CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

T I L ÁiTb l U

KFKI-1 9 8 1 -7 2

H ungarian cAcadcmy of Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

A, SÜTŐ

MODELS OF SUPERFRUSTRATION

— --- ---

2017

л.

K F KI-1981-7 2

MODELS OF SUPERFRUSTRATION

András Sütő

Université de Lausanne, Section de Physique CH-1015 Dorigny, Switzerland

To appear in the Zeitschrift für Physik В

HU ISSN 0368 5330 ISBN 963 371 854 6

On leave of absence f r o m the Central Research I n s t i t u t e for Physics, Bu d a p e s t

ABSTRACT

Earlier studies of the triangular lattice antiferromagnet and the fully frustrated model on the square lattice proved that in these models the pair correlation < ^ 3Sr> decreases asymptotically as r ’ at zero temperature. In the present paper the existence of two and higher dimensional models is shown in which the frustration is so strong that it destroys the phase transition even at T=0: the correlation length remains finite. The influence of this

"superfrustration" on the free energy and on the ground state properties is also discussed.

АННОТАЦИЯ

Исследования корреляционной функции <SQSr> в треугольной антиферромаг- нитной и полностью фрустрационной^квадратной решетке показали, что она асимптотически пропорциональна г 2 при нулевой температуре. В настоящей статье показано, что при размерности пространства два и больше существуют такие сильно фрустрационные модели, в которых нет фазового перехода даже при нулевой температуре; корреляционный радиус остается конечным. Эффект

"суперфрустрации" проявляется и в свободной энергии и свойствах основного состояния.

KIVONAT

Korábbi vizsgálatok, melyeket a háromszögrács-antiferromágnesen, vala­

mint a négyzetrács fölötti teljesen frusztrált modellen végeztek, azt bizo­

nyítják, hogy az <SQSr > párkorreláció aszimptotikusan arányos r"^-del nulla hőmérsékleten. Az alábbiakban megmutatom, hogy két és bármely nagyobb dimen­

zióban léteznek olyan erősen frusztrált modellek, melyekben T=0-n sincs fá­

zisátmenet: a korrelációs hossz véges marad. Ez a "szuperfrusztráció" nyo­

mot hagy a szabadenergián és az alapállapoti tulajdonságokon is.

1. In tro d u c tio n

Isin g models w ith competing in te ra c tio n s have a tt r a c te d a consider­

able in t e r e s t since the possible relevance o f f r u s t r a t io n to the pro­

pe rties o f spin glasses has been pointed ou t by Toulouse [ 1 ] .

F ru s tra tio n is an extreme m a nifesta tion o f the competition among the in te r a c tio n s , leading to an accidental (not-symmetry-imposed) degenera­

cy o f the ground s ta t e . In the most spectacular cases, t h i s degeneracy re s u lts in a macroscopic entropy a t zero temperature (T = 0) and the disappearance o f the phase t r a n s it io n s f o r T > 0: I mention here the antiferrom agnetic model on the tr ia n g u la r l a t t i c e [2 ] (denoted by TAF), the an tife rro m a g n e tic and f u l l y fr u s tr a t e d models on the Kagomé l a t t i c e ГЗ] (KAF and KFF re s p e c tiv e ly ) and the f u l l y fru s tra te d model on the square l a t t i c e [ 4 ] (SFF). In the case of two dimensional models w ith pe riod ic nearest neighbour p a ir in t e r a c t io n s , as in the above examples, the fre e energy can be calculated exactly by Onsager's method and the zero p o in t entropy can be in fe rre d from i t . Spins not being fro z e n -in a t T = 0, the c o r r e la t io n fu n c tio n s may also be subject to in te r e s t.

