 1 6
,
is C a t a l a n ' s
X > I
model
This
point
of t h e m o d e l
of c o m p l e t e
equivalence
with
dimer
yields
the
GS
s i t e 60
So(X=I)
culation
GS
degeneracy
per
J
PUD
(X = I, T = O)
GS
entropy
the
interaction the
The point
where
J
odd model
Q [ ~ ~ - ~ @ _ ~ _ Q ! ~ [ ~
L e t us
the
.
et al.
three
The
.
with
the n u m b e r
valent
For
.
J' GS
chains
(3.59)
constant.
dominates,
the
J'
chains
degeneracy
c a n be o b t a i n e d
which
ordered
have
is c o m p l e t e l y
analog
J'
to the
for
are
f r o m the n u m b e r
chains Id
T = 0
on b o t h
ANNNI-chain,
AF
ordered
of
GS
sides.
The
Section
of cal-
2.1.1,
and yields
So(X>I)
-
1+~ 21 in (\--~----) ~
It is i n t e r e s t i n g an e n t r o p y way,
the
effect.
to n o t e When
intermediate
the c o u p l i n g
the J
0.2406
spins
chain
(3.60)
of n e i g h b o r i n g
on b o t h
has
chains
a finite
GS
J'
chains
are o r i e n t e d entropy
per
the
via same
site.
69
However, tropy
for o p p o s i t e
of t h e
The
PUD
tion
point
given
spin
intermediate
model
has
orientation chain
two phase
(Fig.
3.26).
The
2K s i n h
(K+K')
=
on t h e
J'
chains
the
GS
en-
vanishes.
boundaries,
both
two boundaries
ending
of t h e
at the
frustra-
F and AF
phase
are
by 59
sinh
I
and
(3.61) sinh
Like in there
2K s i n h
(K+K')
the U n i o n
Jack
exists
sublattice
LRO
=
lattice
on o n e
causes
the
- I
also
in the
sublattice,
entropy
PUD
while
to r e m a i n
model
for
the disorder
finite
down
to
I > I
on the other T = 0
.
T/J
PARA
~/
-
Lt
Fig. 3.26
The
GS
ordered
Phase diagram of the
of t h e for
ZZD
i < 1
For
I = I
del,
(I = I, T = O)
For J'
i > I
like
of s p i n s pairs
model too,
the
S
model
markedly
do n o t
interacting
form a fully
(~ick-zack
with
PUD
model (ref. 59).
o
= 0
domino),
via
form J'
frustrated
see T a b l e
3.2d,
is
F
.
it is e q u i v a l e n t
is its f r u s t r a t i o n
it d i f f e r s
interactions
PUD
from the
contiguous
point PUD
chains.
are therefore triangular
to V i l l i a n ' s
odd mo-
too. model, In t h e
coupled
lattice,
as t h e GS
rigidly.
the
GS
strong the pairs As
these
entropy
70
per pair site
is i d e n t i c a l
thus
So(~>1) The
ZZD
I sAFA 2 o
=
model
of t h e
AF
triangular
lattice;
O.1615
possesses
2 tanh
(K+K')
I > I
there
For
to t h a t
So
per
is
tanh
only
2K
=
(3.62)
an I s i n g
system remains 59 diagram is s h o w n
PARA
I < I
at 59
(3.63)
in the f r e e e n e r g y
in t h e p a r a m a g n e t i c in Fig.
for
I
is no s i n g u l a r i t y
the
transition
phase.
The
for
T ~ 0
corresponding
,
phase
3.27.
FERRO-
-
--I =J'lJ
Fig. 3.27
3.3.2b
Correlation .
Consider For
Phase diagram of the
.
.
.
.
GO(r)
.
.
.
.
.
.
PUD
and all of
chain):
.
(without
model (ref. 59).
Functions .
.
.
.
.
ZZD
.
functions models
F o r g a c s 35,
found
q = I/2
G(r)
zero and finite
sults
.
and
(odd m o d e l )
the b e h a v i o r for
.
the c o r r e l a t i o n
I < I
= I
.
ZZD
:
ferromagnetic;
. Wolff
in h o r i z o n t a l ,
signs
T = O
G°(r)
G a b a y 61 a n d P e s c h e 1 3 0
temperature.
changing
are
for
and
have
For
determined
Z i t t a r t z 54 d i s c u s s
diagonal
and vertical
Here we only mention
depending
= I
on the e x a c t
the
in d e t a i l direction
T = O
position
re-
of the
71
const.
• r-~/2
;
r
even
GO(r)
(3.64a) const./~
for horizontal
OOr>
.
For
i > I
for
T = O
functions
-
I 5 (-I)
I > I
and
>
chains
chains the
J'
of t h e
cause
PUD
model
the f i n i t e
and J
chains
GS
are
are
ordered,
entropy.
((b)
The
is o n l y
,
(3.65a)
(3.65b)
apart
and diagonal
f r o m the
entropy
direction
also
G°(r)
is e q u i v a l e n t
of t h e
to t h e
AF
ZZD tri-
result:
o
GZZD(r)
3.3.2c
~
r
-~/=
cos
{2~ ) \~- r
(3.66)
~~!_~[2[~_~2~!~
PUD
tically
and the
ZZD
models
(or h o r i z o n t a l l y )
arranged
translation
period
is
papers 54'62
have
lated
phase
the
)r
T = O
in v e r t i c a l
tition
J'
J
along
o (r) GpUD
The
and
r ~ ~):
(-I
angular
the
intermediate
=
For
directions,
odd
00
o' GpuD(r)
model
r
direction. and
the
correlation valid
and vertical
;
cos
for diagonal
whereas
• r -I/2
their
results
invariant
~ = 2
can be c o n s i d e r e d
layered
within
. Wolff,
investigated diagrams
are summarized
and
models, each
Hoever
general
and
54.
layer
as the the
simplest
interactions
and the
Zittartz
layered models
correlation
in ref.
where
functions.
layer
are
repe-
in a s e r i e s and have
ver-
of
calcu-
The method
and
72
We w a n t
to d i s c u s s
ization
of the
strating
how
little
the t r a n s i t i o n , model ones (Fig.
J1
if
model GS
3.28)
J2
where
changes
of t h e s e with
, for
models.
are related
. Calling
the
J
the
J'
first
one,
one
obtains
only
along
the t h i c k
along
these
lines
lines
and a l s o
%% %% -I %%.
I T
"%%
demon-
to the p r o p e r t i e s
parallel a
a general-
clearly
interactions
interactions
T = 0
I
The
v = 2 , is an e x a m p l e
properties
Tc > O
and m o d i f y i n g
to b e c o m e
guration
two m o r e
PUD
'phase
to the
at PUD
J1
d i a g r a m '63
So > O along
of the
The
spin
the d a s h e d
confi-
part
of
Yl
%k
--1 "I(* %
Fig. 3.28
Groundstate phase diagram of the modified
the d i a g o n a l Fig.
3.29
Y2
But o n l y
9ig. 3.29
the
Yl
+ Y2 = O
GS along
energy
in Fig. Eo
the w h o l e
3.28,
where
is p l o t t e d diagonal
PUD
Yl
model (ref. 63).
Yi = J i / J
as a f u n c t i o n + Y2 = O
The corresponding groundstate energy (ref. 63).
" In
of
one has
Yl
and Tc = O
,
73
everywhere pending
ohly
Fig. 3.30
The
else on
T = O
not have
is f o r G(r)
Ghl (r)
whereas
=
along
Gv(r)
the
and
frequent
,
< ~ r>
r
for
r
occurrence
again
ample
the
de-
c
on
(ref. 63).
o f the Tc
correlation Yl
we
chessboard
=
GS
spin
confi-
It is i n t e r e s t i n g function
< I , only
(-I) r
similar 58
for
the h o r i z o n t a l
model
such
chains
n = I/2
to t h e a n i s o t r o p i c
frus-
are
have
become
to d i s c u s s
different
also
completely
de-
.
q = I/2
systems
now going with
(3.67)
(3.68)
u p to n o w w h i c h
are
,
;
of t h e v a l u e
discussed
. However,
r ~
• r -I/2
a s s u m p t i o n 35 t h a t a l l
q = I/2
direction
the h o r i z o n t a l
for even
systems
changes
(if
Gh2 (r)
lattice
cos
for odd distance coupled,
> I
T
direction
the vertical
=
Tc
influence
and vertical
= - Y2
at a f i n i t e
are exchanged).
I
triangular
trated
occurs
3.30).
indicating
any clear
Yl
the h o r i z o n t a l
trated
transition
(see Fig.
the h o r i z o n t a l
, that
and vertical
The
ly I + y 2 1
'phase diagram' does
to c o n s i d e r
Along
Ising
The corresponding transition temperature
gurations
Tc = 0
a normal
in t h e
examples
no finite critical
Tc
of f r u s -
had
at
Tc = O
as a f i r s t
counter
behavior.
led to with ex-
74
3.3.2d The
Chessboard
chessboard
= 4
shown
frustrated. relation T =O
Model
model 59'54'64
in T a b l e
3.2e where
We mention
length
down
every
this m o d e l
to
T = O
layered model
second
as the
period
elementary
square
one with
a finite
first
, and thus
with
not becoming
critical
is corat
.
Already
Andr&
e t al.
59
and also the existence
tartz
64 h a v e
S
~
o
In ref.
As the
reason
show
rives
criteria.
T = O
GS
systems
r -q
short turn
T = O
the
entropy
So
for
. Wolff
T > 0 and
Zit-
short
correlation
function
correlation
length
by flipping with
decay,
range
which
GS
different
6o
between , that
local!y
whereas
and
is
only GS
systems
other models
demonstrate
from the previous
'superfrustration',
isolated
correlations
to t h r e e
decays
ex-
:
(3.70)
'isolated'
conjectures
for
with
He d i s t i n g u i s h e s
from other
case.
GS
the diagonal
, S ~ t o 65 c o n s i d e r s
without
thus
that
for t h i s b e h a v i o r
Tc = O
We now
of a f i n i t e
of a t r a n s i t i o n
in (I + ~ )
with
only
absence
(3.69)
for
=
spective
the
obtained:
54 t h e y
~I
found
O.371
ponentially
and
is a d i a g o n a l
frustrated GS
a limited
without
e -~/r
systems
, which
to h a v e
systems
for w h i c h
range
isolated
with
cannot
number
long
GS
he dere-
be r e a c h e d
of
spins.
should have
decay.
also exhibiting
the chessboard
model
He
correlations
exponential
decay
to b e n o p e c u l i a r
75
3.3.3 The
~9~__~!~2
hexagon
diagonal
lattice
layered
interaction
(Tab.
square
is o m i t t e d
3.1b)
lattice as
shown
can
be
where
This
way
thus
also
with
the
in e v e r y
in Fig.
/\ k.,,,.,~",~ ~..'~,. /-,,,/\/ Fig. 3.31
regarded
as
a special
second
row
every
and
the
fully
o•" K
i~
~
configuration
of
solved
Ising
interactions
the
system K. = l
general
on
the
"
± K
The dotted
anisotropic
hexagon shown
and
lattice
in Fig.
3.32.
/\/k/\ I I I \/\/\/\.
i- , ~ / ~i / \ / /J Fig. 3.32
The
Configuration of interactions of the fully frustrated hexagon lattice. Thick lines are AF , thin lines F interactions (ref. 66).
system
has
SO
O.214
~-
a finite
GS
,
entropy
66
(3.71)
a
second
\/\/
Z i t t a r t z 66 h a v e frustrated
of
3.31.
Transformation of a square lattice into a hexagon lattice. interactions have to be omitted (ref. 66).
Wolff
case
76
and
is p a r a m a g n e t i c
relation
length
-1 go
=
in
chessboard
become
critical
3.3.4
~H2D_~i~
investigated
of a l a y e r e d
down
to
T = 0
, where
the c o r -
f i n i t e 66
(3.72)
model
at
ferromagnetic
been
all t e m p e r a t u r e s
(2 + V ~ )
As t h e
The
for
remains
the
T = 0
and the
fully
hexagon
lattice
does
not
.
fully
by Waldor,
system which
frustrated
frustrated
Wolff
is s h o w n
AF
pentagon
a n d Z i t t a r t z 67, in Fig.
lattice
as a n o t h e r
have
example
3.33.
S Fig. 3.33
Pentagon lattice; one layer is drawn with thick lines (ref. 67).
Whereas
in t h e
in the
AF
S
o
~
F
case
case the
does
down
correlation
of t h e a s y m p t o t i c
also
the u s u a l
a finite
GS
Ising
behavior 67
systems
as e x p e c t e d ,
entropy
(3.73)
a n d is p a r a m a g n e t i c
the
found has
0.2336
horizontal
Like
they
system
to
T = O
length
oscillations of the
not become
. The
~h(T) of
Gh(r)
two previous
critical
at
temperature
dependence
and of t h e w a v e v e c t o r are
sections
T = O
.
shown the
in Fig. AF
of the
q = e(T) 3.34.
pentagon
lattice
77
I
2
Fig. 3.34
3.3.5
&
6
8
T/J
Horizontal correlation length {h and wavevector lattice as a function of temperature.