While the absence o f a local magnetization and, th e re fo re , the decay of the c o r re la tio n s w ith increasing distance is rather obvious (though, as I w i l l p o in t out l a t e r , the boundary con d itio n s play a n o n - t r i v ia l r o le ) , one cannot simply guess the rate o f t h i s decay. The free energy of any reasonable s t a t i s t i c a l ensemble is s in g u la r at T = 0 - th is follows from the e sse ntial s i n g u l a r i t y of e x p ( - H /k ^ T ) - so one would expect a power-law behaviour o f the c o r re la tio n s as a t a c r i t i c a l temperature. The r e s u l t of Stephenson [5 ] f o r the TAF model supports th is expectation: < SQSr > ^ r ’ n a t T = 0, w ith n = 1/2. In two recent papers, Southern e t a l . [ 6] and Forgács [ 7 ] studied the c o r re la tio n s of the SFF model. In the f i r s t paper, the authors found a power-law decay w ith c = 1/4. Forgács' more careful analysis lead to c = 1/2.

Considering these r e s u lt s , obtained f o r f u l l y fr u s tr a te d models, together w ith the f a c t tha t in the absence o f f r u s t r a t io n there is a long range order a t T = 0, one might th in k th a t the c o r r e la t io n length

2

must be i n f i n i t e in a l l Ising models at zero temperature.

In the present paper i t is shown th a t in certain models the f r u s t r a ­ tio n may be so stro ng that the c o r r e la tio n s decay e x p o n e n tia lly a t any temperature, in c lu d in g T = 0. The term " s u p e r f r u s tr a tio n model" w i l l be applied to models o f f r u s t r a t io n which do not undergo any phase t r a n s it io n and in which the c o r r e la t io n len gth is f i n i t e a t T = 0. The f i r s t examples f o r such models were given in an e a r l i e r paper [ 8] where a method f o r discussing the a n a ly tic a l p ro p e rtie s of the free energy and the c o r r e la t io n s in the case o f f r u s t r a t io n was developed. The present work generalizes the r e s u lts of [ 8 ] and uses a s im p lifie d argu­

ment to show th a t the fo llo w in g con dition s are s u f f i c i e n t f o r the s u p e rfr u s tra tio n :

In the given model there e x is ts a "dominant" subset, S, o f plaquettes so th a t

( i ) S contains t r ia n g le s and/or squares ;

( i i ) S covers the whole set o f bonds d i s j o i n t l y ; ( i i i ) S is weakly connected in the sense t h a t

a) every l a t t i c e s ite is shared by a t most two dominant pla que ttes;

b) Nn , the number of closed chains formed by a given dominant plaquette and n-1 o th e r elements o f S, does no t increase very ra p id ly w ith n ;

( i v ) there are o n ly nearest neighbour p a ir in te r a c tio n s , J . . , defined

* J so th a t every plaquette o f S is f r u s t r a t e d .

One can formulate ( i i i / b ) more q u a n t it a t i v e l y : n

( i i i / b ) E N --- - . XS < e < 1 n n ( 1- e ) n"

can be solved f o r e. Here

x$ = (1 /3 ) a ( 1 / 2 ) 1" a

i f S is a homogeneous mixture o f tr ia n g le s and squares and the proportion of the former is a . These co n d itio n s can be s a tis fie d in two and higher dimensions, th e r e fo r e , s u p e rfru s tra tio n e x is t s in any dimension gre a te r

- 3 -

than one. For a two-dimensional example, I mention th a t ( i ) - ( i i i ) are f u l f i l l e d fo r the Kagomé l a t t i c e i f S is chosen to be the whole set of tr ia n g le s . I f , in a d d itio n , the in te ra c tio n s are defined so th a t every t r i a n g l e is f r u s t r a t e d , the model w i l l be su p e rfru stra te d . In p a r t i c u l a r , t h i s is true f o r the KFF model and also f o r the KAF model where the

hexagonal plaquettes are n o n -fru s tra te d . The example of the KFF model shows that the conjecture [ 7 ] , according to which a l l two-dimensional f u l l y fru s tra te d models form a u n iv e r s a lit y class characterized by the

- 1 / 2

decay <S S > ^ r a t T = 0, does no t hold. I t is worth mentioning

• ' o r

t h a t the SFF model, which is not s u p e rfru s tra te d , v io la te s ( i i i / b ) ; th is shows the "sharpness" of t h i s co n d itio n .

The appearance o f a f i n i t e c o r r e la tio n length despite of a s i n g u l a r i ­ ty in the fre e energy runs against the i n t u i t i o n . These fa cts w i l l be reconciled by showing th a t the s in g u la r it y of the fre e energy a t T = 0 is r e la t i v e l y m ild , e s s e n tia lly of the same type as that produced by a s in g le Ising spin in an external f i e l d .

I t should be emphasized th a t the c o n d itio n s ( i ) - ( iv ) are not necessa­

ry to the s u p e r fr u s tr a tio n : each of them could be weakened but the form ulation o f the con dition s would become more complicated. An im­

p o rta n t feature o f these conditions is th a t they provide us w ith a recipe f o r con structin g sup erfrustra ted models. A non-constructive c o n d itio n f o r the s u p e rfru s tra tio n which is , however, both s u f f ic i e n t and necessary, can also be obtained. At zero temperature, the only c o n fig u ra tio n s

a v a ila b le f o r a system are the ground states o f t h i s system. The d i f f e r e n t behaviour of the c o r re la tio n s at T = 0, found in the models mentioned above, must therefore be the consequence of a q u a l i t a t i v e d iffe re n ce between the sets of the ground sta te s. I believe t h a t th is con sists of the f a c t th a t the fam ily o f the ground states in both the TAF and SFF models is unstable w ith respect to boundary perturbatio ns whereas t h a t of the KFF, KAF e tc . models is s ta b le . The i n s t a b i l i t y means th a t the ground state degeneracy can be s p l i t up by a s u ita b le choice o f the boundary c o n d itio n . In other words, the e q u ilib riu m s ta te at T = 0 is not unique f o r the TAF and SFF

4

models, while i t is apparently unique f o r the KFF and KAF models. The uniqueness o f the zero temperature e q u ilib riu m s ta te , together w ith the non-uniqueness o f the ground s ta te c o n fig u ra tio n s , seems to be s u f f i ­ c ie n t and necessary fo r the s u p e r fr u s tr a tio n .

The paper is organized as fo llo w s . In Section 2 the p a ir c o r re la tio n s in models on the Ragomé l a t t i c e are studied by using vacuum boundary co n d itio n and i t is shown th a t <S Sr > decays exp o n e n tia lly w ith i n ­ creasing r at any T > 0, i f a l l the tr ia n g le s are fr u s t r a t e d . The ex­

tension of the pro o f to a ll models of the type ( i ) - ( i v ) is discussed a t the end o f the section. I omit the pro o f of the exponential c lu s te rin g of general n - p o in t c o r r e la tio n s ; in the case of the Ragomé l a t t i c e models, t h is was done in [ 8 ] . Section 3 presents the re s u lts on the a n a l y t i c i t y pro p e rtie s of the free energy. In Section 4, I discuss the e f f e c t of the boundary c o n d itio n s on the ground s ta te properties and give a q u a li­

t a t iv e argument p re d ictin g a power-law decay of the c o r r e la t io n f o r the TAF and SFF models and an exponential decay fo r the RFF, RAF e tc . models.

F i n a l l y , Section 5 contains the summary o f the paper.

2, Decay of the P a ir C orrelatio ns on the Ragomé L a t t ic e

Let us consider the c o r re la tio n s o f the spins belonging to the sites i and j .