@
of the
AF
pentagon
~2~9_~9~9
As for the t r i a n g u l a r gom6
lattice
tions
(Table
case with
all i n t e r a c t i o n s
fore,
o
the p r o p e r t i e s for d i f f e r e n t
of the Ka-
configura-
Then
is the i s o t r o p i c
all e l e m e n t a r y
but not the hexagons.
case,
in this range.
where
in the
Kano and Naya 68 have
in the free e n e r g y
is p a r a m a g n e t i c
AF
triangles
for
T > 0
The
GS
lat-
found
; there-
entropy
is
and very high:
~
O.5018
(3.74)
A very
similar
tioned
in c o n n e c t i o n
triangles
Sp p
frustration
of a s i n g u l a r i t y
the s y s t e m
finite
lattice studied
Ki .
are equal.
are frustrated,
the absence
S
have been
of the i n t e r a c t i o n s
The s i m p l e s t
tice
and the square
3.1c)
=
value
as being
is o b t a i n e d with
from the P a u l i n g
the t r i a n g u l a r
lattice,
approximation which
treats
menthe
independent:
2 3 in 2 + ~ in ~
~
O.5014
(3 75)
78
The
reason
relation Figure
for
3.35
respective lattice
this
which
we
shows AF
good
shall the
agreement discuss
internal
interactions
r e s u l t s 68
which
#
is p r o b a b l y
energy
of t h e K a g o m &
together
for
the very weak
pair
cor-
further.
T > 0
with look
lattice
with
the corresponding
F
triangular
similar.
-0.4-O.B- 1.2-
-1.6-
-2.0-2.4]
I
-2.8
/
Kano
anisotropic
J1
three
and Naya
Kagom~
= J2 = J > O
magnetic
The
s i g n of
P
s
L'
12-,.I/ILl
1~
calculated
the partition
(with t h r e e
[J[ < 0
J
different
only
function
of the
interactions
Ki
for
G e i l i k m a n 69 for t h e c a s e
shown
in Fig.
functions
3.36a has
along
discussed
the dashed
is u n i m p o r t a n t
in the a b s e n c e
a normal
transition
of
lines a
field. J3
F
and the transition
phase, I
C
and the correlation
For weak
K c-
I
~,
nn-directions),
, J3 = - ~
diagram
3.36a.
have
lattice
different
the phase in Fig.
~-
Temperature dependence of the internal energy of the AF (upper) and F (lower) Kagom& lattice (full lines). The corresponding triangular lattice results are shown as dashed lines (ref. 68).
Although
the
I
"~l
K/011
Fig. 3.35
/
I I I I
_
(l < I)
T J
c
he f i n d s
I In 2
(1-~)
temperature
Ising for
i ~ I
to a s i m p l e
vanishes
linear:
(3.76)
79
7 f\\ ~/ × X \,
£
X - //"
(a)
Fig. 3.36
The
(b)
Frustrated Kagom@ lattice; the double lines correspond to tions J3 "
Only the triangles are frustrated, along the dashed lines the pair correlation function is discussed in ref. 69;
(b)
here the hexagons are frustrated too.
correlation
function
G(r)
completely along 69 for I ~ I : TD J
KDI
2 In 2
-
(I : I, T = O)
triangular
lattice
i > I
the
for For has
three
i > I
the
system
inves£igated
free
models
for
apart
order
and
from
TD
lines
in Fig.
(T D > T c)
3.36a
which
frustration
point
for
but no order
T ~ 0
(according
correlation
of
the pair
to ref.
functions
is p a r a m a g n e t i c
a whole
configurations energy
dashed
is a l s o
(3.77)
is the
chains
all
the line
like
for the
AF
3~16).
direction
T = 0
different the
(Fig.
J3
the p e r p e n d i c u l a r
along
a disorder
(I-I)
The point
For
interac-
(a)
vanishes linear
AF
family
for
finite
function
an a r b i t r a r y
occurs
T ~ O
in
~t ~ O
temperature. Kagom~
S H t o 70
models
with
the b e h a v i o r
of
for
T > O
. In t h e s e
part
of the h e x a g o n s
is f r u s t r a t e d . To the Fig.
right
3°36b
side
shows
of Fig.
3.36a,
one possible
where
no hexagon
realization
;
vanish).
interactions
correlation also
Gi(r)
of f r u s t r a t e d
F and AF
all t r i a n g l e s
69 for
is f r u s t r a t e d ,
of t h e o t h e r
extreme
where
80
all
hexagons
tration
the
the
complexe
the
whole
in
are
of
the
tanh
energy
corresponding
~max-1
In
the
N>
proof
function
with
w
fast;
the
=
=
agrees
=
already
in F i g .
This
means
temperature
independent
outside
the
3.37
there
is can
including
T > 0
(0.74) li-jl
,
a maximal
correlation
the
be
no
the
frus-
area
analytic
T = 0
upper
of
hatched
of
including
singularity
. For
the
cor-
b o u n d 70
(3.78)
length
0.30
(3.79)
expands of
two
<s.s > in p o w e r s o f t h e 1 3 spins on a single frustrated
w I + w + w2
KI
simplest
2
that
for
to
Itanh
-
any
shows energy
- plane
obtains
4 •
he
nn-correlation
U
at he
S~to free
axes.
<
~nn
l~nnl
IJl)
real
function
l<sisj>J
the
(B
positive
free
relation
frustrated.
hexagons
and
=
1 - x 3 + x
x = e
approximation
< --
-21KI
1 ~
the
correlation
triangle
'
This
for
nn-pair
power
internal
(3.80)
series
converges
energy
(that
is,
very the
function)
I~nn
I
(3.81)
to w i t h i n
a few
percent
with
the
exact
U
.
4i
-4
Fig. 3.37
2
In the non-hatched area of the complexe tanh (~ IJI) - plane the free energy of all KagomA models with fully frustrated triangles is analytic.
81
Equation also
(3.78)
for
means
T = 0
tem besides lattices,
. Thus
internal
the pure
AF
(3.81)
is a good
depend
only w e a k l y
We also note
close
energy
Kagom~
U
at
integrating for
S
also
in this
U(T)
function
is the forth
pentagon T = 0
no hexagons
on the f r u s t r a t i o n
to the exact
lattice
and the e n t r o p y
approximation
that
Kagom~
critical
lattice w h e r e
approximation
of the c o r r e l a t i o n
and the f r u s t r a t e d
does not b e c o m e
The exact
decay
the f r u s t r a t e d
the c h e s s b o a r d
which
the Pauling
exponential
sys-
and h e x a g o n
. are known only
are
case,
frustrated. U(T)
for
As Eq.
and S(T)
can
of the hexagons. from Eq.
S O , Eq.
(3.81)
exactly
(3.75), w h i c h
yields
also was very
result.
~ ~ _ ~ _ _ ~ _ _ ~ ~ _ ~ _ ~ ~ _ ~ ! _ ~ _ ~
3.3.6
Transition
at
TG_[_ ~
At the end of Sections trated
Ising
systems
considerations state
3.2 and 3.3 w h e r e we have d i s c u s s e d
solved exactly,
on the c o n n e c t i o n
and the e x i s t e n c e
Hoever,
Wolff
between
If the set of all
2d
more
the d e g e n e r a c y
GS
formulated
frus-
general
of the ground-
is connected,
the f o l l o w i n g
that
the global
symmetry
, cannot be broken.
conjecture
is if any two
into each other by a series
transformations, s I ~ - sl
to m e n t i o n
of a transition.
and Zittartz 71 have
be t r a n s f o r m e d
we w a n t
of p u r e l y
GS
can
local
of the Hamiltonian,
In this
case there
is no phase
transition.
In case of the c h e s s b o a r d 71 and the connected 2 nn
by l-spin
flip processes;
spins m u s t be flipped
another
one.
For all three
systems
do not become
AF
Kagom~
simultaneously systems
critical
at
lattice
in the h e x a g o n
~(T=O) T = O
to obtain one remains
all
lattice
GS
GS
from
finite,
these
in a g r e e m e n t
with
the above
conjecture. If the Hoever
GS
are
always
are not connected, there are no g e n e r a l statements; 71 et al. m e n t i o n examples w i t h and w i t h o u t a transition.
82
S~to has tems
also put
foreward
a conjecture
consistent
with
the t h r e e
sys-
just mentioned:
If a n d o n l y
if a f r u s t r a t e d
no transition
and
~(T=O)
Ising
system
remains
also
finite,
for
the
T = O
has
set of all
GS
is c o n n e c t e d .
S~to
calls
such models
'superfrustrated'.
Both
conjectures
still
have
to b e p r o v e n .
3.4
Frustrated
In this the
section
square
brick model
2d 2d
Sec.
2.1)
GS
point The
and crossing
a n d the
nnn
we also
triangular
interactions
and finally
consider
comment
analog
disorder
and
which,
the
on the
(simplified)
connection
of
Tc $ O
the
and mean
field
Id
++++
the p h a s e
ANNNI-chain direction
(compare
and
introducing
case: = <2>
F
for
for
I = - J2/J1
i > 0.5
< 0.5
; at t h e
frustration
occurs. to t h e
Id
. Of the w e l l
low t e m p e r a t u r e approximation diagram
c a l c u l a t i o n s 12'72
Id
J > 0 b e t w e e n s p i n s on n e i g h b o r i n g o l a t t i c e is s h o w n in T a b l e 3.2i.
to the
difference
with
f r o m the
in o n e p e r p e n d i c u l a r
, and periodic
essential
sitions
ANNNI-model
exactly.
is d e r i v e d
corresponding
are
Id
Interactions
models.
nn-interactions
diagrams
Thus
exactly
to v e r t e x
by repetition
the
(J1 > O)
2d
the A N N N I - m o d e l
ANNNI-model
chains;
nn
Crossing
ANNNI-Model
additional
The
with
With
the
be solved
solved
these models
The
we discuss
cannot
comparison
3.4.1
Systems
lattice with
therefore, For
Ising
too
(does not
(too l a r g e 3.38 low
is the e x i s t e n c e
known methods
series
of Fig.
for not
chain
to d e t e r m i n e
converge
fluctuations)
is c o m b i n e d T
of t r a n -
for cannot
from Monte
phase
d = 2) b e used.
Carlo
, the M~ller-Hartmann/Zittartz
(MC)
83
Poramognetic
2.0
1.0 Antiphase
I
I
0.2
Fig. 3.38
for d o m a i n
a p p r o x i m a t i o n 74 g o o d Adjacent
to the
finite
therefore, rate
especially
point, With
M
location
phase
1.0
Jl = (l-e) Jo
and
for
with
X < 0.5
a free
the
same
respective
order.
Not
is the o c c u r e n c e
between
and
fermion
IJ21
interesting
of the L i f s h i t z
the
<2>
point
L'
common
and,
of an i n c o m m e n s u -
and the 75
X > 0.5
P
, a special
phase;
the
multicritical
is not y e t known.
the m e t h o d
of M H l l e r - H a r t m a n n
can be e s t i m a t e d
F
phases
0.8
energies 12'73'76
T << Jo'
groundstate
temperature
I
¢¢
ANNNI-model with
boundary
for
LRO
modulated
exact
I
0.6
Phase diagram of the 2d J2 = - e Jo (ref. 72).
approximation
are
I
0.4
:
sinh
and
Zittartz
the p h a s e
boundaries
analytically12'73a:
2 ( K I + 2 K 2)
sin
2K O
=
I
,
(3.82)
and
<2>
:
exp
(2K o)
=
(I - e x p
(4K2))/{(I
-exp
(-KI+2K2))
(I - e x p
(K1+2K2))} (3.83)
They
are
in g o o d
The
MC
data
modulated
M
agreement
also phase
with
indicated
the
MC
d a t a 72.
f i r s t the e x i s t e n c e
by a n o t h e r
maximum
of the
of the
specific
incommensurate heat.
84
Contrary
to t h e o r i g i n a l
(I < 0.5,
T > O)
now there
are many
frustration The
free
described
Figure
only
T = 0 3.39
The
shows
M
data,
configurations with
each other,
one
over
such domain
.
1 .
.
.
.
these
analytically
largest
eigenvalue
and thus
determines
visible <2>
to the
phase.
They
in t h e
occur,
start
M
which
a minimal
distance
but
have
don't
from
phase
at
c a n be r = 2 .
to be s t r a i g h t
.
234 .
.
.
.
.
.
.
.
wall
56
.
.
.
.
.
configuration.
.
7890 .
in Fig. phase
'dislocation
and
leads
to a f r e e
is c o n n e c t e d
3.40,
q
q
increases For
wall
fermion
to t h e a v e r a g e
the w a v e v e c t o r
boundary.
free'
FFA
(ref. 74).
configurations problem
distance
where between
can the walls
. continuously
the t r a n s i t i o n s
from the
Villain
F
to
a n d B a k 74 ob-
tain -2K F
:
K I + 2K 2
=
e
o
(3.84a)
and -2K <2>
:
at
a n d B a k 74 is v e r y
phase.
that
down
F and M
A typical domain wall configuration included in
summation
the
of the
walls
to r e a c h
the
of V i l l a i n
MC
spin
phase
occuring
.
by d o n e
As
P
all a r e
between
(FFA)
by the
do n o t t o u c h
.
Fig. 3.39
for the
the b e h a v i o r
such
phase
T = O)
b y a set of d o m a i n
the w a l l s
as for
in e q u i l i b r i u m ,
approximation
supported
low t e m p e r a t u r e
point
, M and P
(i = 0.5,
to u n d e r s t a n d
the a s s u m p t i o n
a s s u m p t i o n 12 of a L i f s h i t z F
indications
point
fermion
helpful
Thus
where
K I + 2K 2
=
- 2 e
o
;
(3.84b)
85
i 112i
I -exp (-2~ Jo)
I
0
X
2exp (-2~3 J0)
Fig. 3.40
Wavevector q of the modulated phase as a function of x = - (JI+2J2)/T (ref. 74).
the f i r s t
one
corresponds
second
one differs
(K ° >>
I)
The
FFA
rection
result
n
-~
r -~ cos
I ~
factor
for
t w o on t h e
K ° >> right
I , whereas hand
site
the
from the
(3.83).
for t h e p a i r
depending
=
(2.82)
of Eq.
of the c o m p e t i n g
G(r) with
b y the
expansion
to Eq.
correlation function 74 is :
G(r)
in the di-
interactions
~ q x
(3.85a)
continuously
on t e m p e r a t u r e
and
~ :
(l-q) 2
(3.85b)
Within
the f r a m e w o r k
walls,
the phase
o f the
boundary
FFA
between
which
assumes
M and P
nontouching
phase
cannot
domain
be
investi-
gated. The
inclusion
fects
where
in t h e also
2d
XY
found
G(r)
of a l o w c o n c e n t r a t i o n
walls
touch)
ferromagnet,
a modulated
=
where
n(T)
phases
in t h e
corresponds
of d i s l o c a t i o n s to t a k i n g
for w h i c h
care
Kosterlitz
(that is of deof t h e v e r t i c e s
a n d T h o u l e s s 77
have
phase with
r -n
(3.86)
is t e m p e r a t u r e two models
dependent.
together
with
The
equivalence
the k n o w n
Tc
of the
M
of the X Y - m o d e l
86
yields
the t r a n s i t i o n
A NNNI- m o d e 1 7 4 .
temperature
between
the
M and P
phases
of the
For ]
q
the
<
M
I
phase
down to This
(3.87)
V~ cannot be stable.
T = O
Therefore,
the
P
phase m u s t
in a n a r r o w region b e t w e e n
the
F and M
reach
phases.
result is c o n s i s t e n t w i t h the d i s o r d er line TD found by Peschel 49 in the 'Hamiltonian limit' (Jo ~ ~' J1 ~ O, J1/J2 = c o n s t . )
and Emery w hich
extends
down
exponentially. in the Using
P
ANNNI-system
matrix
and approaches
G(r)
close
decayes
approach
results
f uncti o n
the finite w i d t h
the finite w i d t h be estimated.
the d e p e n d e n c e
The c r i t i c a l
This
the open question,
c onver g e n c e
deviate
of the finite
extends The
2d
down
to
ANNNI-model
the competing with
In the other isted,
systems
studied
the
on
because
there
for
wavevector
can only
from their
is due to slower
of the p r o x i m i t y
Ising
phase
Tc
of indeed
transition. system w h e r e
modulated
and short range order
up to now b e l o w
of
IJ21/J I ~ 0.3
is a m o d i f i e d simple
because
J2/J1
obtained
in case the p a r a m a g n e t i c
is a r e l a t i v e l y
discussed
supporting
although
the d e v i a t i o n
size analysis
result
always
phase
(SRO) LRO
ex-
at least on a sublattice.
In the chapter differs
varying
is
latter
limit,
i n t e r a c t i o n s lead to an i n c o m m e n s u r a t e
continuously
line TD
they can see the tran-
behavior
B/~ and ~
whether
, or w h e t h e r
thus
is
. They d e t e r m i n e
. The
from the Ising values
transition
T = O
that
In the c o r r e l a t i o n
of the w a v e v e c t o r
exponents
analysis
the p a r a - t o - m o d u l a t e d
N and T
of the strips
to o s c i l l a t i n g
finite-size leaves
of
from this approximation.
from f e r r o m a g n e t i c
transition
(N x ~; N < 13)
in the H a m i l t o n i a n
derived
across
F
and K r o e m e r 73b have
strips
the c o n c l u s i o n s
sition
Pesch
heat as functions
to their
the
exponentially,
diagram.
for s e m i i n f i n i t e
and specific
is quite
T = O TD
range of the phase
a transfer
entropy
to
Along
on
markedly
3d
systems
from the
2d
we shall case near
see that
the
3d
the f r u s t r a t i o n
ANNNI-model point.
87
3.4.2 This
Brick model
nately
can be d e r i v e d
every
competing maining Then
Model the
2d
Jo
(J~ ~ O)
and d o u b l i n g
nnn-interactions
transfer
matrix
ANNNI-model
by o m i t t i n g
perpendicular
(J"o ~ 2 Jo ) , see T a b l e
crossing
with
from
interaction
interactions
ones
no
solved
second
the
alter-
direction
strength
of
of the re-
3.2h.
are
methods
to the
left
and
the
as in S e c t i o n
system
3.3
can be
(Bidaux
and de
Seze78). Although appear GS
the d i f f e r e n c e
quite
small,
degeneracy.
the v a l u e
Thus
of the
So(~ = 0 . 5 )
~ > 0.5
interactions.
Mean
field
the
for
phase, w i t h tanh
GS
wrong
Tc
sinh
diagram
is s h o w n
in Fig. T = O
for
~ > 0.5
lations
of
spin
flips
demonstrates when
per
2d
(4~cJ 2)
model
larger
and half
of the
ANNNI-model the e x a c t from
there
a somewhat direction
again
of
precisely even
site)
the
an
compe-
would
lead
solution
shows
F
P
to a
I
exactly
generalized are
one
either
Id GS
of
long
exhibited
brick J1
J2/J1 at
for
MC
incommensurate
where
J3 > J1
whereas
modulated
.
'
for
the e x a c t
time
Tc % O ; r ~ 2
model
or
degeneracy
transition,
a very
values
is no t r a n s i t i o n
vanishes
to i n t e r p r e t e
and m e t a s t a b l e
(3.89)
Jo = J1 )" F r o m
after still
=
for d i f f e r e n t
direction
the n e c e s s i t y
stable
is f i n i t e
in the d i r e c t i o n
whereas
I > 0.5 Jo
systems
be m e t a s t a b l e .
carefully
only
(also he takes
can o n l y This
For in
possible
may
(3.88)
is a t r a n s i t i o n
(2~cJ I) exp
he o b t a i n s
finite
entropy
model
a by far
l a t t i c e 37
diagram,
in a x i a l
> 0.5
GS
has
given by 78 l
the d i s a p p e a r a n c e
= - J2/J1
and t h e b r i c k ~ > 0.5
as for the
there
considered
the n n - i n t e r a c t i o n s causing
here
phase
3.41.
M o r g e n s t e r n 79 has
for
--~ 0 . 1 6 1 5 3
of the b r i c k
G(r)
ANNNI
the
is o r d e r e d
Bc 1
=
and
for
i = 0.5
triangular
~ < 0.5
(2BcJ O)
The p h a s e
model
approximation
to a c o m p l e t e l y only
the
second
~I s Ao F - A
=
ting
between
for
AF
For
that
the
solution MC
(about phases
calculations modulated
simu120.OOO which
very phases
88 KST/JI-r
~:;:::....5.-'/../2 .........1
......': .........-"
..............
.-i""
".............. ilI/,4-. .......... /"•............ •......... ............. ............... °l"T1 .. ,2 i
,
-5
Fig. 3.41
may
LRO
I
l
0
i
I
1
I
!
2 J21J~
3
Phase diagram of the brick model for different values of function of - ~ = J2/J 1 (ref. 78).
occur.
between
. ....
Thus
phases
in f i n i t e where
; periodical
systems
G(r)
boundary
it m a y
decays
with
conditions
be d i f f i c u l t a power
can
Jo/Jl
to d i s t i n g u i s h
law and o t h e r s
stabilize
as a
metastable
with states.
3.4.3 Field In S e c t i o n (an-)
3.2 we h a v e
isotropic
cal i n t e r e s t , stems,
In this
e.g.
noble
interactions GS
we w a n t
J2
lattice gases
are h a r d l y
further
understanding
as the
there
the J1
on the
section
interactions for
but
because
influence
discussed
nn-interactions
triangular
any
corresponding
even
if t h e y
lattice
is of g r e a t
experimental
are w e a k
with
theoreti-
have
sys-
a strong
degeneracy.
to c o n s i d e r
and m a g n e t i c magnetic
gas m o d e l
frustrated . This model
for
on g r a p h i t e 81.
the m o d e l
field
quasi-2d the
H
systems
study
with
. This
like
of t h i n
additional
is not
only
E r G a 2 80,
adsorption
nnn-
interesting but
layers
also of
89
3.4.3a
Additional nnn-Interactions J2 ...............................
The
GS
been
investigated
of
K a n a m o r i 84 Fig.
the
triangular
who have
3.42.
The
lattice
b y M e t c a l f 82,
GS
found
four
phase
and
I x 4 ferro
give
Whereas
four phases
the
and
J~
interactions
(Fig.
(b)
LRO 3.43)
phases shows
has
Kaburagi
shown
the
GS
and
in phase
(o)
The four LRO groundstates of the triangular lattice with nn- and nnn-interactions: (a) ferro, (b) ~ × ~ , (e) i × 2 and (d) 1 x 4 (ref. 85).
boundaries. the
J1
and Uryu83Zand
different
diagram
(a)
Fig. 3.42
with Tanaka
phases), and
a finite
the
(2-,
~ GS
GS
6-,
along
x ~
degeneracy
6- a n d
12-fold
the b o u n d a r i e s phases)
entropy
per
is f i n i t e for
the
(apart
the degeneracy
site
So > O
in the ferro,
interior ~
x ~,
of I x 2
form the one between is
large
enough
to
.
Jz
-JT
¢x2
Fig. 3.43
I~ ""4,44
GS phase diagram of the triangular lattice with nn- and nnn-interaction~ Jl and J2 •
00
An overview ferro
of the t r a n s i t i o n s
and the
perature
~
series
ceptibilities, havior
(e.g.
ferro phase y = 1.75 y ~ 2.4 higher
× V~
expansions where
X ~
than
cases
((T-Tc)/Tc)-Y)
, for t h e at
in t h e
Fig.
a value
he h a s
The
close
Ising
or
T
order
assumed
parameter
y = J2/J1
temsus-
critical
Whereas
be-
for the
nn-value
= - I
. This value
for t h e
from high
normal
to the e x a c t
(q = 3)
lines
C
determined
in the a n a l y s i s . y
V~ × ~ phase with c K I = 0.505 ± 0.005 nn
3.44.
O i t m a a 85 h a s
of the c o r r e s p o n d i n g
in both
he obtains
± 0.002
shows
phases
of
he g e t s y
is m u c h
Potts model.
QS
XY
o~
0 1 ~
,
,
,
I
-422
-
-0.1,
o
- .2
/oo,'a -o.1
-o.i
/
/
-oL;
B
[
-o.8
i
-~o
Fig. 3.44
Phase diagram of the triangular lattice with nn- and nnn-interactions.
This phase because sality
transition
Domany class
e t al.
of the X Y - m o d e l
an i n t e r m e d i a t e netic
phases.
with
MC
for J1 < O and J2 > O is of s p e c i a l i n t e r e s t 86 h a v e p r e d i c t e d t h a t it b e l o n g s to t h e u n i v e r -
phase
is e x p e c t e d
L a n d a u 88 h a s
calculations
with
done
without
6 th o r d e r between
extended and with
anisotropy.
the o r d e r e d finite
size
a magnetic
For
this m o d e l
a n d the p a r a m a g scaling
field
H
analysis ; here we
91
first
discuss
the
In
a large
range
of
size
x L
L
temperature T = 3.6 within
J2 the
H = 0 of
1
with
(Fig.
(-I/16 L =
3.45);
>
30
I > -8)
exhibits
the
, independent three
results.
lower
of
sublattices
the two
one
is
maxima
J1
At
interaction
• I-ve * I-V16
heat
as
located
with
I 0.5 |
this
specific
systems
a function
very
close
temperature J2
of
SRO
of
to occurs
"
(a)
kT
al (b)
o-1
C/R
k.. T_T
,12 Fig.
3.45a,b
Temperature dependence = J2/Jl . (a) :
I >-i/4
; (b) :
The abscissa in
Whereas of
Wada
and
divergencies
size L ~ ~
analysis , the
determined Landau
(for exponent
from
further
sublattice
the
when
the finds
I :-I)
position that
approaching
this
must of
the
this
system,
that
thus
.
have different scale
interprete
infinite
s
i <-i/4
(a) and (b)
I s h i k a w a 89 in
of the specific heat in a large range of
the
be
maximum
maximum
transition
as
has
and
(Fig.
an
shown
remains
negative
correlation
the
maximum
Landau
(ref. 88).
indication by
finite
finite Tc
for
cannot
be
3.46).
length
~
within
from
above
and
the below
single
92
,o~--I.o2~ /~/(Tt.H,)
IOC
| ~*0,44
~ ~.~__H=Z4 3
3.(3
I,O o
o
Q H'Oo
o
L.
Fig. 3.46
Size dependence of the maximum of the specific heat for
diverges
exponentially
T 2 = 4.26
J1
at d i f f e r e n t
Because law
~O exp
T-TiI~-I/2 ]
of this
size
exponential
scaling
susceptibility very
a \ --~I
88
and the
well when
analysis
J1
and
TI
the w h o l e
range
dependent
on
T
must
inverse
like
order T2
correlation
to a n a l y s e parameter.
are
agreement
shows
~ r -~ with
the o r d e r
for power
, with the
to the u s u a l a modified
Both
inserted
function
T 2 ~ T ~ T I : G(r) , in g o o d
(in c o n t r a s t
in the X Y - m o d e l
be u s e d
respective
of the p a i r
(3.90)
divergence
~ = ~o " ( ( T - T c ) / T c ) - v )
of f i n i t e
T I = 4.89
(ref. 88).
:
{ =
temperatures
I =-i
~
parameter
quantities Tc
power
version
scale
. Finally
law b e h a v i o r being
theoretical
in
continuously
predictions
for the X Y ~ m o d e 1 7 7 ' 8 7 :
~Mc(TI)
=
0.27
± 0.02
,
ath(T1)
=
I/4 (3.91)
nMc(T2) This
=
confirms
triangular
O.15 the
lattice
± 0.02
interesting with
H = O
,
nth(T 2)
phase and
=
diagram I < O
I/9
expected :
for the
frustrated
93
T < T2 :
LRO
phase
with
lim r~
G(r)
=
T2 < T < T I :
XY
finite
sublattice
c o n s t . (T)
model-like
exponents
phase
are
Landau T2
Paramagnetic
has
on
found
l
addition
to t h e
transition
to b e d r a w n ;
The phase
line
it is s h o w n of the
u p to n o w o n l y b y a m e a n
exact
result
authors
used
of t h e
domain
considered
tabulated
the
The phase matic,
boundary
of p h a s e s
Figure
3.47
a
shows
in a d d i t i o n
3 x 3
H # O
for
the
appears.
the
T I and
intermediate
XY-phase
in Fig.
3.44
to the d i v e r g e n c e line
corresponding
by the dotted
in Fig.
in
of t h e to
T2
line.
3.44 h a s b e e n
c a l c u l a t i o n 90 n o t
and by Slotte
investi-
leading
to t h e
a n d H e m m e r 91.
of are
the
These
I x 4
transition
3.44
phase
only
is d r a w n
of the p h a s e
do not yet
Field
in t w o o r
from
their
transition
and val-
exist. in Fig.
3.44
is p u r e l y
sche-
H
magnetic
frustrated GS
to the
respective
of Fig.
available.
of t h e the
are nnn-interactions
line
the order
exponents
on a h o m o g e n e o u s
number
where
Results
Add!~!2na!_Ma~netic
Switching
phase
field
there
transition
critical
no results
3.4.3b
transition
J2 = 0
cases where The
data.
of t h e
but
of
the M O l l e r - H a r t m a n n / Z i t t a r t z m e t h o d for the c a l c u l a t i o n 76 wall energy . Apart from the isotropic case they also
one direction.
ues
for
~ r -q(T)
I . Therefore,
corresponding
I × 2
'critical'
in t h i s r a n g e
. The dependence
schematically
gated
Tc = 0
of
, the
(-r/~(T))
I =-I
range
T I , a second
boundary
G(r)
investigated,
in a l a r g e r
at
~ exp
for
LRO
dependent;
occurs;
G(r)
this behavior
occurs
susceptibility has
phase;
has not yet been
certainly
without
temperature
a line of f i x p o i n t s
T > TI :
magnetisation;
phase
field
triangular
diagram
four phases
1 x 3 between
phase the
H
for
increases
the
lattice.
of K a b u r a g i H = O
occurs. ferro
further
also
and Kanamori a
On t h e o t h e r
and paramagnetic
2 x 2 hand
92
and
for
phases
dis-
94
lJ~h
oxol-I
i
Fig.
3.47
GS phase diagram of the triangular lattice with nn- and nnn-interactions Jl and J2 in a magnetic field H (ref. 92).
Consider with For
the
spins this
sition For
by
Monte
phase
Alexander to
(J2 = O)
3.48a,b.
For
I < 0
already order
been up
order.
at
Figure
value y =
is 1.42
values:
phase
3.46
H = 2.43
> O)
the
with
an
= 0.40).
, v = 0.87 ~ =
I/9
, T =
Finite
size
n = 0.27 ~
1.444
predicted
the
q = 3
finite, phase
by
the
< H
are of
~ 0.44
the (the
for
specific 'best' yields:
, very
to
close
and
M~I-
shown
in
the
~ 0.867
I
is
second
it
is
i = by
iJ11
X = -
< Hc
scaling 40-42
, ~ = 5/6
94 are
case
marked
tranmodel.
determined
transition
Ht
L a n d a u 88
points
= 6
curves
and
the
Potts
< H < Hcl
(H)
exist
antiparallel.
c renormalization
, for
divergence s/~
13/9
the
H = Ht
exponent
and
is
GS
one
have
0 T
three
on
qualitatively
H > O
determined
of
for
. The
Tc
tricritical
and
al. 86
field
agree
only
H
class
c real space
H = 0
the
. Then to
et
T
88,
For
demonstrates
s/v
finite
point
diagram
where
J1
magnetic
and
discussed.
3.49
Domany
m e t h o d 95
a tricritical
The
in F i g .
and
calculations
(J2
H > O
parallel
universality
the
ler-Hartmann/Zittartz Fig.
93
the
a transition Carlo
for
sublattices
belong
~ = 0 to
x ~
two
case
to
leads
~
on
I
has
first is
shown
crosses. heat Potts
at model
B = O.11 'exact' and
,
Potts
~ ~ 0.266
.
85
1.6
[
,
,
~
J
i
1.4 kT/J
1.2 1.0
-- o.a 0.6
0.4
0.2
o;
i
-6
H/IKI
(a)
Fig. 3.48
HIJ
0
(b)
Phase diagram of the AF triangular lattice in a magnetic field. (a) RS-RG and MC results ; (b) MHZ and MC results (refs. 88, 94,95).
Jnn
o:
.: I I I I ! I I J
2
L
I
Fig. 3.49
The
MC
3
0
,
,
,
,
=., J,n
Triangular lattice in a magnetic field for J2/Jl = - i . At IHI = H t tricritical points (crosses) occur where the order of the transition changes from second (below) to first (above) (ref. 88).
results
transition q =
,
I
-2
-~
for
Potts
Landau
also
and
at
the
but
this
we
are 0
< H
thus
consistent
< Ht
with
belongs
to
the
the
prediction
universality
that class
the of
phase the
model.
examined crossover do
not
the
critical
from discuss
XY
to
here.
exponents q
=
3
at Potts
the
tricritical
behavior
for
point small
H ,
96
A good
experimental
gular
lattice
which
Doukour~
scattering.
with
At
finally
for
sitions
, for
H > Hc2
interactions
Hcl
(see Fig.
of the
they
I × 2
monotonous
expected
3.47). and
and J2
the
I × 2
= 20 kOe (Fig.
for
T = O
trian-
ErGa 2
, for
and n e u t r o n
phase 2 x 2
3.50).
This
for phase
and
is e x a c t l y
in the r a n g e (above
the
the m a g n e t i s a t i o n
of
(Fig.
ErG2o/
is
the
temperatures
phases)
saturation
s y s t e m on the
magnetisation
phase
At h i g h e r
2 × 2
until
Ising
J1
find
< H < Hc2
the f e r r o / p a r a
of t r a n s i t i o n s
I/4 < ~ < I
increases
negative
low t e m p e r a t u r e
= 6.8 kOe
sequence
for a f r u s t r a t e d
and G i g n o u x 80 h a v e m e a s u r e d
H < Hcl
the
example
tranErGa 2
3.50).
K'~
N
0
Fig. 3.50
A more
precise no p h a s e
boundary
test
between
the
approximation as has b e e n
in the
chapter
itatively
for this
diagram
3.47 N a k a n i s h i
field tems
8 12 t6 20 2& APPLIED FIELD (kOe)
28
Magnetisation M of ErGa 2 as a function of the magnetic field (ref. 80). For low temperatures two critical magnetic fields occur where M rises abruptly.
H % 0
Fig.
&
been
2 x 2
and
on
is not
which,
3d
3 x 3
however,
where
For
(or
yields
this
because
the v i c i n i t y
I × 3) p h a s e s
modulated
for the A N N N I - c a s e . systems
yet p o s s i b l e ,
determined.
and S h i b a 96 d i s c u s s
proven
correct.
system
has
wrong
phases results
We come
back
approximation
of
for the
in
within for
mean
2d
sys-
to this
paper
should
be q u a l
97
~2K~2£H2~i~
3.4.3c A
2d
Ising
configuration represent tion
of a s u b m o n o l a y e r
causes
ic field becomes age
8
Gas M o d e l
spin c o n f i g u r a t i o n
occupied
J1
Lattice
sites,
of adatoms
si = - I
repulsion
between
the c h e m i c a l
(~ m a g n e t i s a t i o n
can be taken
M).
as a r e p r e s e n t a t i o n
on a surface:
vacant
ones.
two adatoms
potential
The
spins
AF
nn-interac-
on nn-sites,
determining
Such an a d s o r p t i o n
the magnet-
the average
model
of a si = + I
cover-
is called
a lat-
tice gas model. For a m o r e Kr
realistic
on h e x a g o n a l
nnn-interactions
description
graphite
of the a d s o r p t i o n
layers
(Fig.
as an a p p r o x i m a t i o n
3.51)
of noble
one needs
for the better
gases
like
at least also
Lennard-Jones
po-
tentials 97 .
Fig. 3.51 Lattice gas model with nn- and nnn-interact±ons for the description of adsorption of noble gas atoms on hexagonal graphite layers (ref. 97).
It m a y well be p o s s i b l e necessary
refer
d i agram s
lead to v e r y
ranges
the m a p p i n g
preted;
further
reaching
of a d s o r p t i o n
complicated
phase
interactions
~th
first
of the p a r a m e t e r s
on the
one only has
lattice
lattice
to note
gas
found
are
layer m e a s u r e m e n t s ,
diagrams.
Therefore,
to the p a p e r by K a n a m o r i 98 who has d e t e r m i n e d
for the h e x a g o n
and in special havior 99 . After
still
for the i n t e r p r e t a t i o n
these w o u l d here
that
GS
but
we
phase
to third n n - i n t e r a c t i o n s "devil's
the Ising results
that e x p e r i m e n t a l l y
staircase"
be-
can be r e i n t e r -
in a d s o r p t i o n
layers
98
the
coverage
8
(~ M)
independent
variable.
(Fig.
this
3.52)
and
not
For
leads
the
chemical
i = - I
to
large
.°. ~
in
potential
the
8
coexistence
- T
u
(~ H)
phase
is 88
diagram
the
regions.
LL
_]~9_o. • (13
0.5
e
~co~
J- :<
_.
[ ,,.,,,-I'Y,9. =
{.~.
o'o ~
g ~
X 10.80
.o
-
r,
~
k--~T-7 2 5
-ZO.F+C 2kT
,o
2.0
3,o ,.o k.T
5!o
6'.o
Jnn Fig. 3.52
Phase diagram in the coveragetemperature phase for I = - 1 (ref. 88).
The
adsorption
Tt
corresponding
For
T > T 2,
T >>
IJnnl
isotherms
Tt
to
physical alloys,
we
postpone
I
by D a s h 100 systems
where
for
the
respective discussion
to
Chapter
sublattices.
noninteracting
of
the
thin
is
. As 4.
A alloys
in
the
adsorption
still
Ising
species 1
three
for
which of
the
for
isotherm
field
si = -
of
steps
and
the
atoms
exhibit
out
experiments
given
Adsorption isotherms for i = - i . The thick line is Langmuir's isotherm (T >> IJnnl) (ref. 88).
washed
and
is
3.53
covering
become
Langmuir
surfaces
si = +
successive
steps
the
theories
by
in F i g .
approach
on
binary
the
these
A review
Further
shown
Fig. 3.53
in
results and
B
are
limit
rapid
can
adatoms.
layers
can
on
solid
development be
be
usually
T < T 2,
applied
are
represented 3d
systems,
99
~{2_~b!£S_~!~h_~2~£~!~_~z_~_~:!~2~£~2£!2~£z_~2!~:
3.4.4
tion .
In this arise
.
.
.
to V e r t e x .
.
.
.
.
section
.
.
.
.
.
Models .
but
.
.
F irst
Dalton
lattices
with
J1 > O
occur,
model.
on the
8-vertex
is b r i e f l y
discussed,
generacies
of the
additi o n
we refer
who studied actions model,
as well
especially
expansions
the critical
respective
2d
surfaces
points
between
of a
exponents emerge.
16-vertex
6-vertex
are
this has
critical
the
for
the
models GS
de-
model. In of M i y a s h i t a I02 and Fujiki et al. I03
of a d d i t i o n a l
regard
odd
exponents
. However,
field t r i c r i t i c a l
and the
frustrated
with
series
as the c o n n e c t i o n
to two papers
the effect
from
one n o n u n i v e r s a l
(Baxter)
latter m o d e l
in the fully
in V i l l a i n ' s
not.
I = J2/J1
magnetic
does not
nn- and n n n - i n t e r a c t i o n s
for one of the two c r i t i c a l
On the other
and in an a d d i t i o n a l
The m a p p i n g
or
, that
of the ratio only
frustration
as for example
concluded
J2 > 0
where
between
field p r e s e n t
and
independent
general
Ising m o d e l s
and Wood I01
turned out to be true more
.
by c o m p e t i t i o n
in a m a g n e t i c
completely
.
nn-interactions
is caused
J1 and J2 e.g.
.
we consider
from c o m p e t i n g
model,
.
third
nn AF
(ice)
and fourth
triangular
to the o c c u r e n c e
nn-inter-
and V i l l a i n ' s
of an
XY
odd
like phase
transition.
3.4.4a The
S[s~2m_W!~h2~_Ma@netic
GS
phase
diagram
and n n n - i n t e r a c t i o n s corresponding Fig.
of the square
shows
to the
Field
three
I x I , ~2 × ~2
3.54 the t r a n s i t i o n
lines
Because
the order p a r a m e t e r
and M u k a m e l 7 have p r e d i c t e d to the u n i v e r s a l i t y
of the
class
renormalization
space
block
spins g e n e r a t e
fore,
the m o d e l
(J1' J2'
additional
has been
J4 ) . This
space
studied
should
group
formulas
structures IO4
with
transition
SAF) In
cubic
(RS-RG)
anisotropy
m e t h o d s IO4-IO6
interactions
in the e n l a r g e d
to
occur.
for the i n t e r a c t i o n s
four-spin
also c o n t a i n s
nn-
(F, AF,
is t w o - d i m e n s i o n a l ,
of the X Y - m o d e l
When
real
I × 2
phase
exponents
out that the r e c u r s i o n
system with ranges
the c o r r e s p o n d i n g
critical
using
Ising
schematically.
SAF
and thus n o n u n i v e r s a l
it turns
and
are shown
Krinsky belong
lattice
low t e m p e r a t u r e
J4
parameter
the special
case
between There-
space J1 = 0
100 K2
AF
"~
s..__Z
YT
~YT
KI
SA~
SAF Fig. 3.54
which
Kadanoff
8-vertex J2'
Phase diagram of the square lattice with nn- and nnn-interactions (ref. 104).
model
J4 ~ O
a n d W e g n e r I07 h a v e solved
The B a x t e r m o d e l tical
exponents
dicating 3.55
a whole
shows
exactly
is a l s o c a l l e d has
line
the t w o
of
sheets
t o be e q u i v a l e n t
b y B a x t e r I08.
Baxter
a second
depending
proven
the
Ising model
with
nonuniversal
cri-
model.
order
transition
continuously fixpoints of t h e
Thus
to the
like
with
on the r a t i o : in the
critical
2d
surface
~ = J4/J2 XY-model.
in
, inFigure
( K I , K 2 , K 4)
space.
K2 Kq
Fig. 3.55
The two sheets of the critical surface of the square lattice with nnn and 4-spin interactions (ref. 105).
nn ,
101
Figure
3.54
sheets
there
upper J1
corresponds
of
these
fixpoints
one
finds
nonuniversal
on the
the r e a s o n
lower
why
J1
# 0
As
an e x a m p l e
sheet
specific
heat
tubation
theory
3.56
is s h o w n
nn and
therefore,
behavior
thus
On b o t h
case).
only
for
of f i x p o i n t s
sheet,
3.55.
(Baxter
In the
exactly
J4 # 0
is a t t r a c t i v e .
also
for
for
. Contrary This
J4 = 0
is
and
is f o u n d I06
the d e p e n d e n c e
as f u n c t i o n
by B a r b e r I09.
independent
garithmically,
line
lower
in Fig.
KI = O
critical
the
behavior
in Fig.
at
are r e p u l s i v e ,
on the w h o l e
nonuniversal
K4 = 0
fixpoints
sheet
= 0
to this
two
to the p l a n e
is a line
For
Ising
square
s(J1=O)
= 0
of J1
of the e x p o n e n t
J1/rJ21 = O
of the
, as d e t e r m i n e d
the
lattices.
~
system
Then
c
in p e r -
decomposes diverges
into
only
lo-
.
01.
C~ (}3 02 01
o~
Fig. 3.56
For
d:3
d~.
ds
Nonuniversal variation of the exponent i = Jl/IJ21 (ref. 109).
finite
ate
' a'z
magnetic
2 x 2
phase
(Fig.
3.57).
order
in e v e r y
The
field
emerges GS
chain
field)
just
chains
are not
correlated,
chains
are
ordered
F
and
of this
second
as the
H
between
SAF
2 x 2
(with
phase.
spins
of the specific heat with
an a d d i t i o n a l
F, AF and phase
spins
in the
the
Id
degener-
SAF
phases
exhibits
perfect
parallel
However,
whereas
(with
T = O the
e
intermediate
SAF
antiparallel
ferro
to the m a g n e t i c
phase to the
the
AF
ordered
intermediate
field).
102
Fi 9. 3.57
The is
GS
We
=
diagram
to
the of
diagram
now
(J1
In
phase
degeneracy
phase
to
consider , where and
~ >
first
case
shown
field
in F i g .
result
the
the
I/2
critical
(Fig.
3.57)
iota. ~ ,_%'.-,,.
× 2
phase
the
phase.
Hc2 3.58
have
(l < O) =
4
(ref.
from
11Oa,
a correction
of
who
did
not
Brandt 11Ob
has
extended
yet
scale
discuss this
GS
neighbors.
transitions
cases
I/2
apart
of K a n a m o r i
2
third-nearest
< O)
the
,AL
-5_/_~ T
GS phase diagram of the square lattice with nn- and nnn-interactions Jl and J2 in a magnetic field H .
identical
the
,
-~
-'
for
I = J2/J1 to the
IJ11 111).
. . . . . . . . .
be
AF
< O
nn-interactions
, ~ = O
, 0
< ~ < I/2
,
distinguished. GS
changes
from
AF
. The
phase
diagram
for
For
small
H/J I
there
to
at
the
I = - I
is
is
F
an
Ising
Tncr~ti ca! I~nt ~ramagn~tt
IOrderecl lanfifmromgnetl i I/ r
Fig. 3.58
Phase diagram of the square lattice with (ref. iii).
I = - i
in a magnetic
field
103
transition
with
normal
field
with
modified
Ht
first
order
(dashed
been
treated
For
I > O
with
on t h e
lieved with
interesting
(that is
of M ~ l l e r - H a r t m a n n
calculation
also
of d o m a i n
We mentioned lattice
series
Above
the
the
tricritical
transition
tricritical
becomes
behavior
has
of a u t h o r s 112
I = O
to b e exact.
3.59)
The
line).
exponents
a n d L a n d a u 113 h a v e
the triangular
(Fig. are
line).
a n d for
the r e s u l t s
based
(full
critical
by a number Binder
culations,
exponents
carried J2 = O)
out
and
Zittartz
wall
energies
this method
MC
a n d at f i r s t w e r e
be-
in c o n n e c t i o n
In the p h a s e
r e s u l t s 114 a n d
cal-
excellent agreement 76 (MHZ) which are
already
and the ANNNI-model.
expansion
extensive
found
RS-RG
diagram
r e s u l t s 115
included.
10
20
kBT/ IJnn~
In t h e m e a n t i m e determined square
z
=
Contrary Fig.
=
116
the
e
by high
critical
order
activity
series
expansion
zc
of t h e h a r d
± O.0001
have
the
MHZ
(3.92)
result
= d(H/J1)/d(T/J I)
a value
MHZ
zc
precisely
to t h i s m
et al.
gas:
3.7962
3.59,
yields
Baxter
very
lattice
c
square lattice in a magnetic field. Circles method; dashed line: series expansion;
Phase diagram of the nn AF MC results; full line: MHZ points: RS-RG (ref. 113).
Fig. 3.59
for -2m
the
=
critical
4
for t h e
slope
at t h e p o i n t
of the
(H = 4
curve
in
IJ1 I, T = O)
activity
(3.93)
104
Therefore,
the
analytical
approximation.
In
the
range
at
two
critical
MHZ
0
method
< I < I/2 fields
p h a s e as c a n b e s e e n d a u 113 f o r i = I/4 and
H e m m e r 117
show in
that
ly w r o n g
T > 0
the
SAF
T = O).
and
Fig.
(Fig.
be
it
increasing
field
H
Hc4
AF
from
3.57.
3.60a)
exact,
MC
and
agree
quite
line
Hc3
field
approximation
the
results
the
3.61)
MHZ
well
P
to
good
the
GS
changes
to
F = P
of
Binder
the
of AF
reaches
which
a very
2 x 2
results
for
phase
is
and
phase
down
yields
Lan-
Doczi-Reger and
to
T = 0
a topological-
diagram.
there
is
phases
no
as
Therefore,
sharp
both
at
order-disorder
in F i g s .
but
with
Hc3
from
the
to m e a n
phase
For
an
(Fig.
along
contrast
cannot
transition
have
T = 0
the
same
between
transition
occurs
between
the
2 x 2
symmetry
(as
opposed
Hc3 shown
respective as
thick
line
and to
Hc5
and
at
T = O
Hc4
3.6Oa,b,c.
Ca) H
ciegenerote tructure
I = 1/4
(b) i.~..i~
5.(]
X = 1/2
dege~ote
UNN'
structure
H
C
2.5
5.0 t ~ ' ~ .
OC
(=)
H
I
2
degenerate
t
IJ.NI structure 10.0 ~ /
00 .
kBTI,JA,
=
0
I =
I/2
paramagnetic (T = O) sition
and to
the
,
~
(Fig.
SAF P
kBTl,J ~,
3.60b)
to
.
.
.
1
ksTIIJ~l
Phase diagram of the square lattice with nn- and nnn-interactions in a magnetic field for (a) : I = I/4 ; (b) : I = i/2 ; (c) : I = i . All phase transitions are first order (ref. 113).
r
down
.
05
Fig. 3.60
5.0 2.5
For
.
P
1
7.5
00
.
"
and
T = O
phase phase.
H
< Hc3
, between
(T > O)
= Hc5 Hc3
occurs,
and with
the
system
Hc4
the
a second
remains 2 x 2 order
phase tran-
105
70 O8 06 OL O2 0
Fig. 3.61
For
I > I/2 SAF
second
order
(Fig.
agreement
size
with
In the
limits the
H = 0
and
•
the
and
in l o w f i e l d s
H > O
one contiguous
and Landau
transition,
for
have
whereas
find nonuniversal
~ = i/4 .
the
system
transition ~ < I/2
found
.
normal
for the
critical
is
line of
2d
SAF-P
exponents
in
results.
and
(for f i x e d
l)
H ~ ~
the exponents
ap-
values.
to V e r t e x
to n o t e
field
tensively
I Z H'
only
Binder
they
I/~ ~ 0
Ising
Here we want
field
I I
to t h e b e h a v i o r
AF-P
H @ O
Connection
without
contrary
analysis
the
already
T > O
at the
for
proach
3.4.4c
for
occurs,
exponents
transition
3.6Oc)
phase;
finite
Ising
I "1
The AF phase transition in MHZ approximation (line) for The points are the MC results from Fig. 3.60 (ref. 117).
in t h e
From
! "2
the
the
close
8-vertex
16-vertex
and w h i c h
Models
model,
connection model
and
which
Lieb
we mentioned
between
the
that between
Ising the
a n d W u 118 h a v e
already
when
of t h e
Ising
system
system with described
discussing
ex-
the Baxter
model. Lieb tions
and Wu J1
show the
a n d J2
equivalence
' the n n n - i n t e r a c t i o n s
system with
J a n d J'
, the
nn-interac-
four-spin
106
interaction the
first
J
and
eight
responding
a constant
vertices
to the v e r t i c e s
interactions
J
of Fig. are
O
, with
the
3.62 m a y
linear
8-vertex
occur.
reversible
The
model
in w h i c h
energies
e. c o r l of the I s i n g
functions
J. 3
+ + + + + + + + .....- -
{91
00)
(ll)
.....
(ia)
(1~)
~ ....... F
(:4)
05)
06)
+ + + + + + + + + u .....~...... , a, ,,+ + + Fig. 3.62
Lieb
The sixteen vertex configurations of the general ferroelectric model on the square lattice and the corresponding bond configurations using vertex (i) as basis (ref. 118).
and Wu also
interactions special Fig.
of the
16-vertex
occur.
These
KDP
six v e r t i c e s corresponds
of Fig.
3.62
lattice
of
T = O
fully
in T a b l e
square,
where
cases
all
3.2f,
where
nnn-
to be e q u i v a l e n t
to t w o
sixteen
of
are called
Ising
which
determined
for
T = O
requires
vertices
the g e n e r a l i z e d
system This
. The
system
two
ice rule. 120 exactly :
GS
because
3.2f
four
F
away
all pair
spins
first cases
six v e r t i c e s
arrows
at e a c h
from
it 121
interactions
c a n be i n t e r p r e t e d tetrahedra
the
in t h e s e these
t w o to p o i n t
cornersharing
tetrahedron to the
of T a b l e
such that only
t w o of the
it a n d the o t h e r
frustrated
in e a c h
c a n be c h o s e n
ice m o d e l 1 1 9 ' 1 2 0 ,
strength.
just corresponding Lieb has
occur
square
towards
In t h e e q u i v a l e n t a n d of e q u a l
model
special
the p a r a m e t e r s
to the
to p o i n t
shown
other
model.
o b e y the ' i c e - r u l e ' ,
vertex
For
Ising model
in e v e r y
cases
For both models
AF
only
3.62 m a y
respective
must
show the
occur
as a
(see Fig.
are
2d 3.63).
are u p a n d t w o a r e down,
The entropy
of t h e
square
ice m o d e l
107
Fig. 3.63
whereas single
3
4 in
the
~4
~
simple
tetrahedra
NP g
I I
I
I
The square lattice (bottom) is equivalent to the sharing tetrahedra (top).
_
So
i I
O.2158
Pauling
2N {3~ N/2 <~]
=
2I In ~3
lattice of corner-
,
(3.94)
a p p r o x i m a t i o n 39 from the
(w~ et = 3/8)
=
2d
yields
the total
GS
degeneracy
of
degeneracy
,
(3.95)
0.2027
(3.96)
and thus SP o
We shall tropic
find the P a u l i n g 3d
pyrochlore
cornersharing In g e n e r a l generacy
the
inverse energy.
lattice
frustration GS
shows
of several
IF
to be even better
in Section
4.3, w h i c h
for the iso-
also consists
of
in the
F
respective
The r e d u c t i o n
into the r e d u c t i o n
of an Ising m o d e l
leading
to m u l t i p l e
with
v e r t e x m o d e l as the min equal m i n i m a l energy e. l
all six v e r t i c e s
have
respective
KDP
model
IKDP
only
four v e r t i c e s
model
of the d e g e n e r a c y
of pair
de-
up in the e q u i v a l e n t
vertices
in the ice m o d e l
frustration),
approximation
tetrahedra.
of the
existence Whereas
~
interactions
equal only
of the v e r t e x w i t h equal
energy two,
(maximal
and in the
have
lowest
energies
strength
maps
in the
108
corresponding
Ising model.
interactions spective GS
in t h e
When
lattice
the h o r i z o n t a l
of Fig.
J
and the d i a g o n a l Y d i a g r a m (Fig. 3.64),
phase
parating
doublelines
p h a s e s 118.
Along
paramagnetic
nnn
T = O
-3x/
ones
J
after
the
the
; for
LRO
vertical
called
models
Id L R O
nn-
re-
x Ising
and
vertex
vertex
only
J
the
phases
corresponding
(inverse)
T ~ O
-
are
, on o b t a i n s
the t h r e e
are named
to
respective
(bottom)
where
the doublelines
down
3.63
se-
model remain
emerges.
f F
GS phase diagram of the Ising square lattice with crossing nnn-interactions in every other square (Tab. 3.2f) for Jx + Jy + J = const. .
Fig. 3.64
We
cannot
vertex
models
sitions these
discuss
phase
the
KDP
function
as w e l l
obey
the
do no
systems
as
transitions
in the
F
obey
r -q
multi-spin
leading
frustration
points.
Here we mention
H
, which
in g e n e r a l MHZ
was
has been
method.
for the
solved
We
fcc-lattice
interactions
studied
for
for
only
only
and phase
tran-
add
the pair
surface
the
of
heat), that
in
correla-
models
which
of 8 - v e r t e x
Tc
also
mo-
can
show
at s p e c i f i c
triangular
J2 a n d J3
lattice
in a m a g n e t i c
with
field
J2 = H = O
an e x a m p l e
in the n e x t
of
specific
6-vertex
interactions
by Doczi-Reger
discuss
We
T > Tc
. Both
to the v a n i s h i n g
exactly
shall
a n d Wu.
properties
ice r u l e 118
effects
and three-spin
order
diverging
on the c r i t i c a l the
frustration
nn-pair
of i n f i n i t e
by Lieb model
like
lie e x a c t l y
general
thermodynamic
exponentially
decays
longer
with
interesting
discussed
G(r)
ice r u l e
dels which
the
order with
extensively
tion
Ising
(e.g.
of f i r s t
are
further
chapter.
b y B a x t e r a n d W u 122 a n d 123 and Hemmer u s i n g the
with
four-spin
interactions
109
4.
Three-Dimensional
In the
last chapter
frustrated three
Ising
Ising
hcp-lattice
on the
fcc
lattice,
a l m o s t no exact
three
, the simple
which
4.4 the
3d
from the
4.1
ANNNI-model 2d
of results
cubic
are known.
First
(sc)
with
for
2d
to this in
An e x c e p t i o n is the on the fcc-
in Section
systems
transitions follows,
Contrary
interactions
4.5.
frustrated
show phase
markedly
results
four-spin
in Section
fully
a multitude
m a n y of them are exact.
system with
discussed
4.3 we treat
Ising Systems
we have d e s c r i b e d
systems,
dimensions
self-dual
Frustrated
4.1
nn-pair
and
to Section
interactions
and the p y r o c h l o r e
(=B
of d i f f e r e n t
In Section
which
order.
has p r o p e r t i e s
spinell)
differing
case.
f_cc A n t i f e r r o m a g n e t
For this
system with
the
to be only
GS
can be stacked
AF 2d
nn-interactions ordered:
arbitrarily
J1
perfectly
on each
other.
D a n i e l i a n 124 has shown AF
ordered
This yields
100
a
GS
planes degener-
acy Ng
~
of course and high
2 (NI/3) the
GS
temperature
for
series
GS
The i n t e r e s t i n g ly has been
Different
question
diverging
series,
for
number
vanishes
(~ N-a/3)
Danielian
gave a rough e s t i m a t e
rJ1r
. Betts
. From
averaged
(1.83±O.O2)
over
The spin GS
how to do the low t e m p e r a t u r e e x p a n s i o n correctauthors 126-128.
for the e x i s t e n c e
N ~ ~
must
differ
They have
formulat
of such an expansion:
from each other by a
of spins with d i f f e r e n t
c o n f i g u r a t i o n s of low e n e r g y
m a y differ
a small
from the c o n f i g u r a t i o n
(finite)
all
fJ1 r
orientation. (4.2)
(b)
low
and E l l i o t t 125
but u n c o r r e c t l y
Tc ~
by several
conditions GS
T c ~ 1.2
and o b t a i n e d
investigated
ed the f o l l o w i n g
N ~ ~
expansions
temperature:
the low t e m p e r a t u r e
(nonequivalent)
(a)
(4.1)
entropy
of the t r a n s i t i o n extended
,
number
excitations of this
GS
'near'
one
for only
of sites. (4.3)
110
When
these
fa(T)
conditions
starting
the e x c i t a t i o n Mackenzie that
for
phases All
fulfilled
GS
small
J2
127
(a)
have
only
with maximum phases
can o n l y
are a
two
(with
Id
that
these metastable appear
equally
several whereas
2d for
of the
free
sality
class
This
phase
the
free
depends
energy
on
s
via
and
showed
2d
transition
sc-sublattices
have
spin
disorder.
between
phase
model
the o r d e r e d
up and
the o t h e r
but
n
two
so
cubic
spin
systems
d = 3
and
in the u n i v e r -
anisotropy.
where spins
tem-
the L a n d a u
. For
transition
phase
finite
and
frustrated
with
and
are
at low t e m p e r a -
properties
dimension
2d
Heisenberg
spins
at low,
Id GS
energy
phases.
occurs
the
an e f f e c t i v e l y
of the
simulations
for g e n e r a l i z e d and
free
differences
equilibrium
is
degenerate
thermodynamically.
a higher energy
MC
LRO
there
fcc-lattices
expect
the
J2
twelfefold
stable
have
in
studied
energy
are
as the
systems
expansion
on d - d i m e n s i o n a l
phases
T = O
and P i n c u s 8 h a v e
they
fa(T)
respective LRO)
However,
stable
Alexander
n = I
(and
disorder)
will as for
can d e t e r m i n e
a nnn-interaction
sixfold
be m e t a s t a b l e .
tures
peratures,
included
symmetry
small
Just
one
; in g e n e r a l
spectrum.
and Y o u n g
other
thus
from
two of the down,
four
and the
P
p h a s e has b e e n i n v e s t i g a t e d in a series of MC p a p e r s by P h a n i et 129 130-132 al. and B i n d e r (et al.) , w h o also i n c l u d e d a n n n - i n t e r a c t i o n J2
and
For
a magnetic
H = J2 = 0
T/IJ1l
= 1.75
critical At
these
remains
For Hcl
a first
. With
fields
Hcl
critical
from
~
< H < Hc2
transitions del 133 "
the
the
=
transition
field and
GS
to
H
Hc2
the = 12
entropy
T = 0
occurs GS
at
changes
IJ1i
is f i n i t e 132 and
the
system
(4 4)
transition
is t h r e e f o l d
Corresponding
be as in the
4.1).
:
the p h a s e phase
at two
(see Fig.
~I in 2
ordered
fourfold.
should
phase
IJiL
two p o i n t s
0 < H < Hcl
.
order
down
So(Hc2)
these
H
increasing = 4
fields
paramagnetic
So(Hcl) Apart
field
q = 3
to t h e s e
is a l w a y s
first
degenerate, degeneracies,
respective
q = 4
order.
for the p h a s e
Potts
mo-
111
H¢2113.1 10 HIl.l"..I 5 1 Fig. 4.1
Domany
MC phase diagram of the (ref. 130).
e t al.
133
a nnn-interaction come
together
this
behavior
tions, Fig.
had
2 keT/Lf..I i
AF
predicted
J2
(> O)
fcc-model in a magnetic field
a new kind
is added.
at a m u l t i c r i t i c a l p o i n t 132 Binder has f o u n d for
although
without
determination
H
of c r i t i c a l
behavior
two phase
transition
The
(Tm > O,
H = O)
J2 = - J1 of the
in
critical
MC
when line~
Exactly calcula-
exponents
(see
4.2).
H¢2 10 H
R=-I ,J
I]...,I
"c_,_S
I]n.I
ill/
,
,
The
fcc
describe are
MC phase diagram of the J2 (= - Jl ) (ref. 132).
Ising AxBI_ x
contained
system with alloys.
in the
MC
,
]
_
& kmTiiJn. I 6
2
Fig. 4.2
,
AF
J1
II
fcc-model with additional nnn-interaction~
< O
and
J2
> O
Correspondig
results
papers
here.
cited
and
has
been
further
used
to
reference~
112
4.2
Fully
and P a r t i a l l y F r u s t r a t e d
The p r o p e r t i e s (sc) As
Ising
square
of the
system
along
are
of
frustration
of e i t h e r
ented
F and AF
contains
for
to look
4.3
respective
frustrated
such
simple
for s i m p l e
or
cubic
as in the f c c - c a s e .
an odd n u m b e r
on the
all p l a q u e t t e s ,
In Fig.
four
only
one has
Lattice
established
nn-interactions
(a)
in the X Y - p l a n e . cell
as w e l l
frustrated
the edges,
figurations
Cubic
and the p a r t i a l l y
are not y e t
plaquettes
interactions
unit
fully
Simple
of
sc-lattice (b)
only
configurations
two e l e m e n t a r y
AF
periodic
con-
leading
those
are
to
ori-
shown.
The
cubes.
(a) "
Yt, Fig. 4.3
et al.
frustrated because
X
134
have
system
one
just
GS
diamond
other
SO
to one
GS
~
from
N-I/s
the
they
for
must
d > 3 or
effect
d i s o r d e r 136 : T h e r e parallel
reasons
interaction
degeneracy
frustrated
determined
for d - d i m e n s i o n a l
as for
of f r u s t r a t i o n The
)
of t o p o l o g i c a l
than
state,
(c)
(a) Comb representation of the fully frustrated square lattice. Straight (wavy) lines represent F (AF) interactions. Elementary cubes of the partially (b) and the fully (c) frustrated sc-lattice are obtained by stacking the square lattice of (a) in different ways (ref. 135).
Derrida
more
(b)
be
4
have
is h i g h e r lattice58).
1OO
the p r e v i o u s
(sc)
hypercubic d > 4
lattices
in p a r t
and
'superblocking'.
GS
of
where
strong
AF
fcc-lattice
Id
disorder
one
every
fourth
chain
can be f l i p p e d leading
found
that
unfavourable
called
in the
fully
of the p l a q u e t t e s
f c c - l a t t i c e 8. This
one,
,
of the
in the e n e r g e t i c a l l y
Instead
direction
properties
in the
then
is a set of
GS
as a w h o l e
kind
(and in the finds
2d
of spins
to get an-
to
(4.5a)
113
in c o n t r a s t
S
~
o
to
N-2/s
(fcc,
This
has an i m p o r t a n t
When
only a finite
one
GS
bE
that
=
4
of these L
MC
of length
L
additional
energy
low e x c i t e d
states.
of such a chain arises
is flipped
a second
from
only at the ends
of
(4.6)
'one-dimensional'
expansion
in good
for the
IJJ
calculations
cate
part
(4.5b)
is the d i f f e r e n c e
. But this way
series
consequence
configuration,
this part;
diamond)
excitations
condition
(4.3)
is i n d e p e n d e n t
is violated,
of the
length
and no low t e m p e r a t u r e
exists. of B h a n o t
order
and Creutz 137 and of K i r k p a t r i c k 136 indi-
transition,
but the c r i t i c a l
temperatures
are not
agreement:
Tc/IJl
~
0.8
(ref.
137)
, (4.7)
Tc/IJl
~
1.25
(ref.
w h e r e a d d i t i o n a l data are in favour Chui et al. mation tain
138
using
a much
This ing
~
that
to d e t e r m i n e but
and specific
Tc
from a mean
(contrary
(ref.
heat
to their
(Fig.
4.4)
field
approxi-
claim)
they ob-
systems
138)
with
are not well
and a p a r t i a l l y
(4.8)
strong
frustration
described
et al. 135 have d e t e r m i n e d
the f o l l o w i n g
Fully
factor
value
fluctuations
for the fully predict
tried
,
value.
sublattices,
2.4
demonstrates large
Blankschtein
(a)
have
higher
TMF, c /IJi
for s t r u c t u r e
of the higher
eight
136)
by m e a n
and c o r r e s p o n d field
the G i n z b u r g - W i l s o n
frustrated
sc-system
theory. Hamiltonian
(see Fig.
4.3)
and
behavior:
frustrated:
The order
parameter
expansion
up to
~2
has
four c o m p o n e n t s
is c o n s i s t e n t
with
(n = 4)
; ~ (5 4-d)-RG
a transition
of
(weak)
114
1.2
1.0
2
: ~<<.
/
o,4
....... ~4,.
...
•
~, ~."~;.-
/ °'Co.o
I
/ ~ I
o.s
~,o
1.8
I
I
2.0
~t
T~,qo~l~re kT/Ji
Fig. 4.4
Specific heat of the fully frustrated sc-lattice. MC results for different sizes L 3 [L = 2 : (dashed), L = 4 : ~ , L = 8 : [] , L = 20 : o and L = 30 : ¢] show the increasing maximum of c with increasing L . The dashed line corresponds to a disordered system (ref. 136).
first
order
Partially
(b)
(contrary
frustrated
orientations The
order
sion.
should
Lallemand
a different Note two
that of
and
the
their
three
From
finite
size
ture
of
fully
the
Tc/IJl
close
to
transition
has
=
the to
MC
the
yield
Nagai
data).
plaquettes
± 0.002
of
be
second
'normal'
of
only
one
of
the
three
in
with ~-expan-
behavior.
sc-systems model
the
using all
the
f u l l y 139
MC
and
techniques.
plaquettes
of
frustrated.
estimate
system
for
the
transition
tempera-
is:
,
Kirkpatrick, order.
XY-model)
irrelevant
investiagted
frustrated are
(n = 2,, order
XY
further
their
frustrated
value
eighteth
frustrated
orientations analysis
components
of
have
partially
1.355
two
term
partially 140 in
(with
parameter breaking
Diep,
the
frustrated)
a symmetry This
to
(4.9)
and
Their
like
results
him for
they the
also
conclude
specific
heat
the
115
derived tion
from
dU/dT
at f i x e d
equilibrium shown
and from
temperature
(Fig.
in Fig.
4.5a).
4.5b.
fluctuations
agree very well The
Below
corresponding
Tc
there
seem
of
U
during
demonstrating internal
MC
simula-
thermodynamic
energy
U(T)
is
to be t w o t e m p e r a t u r e
ranges
1.6
1.2
L,a
• "'j,~
0,4
0.5
1.0
1.5
2.0
kT/J kTIJ 0.5
1.0
t5
-1.0 .."
E
/
-1.;
!
/ o-
-I.L
Fig. 4.5
where
MC results for a) (top): the specific heat c and b) (bottom): the internal energy of the fully frustrated sc-lattice. In a) the full line is dU/dT , the points are derived from fluctuations of U during constant temperature MC runs (ref. 139).
the ordered
disorder (1OO)
j u s t as f o r
in t h e s e
to r e m a i n For
is c o n c e n t r a t e d
chain
order
well
T ~ 0.5
ferent
phase
chains
has
different
in o n e
sublattice
the class
forces
properties.
the
of
GS
chains
For
T ~ 0.5
containing mentioned
forming
]Ji
every
forth
before,
The
the o t h e r
dis-
sublattices
ordered.
rJl
the d i s o r d e r
sublattices.
The
is d i s t r i b u t e d
(continuous)
crossover
more
evenly
between
on t h e d i f -
both
ranges
116
can be seen in the averaged along
(100)
directions
pair
correlation
in Fig.
function
for n n n - s i t e s
4.6.
1.0
0,5
•
'
'
!
.
.
.
.
I
05.
,
,
,
,
1.0
I
,
i
i
1.5
kTIJ
Fig. 4.6
Diep
et
Temperature dependence of the averaged correlation function between nnn-sites along (1oo) directions (ref. 139).
al.
deviations
also d e t e r m i n e d
from the usual
For their p a r t i a l l y GS
properties.
frustrated
The
GS
ordered
plaquettes
GS
previo u s
section.
that M a c k e n z i e finite
energy
agreement With L
increasing
pha s e during
this happens occurs
at
at
GS F
the i n f i n i t e
gests
a second
to the
first discuss
because
the
any stacking
to the n o n f r u s t r a t e d
fcc-case
lead to m e t a s t a b l e phase.
Figure
the
MC
the c o r r e s p o n d i n g temperature
discussed
T ~ 1.7
order
phase
IJ1
phases,
3.7 shows
runs with
in the
IJI
series
The
transition.
the simplest MC
T ~ 1.5
confithe
is very good. phase
changes
into the
size
N = 123
to the p a r a m a g n e t i c
system.
A more precise
of the transition
behavior
for the
F IJl
L
being
results
for the system
transition 123
as the order
the m o n o t o n o u s
F
they call
respective
results
time used;
for the
L
is the one with
which
the
up to
the m e t a s t a b l e
the simulation
although
stable
at t e m p e r a t u r e s
system as well
yet,
disorder, parallel
140
spin up and two spin down,
starting
T c = 2.5
known
Id
planes
; similar
For both cases with
Diep et al.
the t h e r m o d y n a m i c a l l y
two planes
The other
internal
model
and find strong
With a low t e m p e r a t u r e series e x p a n s i o n s just like 127 have p e r f o r m e d they show that for low
the f e r r o m a g n e t i c
gurations.
exponents 139
and Young
temperatures
alternating phase.
a
exponents
Ising
exhibit
of the f e r r o m a g n e t i c yields
the c r i t i c a l
3d
of the
U(T)
phase
value
are not data
sug-
for
117
-18
u
-19
-20
i
05
I.o
1.5 T
Fig. 4.7
Diep
MC results (dots and circles) for the internal energy U starting from the stable L phase and the metastable F phase. The full and broken curves denote the series expansion results. The free energy of the L phase is lower than that of the F phase, although U is higher.
and Nagai
141
Ising
system,
fully
frustrated
lead beyond
4.3 In
AF
the
case scope
Pyrochlore
CsNiFeF 6
t u r e 143, (Fig.
have
and
This
the effect
Ghazali
to
XY
of b o n d d i s o r d e r
and Heisenberg
of t h i s
for the
a n d L a l l e m a n d 142 the e x t e n s i o n spins,
but both
sc
of t h e
topics
review.
Model CsMnFeF 6 , both
the magnetic
4.8).
studied
and Diep,
ions
lattice
of the
form a lattice
is e q u i v a l e n t
(modified)
pyrochlore
of c o r n e r s h a r i n g
to the
lattice
struc-
tetrahedra
of t h e
B
sites
in s p i n e l l s 1 4 4 ; w e s h a l l c a l l it p y r o c h l o r e (PC) l a t t i c e here. S t e i 145 n e r et al. for b o t h s u b s t a n c e s h a v e m e a s u r e d t h e s t r u c t u r e f a c t o r in t h e weak
110
Bragg
plane peaks,
short-ranged are
strong,
and the
site
for
AF
neutron
pronounced
order
system
(Fig.
scattering
diffuse 4.9).
, which
MC
from a one-tetrahedron
apart
caused
from by only
AF
nn-interactions J1 146 calculations of the
show a completely agrees
and,
scattering
The
is f r u s t r a t e d .
Ising model
O < T < ~
obtained
inelastic
found
magnetic
corresponding energy
by
monotonous
very well with
the
a p p r o x i m a t i o n 146
internal
energy
per
118
Fig. 4.8
The lattice of the magnetic ions in the pyrochlore lattice is formed by cornersharing tetrahedra.
01)"
~ Fig. 4.9
~-~! //I/
Lines of constant structure factor in the (ref. 145).
IiO
plane of
CsNiFeF 6
119
~(T)
with
=
- 3
x = exp
S(T)
for
(I-x~)/(3 +4x+x
(-21J1 i/T)
. The
I in 2 + [ in
=
T = 0
reproduces
~)
,
(4.10)
corresponding
entropy
(4.11)
((3 + 4 x + x ~ ) / 8 )
the
Pauling
approximation 39'144
for t h e
GS
entropy
SP o in g o o d
agreement
SMc o As
=
the
that
series
and
AF
Another tonous Fig. As
decay
Hc2
the
= 6
4.11
=
reproducing for
to r e m a i n
TI . 2
well
more
can be m a p p e d
compare
with
exactly
to
the h i g h o r d e r
(4.14)
degenerate
and
. MC 146
absence
as t h e
the i n t e r n a l
LRO
proportional
which
also
the p r o b a b l e paramagnetic
for
frustrated
energy
at t w o
results
for
. The
lower
transition
to
critical
M(H)
full
exp
T ~ 0.28
can be built
changes
gives
fcc-
no i n -
and
curve
is t h e m o n o -
(- AE/T) 146
(see
fJ1 j
from
tetrahedra,
fields X(T)
in Fig.
Hcl
= 2
= dM/dH 4.11b
the JJ1 j
are
represents
approximation
(x+x~)/(3 +4x+x
the
of a p h a s e
simulations
lattice
single-tetrahedron
~(T)
tion
MC
IJ1J
in Fig.
model
also
in 2
before,
for t h e
fcc-lattice, PC
PC
can
trantision.
of o r i g i n a l
during
of the
shown
AF
one
is m u c h
discussed
indication
4.10)
and
of t h e
model
of a p h a s e
in t h e
GS
(4.13)
of N a g l e 1 1 9 :
PC
sc-lattices
dication
(4.12)
I
in 2
(0.29577 ±0.OO007)
the
• in 2
M C v a l u e 146
ice 1 4 4 ' 1 4 7 ,
expansion
=
--~ 0 . 2 9 2 5
the
degeneracy
'cubic'
SN o
with
0.293
GS
of
Thus
I 3 in 2 + 2 in 8
=
MC
data
absence down
to
for
~)
,
(4.15)
low temperature,
of a p h a s e T = O
a further
transition
for
H = O
and .
indica-
for t h e
system
120
0
~0
o
Fig.
4.10
.,o
zo
.,o
~o
,~o
2.0
'~5
i
as"
3.0
E x p o n e n t i a l decay of original longe range order during ferent temperatures (ref. 146).
3.5"
MC
runs at dif
t M
M/H
Z T~ 0.~'~
Z
~ T~ 'LO0
ttt~
tttt #
I
,
i
,
o
H--p
Fig. 4.11
MC results for the m a g n e t i z a t i o n M(H) (a, left) , and for the susceptibility x(T) (b, right) in the nn AF PC m o d e l (ref. 146).
121
Figure
4.12
shows
same
110
ridge
of
ever,
the position
Fig. 4.12
plane S(q)
with
MC
as for extends
a weak
structure CsNiFeF 6
along
MC
of
the
4.9.
the Brillouin
zone
does
(see Fig.
nnn-interaction 4.13,
AF
Like
where
PC there
(BZ)
not agree with
structure factor in the
additional
experiment
factor in Fig.
of t h e m a x i m u m
Lines of constant (ref. 146).
Introducing ment
the
J2/IJ11
leads
PC
in the
a very broad
boundary. Fig.
nn AF
J2
model
How-
4.9.
model
to b e t t e r
agree
= 0.1)
J
Fig. 4.13
MC
structure factor as in Fig. 4.12, but with additional nnn-interactions J2 = O.i IJII (ref. 146).
122
For a more take
complete
account
a n d Fe 3+)
distributed
we do not nn AF
description
model,
which
as f a r
entropy
per
site,
We add that
the
t u r e of C h u i
AF
3d
Ising
to
in c o n t r a s t PC 138
e t al.
for all
down
model
system with
nn-interactions
has been
sharing
octahedra,
to t h a t
of t h e
sites
It is e a s y
completely
at
netically the
free with
system
mains
opposite
to e x h i b i t
ANNNl-model
The
3d
(in
x and y
to t h e
frustrated
thus
lattice.
is f o c u s s e d
data performed
and which
possesses fcc-
phase
transitions
Here
on t h e
can tell, a finite
and
is a c o u n t e r e x a m p l e
finite
GS
by Chui
GS
sc-lattices. to the
at f i n i t e
conjectempera-
This
lattice
corresponds
and c a n be d e c o m p o s e d
to see t h a t , when
e n t r o p y w h e n r e s t r i c t e d to AF 148 . It c o n s i s t s of c o r n e r -
lattice.
AuCu 3
spins
the other
orientation
an o r d e r e d
in
ANNNI-model
z
is s h o w n
direction)
direction
on one
two
(S O > I/3
into three
sublattice
are o r d e r e d In 2)
low temperature
J2
. The properties
the
2d
case discussed
vestigated survey
very
temperature
from
series
MC
(in
of t h i s
The
basic
ferromagnetic
to
The point
<2>
4.14.
. Chui
phase,
but
are
ferromagassumes this
re-
z
The nn-interactions
direction)
interact
3d
in S e c t i o n 149-151
system
3.4.1.
phase
occurs
For at
(T = O,
a multiphase
antiferromagnetically
This
diagram
T = 0 ~
point with
Id
system
different has b e e n
can g i v e is s h o w n
high -154
a change
(= - J2/J1)
K = I/2)
J
o ferromagnet-
are
are quite
, and here we
calculations 152'153,
expansions.
d = I and 2 precisely
J1
extensively
of t h e r e s u l t s .
It is d e r i v e d
and
in Fig.
also nnn-spins
via
more
MC
PC
to N i 2+
to be p r o v e n .
4.4
ic;
on t h e
interest
as the
a cubic
T = 0
(e.g.
systems.
in
sc-sublattices.
one has
ions
T = 0
studied
forming
Cu
data
of m a g n e t i c
randomly
as o u r
predicting
frustrated
experimental
kinds
less
further,
paramagnetic
A related
or
this
remains
ture
more
discuss
PC
of t h e
of t h e t w o d i f f e r e n t
only
from
ina brief
in Fig. 4.15. l o w _155
and
of t h e
= I/2
, j u s t as
is a f r u s t r a t i o n degeneracy.
GS
from for
point,
or
123
Fig. 4 . 1 4
Unit cell of the
3d
ANNNI-model.
I
~;IJ,
!
p,RA
L
*'~z MODULATE0
z
"~ / / FERRO
'
Fig. 4.15
Fisher
Selke
nite
sequence
<2>
phases
156
have
(Fig.
4.16)
to t h e
and only
one modulated
Figs.
wavevector The
steps
'
o'.,,
3d
3.38
2d
which
! I
T
- 3.40).
that
in Fig.
,:_j~/j,
near
&
! 1.0
i
the
phase
with
the multiphase
occurs
all extend P
between
down phase
to
is s h o w n
the
T = O
extends
continuous
The discontinuous
q at l o w t e m p e r a t u r e s
its o r i g i n a l
'
phases
case where M
o'.,
ANNNI-model (ref. 151).
found
of c o m m e n s u r a t e
contrast
(see.
II
Phase diagram of the
and
12,21 ANTIPHASE
t /
o',2
z
point F
. This
down
to
wavevector
variation in Fig.
an i n f i
and
of t h e
the is in T = 0
exists average
4.17.
4.17 d o n o t f o r m a c o m p l e t e " d e v i l ' s s t a i r c a s e " 157 because not all rational numbers q/q<2>
meaning
in
124 (zh)=I2,z,~)=~.,.ttlltltlllllll... Ic2(TJ
xI(T)~ (3,3) Antiphosl
12
3)
x3(T ) (2~3)
(2,21 Antipho~e
I{ • -Jz/Jt
Fig. 4.16
Schematic phase diagram of the point (ref. 156).
3d
ANNNI-model near the multiphase
I 10
:V-
q
q. q
o9
<®;'
O>
:
(3,3)
Ferro- ii -macJnet~c I
Ant~phose
O,e 1 12,21 . i An~hose.
06
I ,,
04
q
.lI
0
Fig. 4.17
occur.
On
the
i
,
i
,
i
Wavevector at low temperature as function of
Fisher
a "devil's
I
and
top
Selke
step"
transition
have
called
to t h e
P
the
behavior
<
(ref. 156).
of
q
near
q<2>
158
line
phase
at
there
is
a Lif-
shitz point L where the F, P a n d M (modulated) phases meet 75'152' 154 • For < < < < ~ q is e x p e c t e d t o r i s e c o n t i n u o u s l y from 0 to L 159 n/4 a l o n g the M-P transition line. S e l k e a n d D u x b u r y have investigated the
the
discrete
complexe
phases
at
behavior low
at
intermediate
temperatures
and
the
temperatures continuous
between q
varia-
125
tion tion. Fig.
near
Tc
They
find
using
an e f f e c t i v e l y
a very
Id
complicated
mean
branch
field
pattern,
(MF)
approxima-
sketched
roughly
in
4.18.
ksTIJo
PARA
Tc
3
I
Fig. 4.18
Of
r
I
o.~
f
r
0.6
I
o'.8
I
I
1'.o ~ =-J~,Ji
Mean field phase diagram of Selke and Duxbury (ref. 159).
special
which is
I
0.2
interest
shown
is the b r a n c h
in Fig.
pattern
they
found
for
< > I/2
4.19.
ksT/Jo
o.5~'°~ 0 0.5
Fig. 4.19
~,"~'~~ x (schemotic)
Branch pattern generated from combining adjacent structures (ref. 159).
126
The
MF
phase
determined Merwe
(FVdM)
tions 160.
which
This
and A u b r y 161 models
(Fig.
ANNNI
then have
interactions
similarity
lattice model
with
of Frank
in detail
the conditions
and Stratt162;
that the
transi-
by Bak 149,
field results
is too e x t e n s i v e
at first glance
e.g.
however,
3d
rich phase diagram m a n y details
and Axel
for m a p p i n g
have been
ANNNI-model
both
obtained
as we noted
to give a c o m p l e t e
looks
the one
and Van der
commensurate-incommensurate
Further mean
we just state, which
has great
Id
has been d i s c u s s e d
shown
by De Simone literature
In conclusion
prisingly
4.18)
describes
connection
into each other.
for instance the
diagram
by Aubry 157 for the
already,
survey here
with
competing
so simple,
exhibits
of which
still have
a surto be
clarified. At the end of this similarities triangular necting
to the A N N N I - m o d e l ,
or hexagon
adjacent
The first model layers. 3.43).
The
GS
Nakanishi
diagram "devil's
section we want
(Fig.
lattices
to m e n t i o n
which
with
are formed by stacking
ferromagnetic
has nn- and n n n - i n t e r a c t i o n s phase
diagram
is identical
and Shiba 163 in
4.20)
staircase"
2d
nn-interactions
con-
similar
MF
with
the t r i a n g u l a r
the
approximation
to the ANNNI-model,
2d have
case
(Fig.
found a phase
exhibiting
a complete
711~
po,o
i, ~ : I~,311x41
,,~
~0 "':'
\I I
20
within
behavior.
TIU~
MF
with
layers.
~o
Fig. 4.20
two other m o d e l s
-I
o
V-Z t ,..
OI
@15 I
.WUzl
)
~
phase d i a g r a m of the h e x a g o n a l model of Nakanishi et al.
(ref. 163).
127
The m o d e l gular
without
layers
nnn , but with
has been
al. 164 and Berker RG
expansions.
phases
with
expansion
as mean
theory
P
they
However,
in this m o d e l
field
the trianet
theory
find two p a r t i a l l y
the
to d e s c r i b e
and
ordered
C o p p e r s m i t h 165b has conventional
theory u s i n g
are not r e l i a b l e
within
by B l a n k s c h t e i n
Ginzburg-Landau-Wilson
phase
symmetries.
of f r u s t r a t i o n
as well
and L a n d a u
from the
nn-interactions
for i n s t a n c e
et al. 165a w i t h i n
Apart
different
that b e c a u s e
AF
investigated
shown
low t e m p e r a t u r e
method
of ref.
163
the low t e m p e r a t u r e
phase. The
second m o d e l
layers
and has been
diagram phase not
has nn-
(with a d d i t i o n a l
boundary
between
fcc
As the
Four-Spin
last
3d
situated
interactions)
A and C
phases
a
GS
where,
(Fig.
phase
however,
3 of ref.
166)
the is
In this
J4
a term
system
at
interactions.
here b e c a u s e
Model
system we now discuss on the
at the corners
contribute
fcc-lattice,
of an e l e m e n t a r y
4 J4 i~I
Si
T = O
there
where
any four
tetrahedron
four-
spins
Si
of the lattice
to the Hamiltonian.
Nevertheless
like m a n y
the system w i t h
is no c o m p e t i t i o n we have
frustrated
i nc l u d e d
systems
between
neighboring
this q u a r t e t m o d e l
it e x h i b i t s
a high
(Id)
degeneracy
N
=
g
similar model Ising with Based
N ~/s
to the
,
AF
(4.]6)
fcc-model
is of special
interest
system w i t h g l o b a l local
(+/-)
with n n - p a i r
(+/-)
symmetry,
the
either
both
T c0 = 2/in duality.
(I+~)
a single ~ 2.27
From e x t e n d e d
or
phase (II)
The quartet
between
the
2d
synametry and the of w h i c h series
system to be self-dual. (I)
interactions.
as it is i n t e r m e d i a t e
on high and low t e m p e r a t u r e
assumed have
the
(Quartet)
Ising
spin i n t e r a c t i o n s
GS
four-spin
the hexagon
correct.
4.5
J4
and n n n - i n t e r a c t i o n s w i t h i n 166 studied by Rujan . He p r e s e n t s
4d z~ gauge m o d e l -32,167 are s e l f - d u a l
expansions
As a c o n s e q u e n c e
transition
Wood 168 had first the system
at the Onsager
a pair of t r a n s i t i o n s
low t e m p e r a t u r e
expansions
linked
Griffiths
should
value
and
by
128
W o o d 169 c o n c l u d e d c a s e (II) to be true. M C 170 sen et al. d e f i n i t e l y s h o w e d the e x i s t e n c e but
their value
To resolve first the
these
proved MC
Tc
was
far a b o v e
contradictions
that
result
effects,
for
the q u a r t e t
of ref.
170.
model
of o n l y
one
Tc ° ' excluding
L i e b m a n n 171
Taking
calculations
in f a c t
and Pearce
self-duality. a n d B a x t e r 172
is s e l f - d u a l ,
c a r e of v e r y
of M o u r i t transition,
strong
contrary
to
hysteresis
in f u r t h e r MC c a l c u l a t i o n s L i e b m a n n 173 a n d t h e n a l s o A l c a 174 175 and M o u r i t s e n e t al. o b t a i n e d f u l l a g r e e m e n t of t h e
raz e t al. MC
value
The phase of the main
Tc
with
transition
low and high
reason
Figure U
for
4.21
a n d the
T°c
required
is of f i r s t temperature
for t h e d e v i a t i o n shows
the
entropy
order phases
, the
self-duality.
with
pronounced
beyond
Tc
of the o r i g i n a l
temperature
S
by
dependence
discontinuity
value of t h e
at
Tc
metastability
, which from
was
the
T° c
internal
is u n u s u a l y
energy strong.
t 08 In2 AS/K s
i
i
H 0
-1.0
:-7 lonh
I
t
I |
U
I -7.0 0
Fig. 4.21
- -~02
, 0.4
i 06
, 0.8 K _.
MC results for internal energy U and entropy S of the quartet model as function of K = J4/T (ref. 173).
129
In
the
raz
MC
calculations 174 al. found good
et
B O - BN
where
for
0f ~ 0 . 9 sites.
~
a
. NI/3
finite
agreement
size
of
effects
B =
I/T c
are
observed.
with
the
boundary is
4.22
the
Alca-
scaling
law
(4.17)
• N-I/3
periodic
Figure
also
conditions
linear
represents
s
dimension the
second
~ O.1 P of
the
, and finite
for
frozen
lattice
ones
with
N
case.
0.45
0.40
0.35
0.30
0.25
I
|
O
!
20
40
(2N-I) I/3 Fig. 4.22
We
note
Finite size effect on (ref. 174).
that
equivalent,
in
the
quartet
contrary
with
high
been
generalized
symmetry
to t h e
are
in
four-spin
models
the
effect
additional
critical
of
point
where
model AF
for
on
the
the
ways.
and
the
results of the quartet model
phases where stable.
Mouritsen
bcc
nn-pair
MC
all N g fcc-model
thermodynamically
several
ied
8 = I/T e
completely
The
the
sc-lattice
charges
These
from
LRO
quartet
et al. 175
interactions.
transition
are only
also and lead
first
to
phases
model
have
has
stud-
determined to
a tri-
second
order
130
Alcaraz with
et al.
ZN
for
Zn
have
symmetry
first order tinuous
176
(N = 2
transition
transitions symmetry
studied
generalized
corresponds
for
N < N
occur w h i c h
fcc-quartet
to Ising
~ 5 , whereas
c (may be)
the system a p p r o x i m a t e l y
that the Ising q u a r t e t m o d e l the hcp-lattice,
et ai.176).
the free e n e r g y
The d i f f e r e n c e s
are minimal.
for
remains
packed caraz
spins).
They
find a
two conc order. Also
self-dual.
on the other
also if self-dual
for spins
N > N
are of infinite
F i n a l l y we note lattice,
models
3d
close
(Liebmann 173, AI-
to the fcc- case , for instance
in
131
5.
Conclusion
This book has
intended
to give
theory
of p e r i o d i c
sions.
We first d i s c u s s e d
cannot
exhibit
mon
effects
frustrated
finite
a review !sing
of the p r e s e n t
systems
the r e l a t i v e l y
temperature
of f r u s t r a t i o n
simple
transitions,
like finite
state
of the
in one to three dimenId but
systems
which
show a l r e a d y
groundstate
(GS)
com-
entropy
per
site. In two d i m e n s i o n s interactions because thods.
these
systems
of ref.
of the
is strong
2d
enough
ferromagnetic
c ritic a l
T = O
tional
at
to
r-I/2
second
class
finite
down
Inclusion
havior and,
for instance, of these
reaching
This
2d
rely on a p p r o x i m a t e
ones have
been
results
true
diverges
length
exponents,
and n u m e r i c a l 3d
up to now. Here
proporIn the remains
and of a m a g n e t i c
in g e n e r a l
systems, But these
in the next
field
multidimensional
wealth
of critical
multicritical
be-
points
transitions.
can no longer be solved e x a c t l y
for the
frustration
critical.
a corresponding
critical
properties.
However,
and one has to
methods. where
only
few ones
the simplest
show a l r e a d y
few years m a n y
further
can be expected.
For c o m p a r i s o n
with e x p e r i m e n t a l
sion of further
interactions behavior
weak.
of i n t e r e s t
Another
point
to ones w i t h higher
This will
also be very
frustrated
systems
will be necessary,
the low t e m p e r a t u r e
systems
with
the ex-
are two n e w
exponent).
the c o r r e l a t i o n
interactions
analytical
studied
different
length I/4
become
systems
causes
there
in
temperature
type w i t h
however,
commensurate-incommensurate
systems
is even more
systems
, they never
of
m a t r i x me-
of the first one b e c o m e
to the usual
like n o n u n i v e r s a l
most
Systems
the c o r r e l a t i o n
of further
a finite
Ising
When,
, where
lead to a m u l t i t u d e order parameters.
case.
(contrary
T = O
accumulated,
by transfer
such a transition,
of systems.
of f r u s t r a t e d to
with noncrossing
has been
as some kind of layered m o d e l s
is of the usual
to suppress
classes
of systems
of k n o w l e d g e
long as they e x h i b i t
it always
universality
very
bulk
can be solved e x a c t l y
54. As
transition,
ponents
This
large group
They all can be d e s c r i b e d
the sense p hase
for the
a considerable
drastically
the inclu-
as they m a y
even when
from
(XY , H e i s e n b e r g
for u n d e r s t a n d i n g
change
they are quite
is the g e n e r a l i z a t i o n
spin d i m e n s i o n
important
often
Ising case).
experiments.
132
Finally r e p l a c i n g classical by q u a n t u m spins may be of importance, since frustration effects seem to be quite d i f f e r e n t 177 in both cases
References I.
E. Ising, Z. Phys. 31, 253 (1925). H.N.V. Temperley, in: Phase T r a n s i t i o n s and C r i t i c a l Phenomena, Vol. I, Eds.: C. Domb and M.S. Green, A c a d e m i c Press 1972.
2.
e.g.: K. Binder, in: F u n d a m e n t a l P r o b l e m s in S t a t i s t i c a l Mechanics, Ed.: E.G.D. Cohen, N o r t h H o l l a n d 1980, and: H e i d e l b e r g C o l l o q u i u m on Spin Glasses, Eds. J.L. van Hemmen and I. Morgenstern, Lecture Notes in Physics 192, S p r i n g e r 1983.
3.
G. Toulouse, Commun. Phys. 2, 115 (1977). J. Vannimenus, G. Toulouse, J. Phys. CIO, L537
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Acknowledgements
The author thanks Prof. Dr. H.G.
Schuster and the other members of the
Institute for T h e o r e t i c a l Physics,
U n i v e r s i t y Frankfurt,
FRG, for
stimulating discussions. He also thanks Prof.
Dr. K. Binder,
then at the IFF, KFA J~lich, FRG,
for introducing him to M o n t e Carlo calculations, Steiner,
HMI Berlin,
and Prof. Dr. M.
for c o l l a b o r a t i o n c o n c e r n i n g the i n v e s t i g a t i o n
of p y r o c h l o r e systems.
Part of this work has been supported by the Deutsche F o r s c h u n g s g e m e i n s c h a f t through the S o n d e r f o r s c h u n g s b e r e i c h 65 F r a n k f u r t - D a r m s t a d t . The m a n u s c r i p t was finished at the M a x - P l a n c k - I n s t i t u t forschung,
Stuttgart,
Hanna-Daoud, Darmstadt,
FRG. For the very e f f i c i e n t typing I thank Mrs.
Institute for T h e o r e t i c a l Physics,
FRG.
fHr Festk6rper-
Technische Hochschule