<S.S.> = E S .S . exp(ß E J . . S . S J / E exp(3 E J . . S . S J (1) 1 J {S} J <kl> Kl K 1 { S} < k l> Kl K 1

Applying the usual transformation

expiBJ^^S^S-i) = cosh ß J ^ ( 1+S^S^ tanh ß )

= cosh BJk1 ( 1+S|<S1 zk l ) (2)

and summing over the spin configuratons we obtain

< ^ у

= L’

П

J g6G. . <kl>€g Kl g€G < k l> « g K

* J

- 5 -

Here G^. and G contain c e r ta in graphs b u i l t up from the edges o f the l a t t i c e : the graphs of G are unions o f closed loops and those o f G.^

are formed from closed loops and from a s trin g connecting the s ite s i and j .

Every edge o f the Kagomé l a t t i c e belongs to a unique t r i a n g l e , the re ­ fo re the set o f a l l edges can be considered as the union o f tr ia n g le s .

1 N

Let us assume th a t our system consists of N t r ia n g le s : В . The tria n g le Ba (a = 1 , . . . , N ) is the c o lle c t io n o f three bonds

<k 1 > , <1 m > and <m к > . Dividing the numerator and denominator o f

a a a a a a

(3) by N

П

a=1 ( 1 +

1a m

a

and performing some algebra we a r r iv e at

<S.S.>

1 J Г П

z

g * G .. < kl > € g g <G < kl > t g

a

J э

(4)

Here

z. , + z. z , J k j lm mk_

1 + z . , z, z . kl lm mk

< Sk S1 Ba ⁽⁵⁾

is a c o r r e la t io n fun ction on the t r i a n g l e Bu containing the bonds <kl

<lm> , and <mk> and the prime in d ic a te s that the summations are re ­ s t r i c t e d to graphs using a t most one edge from any tr ia n g le . Our gain is twofold w ith the in tro d u c tio n o f the new v a ria b le s (5 ). F i r s t l y , the number of terms in the sums of (4) i s less than th a t in Eq. (3 ) .

Secondly, i f the t r ia n g le Ba is f r u s tr a t e d then f o r any <kl><sB(‘ the magnitude o f varies in the in t e r v a l [ 0 , 1 / 3 ] with ß running from 0 to + 00 , w h ile | z ^ j covers the whole in t e r v a l [ 0 , 1 ] . Indeed,

i < k i ‘ ■ i t s * . j k ) ) — * — г I * i w 1 + w + w

( 6 )

- 6 -

i f

s9" < J kl ° l m Jmk > * and

tanh В IJ ki I = w

f o r any <kl>feBa . One may, th e re fo re , hope f o r a b e tte r convergence of the sums in (4) than of those in (3 ). Indeed, th a t i s what we f in d .

The summation in the numerator o f (4) goes over a ll the "two-leg"

graphs having t h e i r legs in i and j . Each such graph c o n s is ts of a connected two-leg graph and a number of closed loops. The c o n trib u tio n o f these l a t t e r cancels a s u ita b le fa c to r o f the denominator, thus leaving

< S.S. >

1 J n>1 { <k 1 a 1 a 1

^ к 1

■ a 1 a 1

, <k 1 >}«G. . 1 + t “ 1

a a

a n an l+t na„

"01 >| *"06 2 • (7) The double prime in d ic a te s th a t one has to sum over the s trin g s con­

necting i and j and containing a t most one edge from any given t r i a n g l e . Furthermore,

Z

<kl> 6 Вa j 4 l <Sk V ( B a 1 u . . . u Baj ) C ⁽⁸⁾ th a t i s , t J a i is the l i n e a r combination of three f i r s t neighbour

- a . . . . - a .

' J

c o r r e la t io n fu n c tio n s which do not depend on the in te ra c tio n s belonging to the tr ia n g le s B°^ , • • • »B°^ .

Let us apply (7) to { i , j } = < ij> , i . e . , to a bond. Then a f i r s t

1 N

neighbour c o r r e la t io n , depending on the in te ra c tio n s o f В , . . . , B , is expressed in terms of other f i r s t neighbour c o r r e la t io n s , each depending on the bonds of a number of t r ia n g le s smaller than N. I f (7) is w r it t e n f o r the three edges o f a t r ia n g le and a l i n e a r combination

of these equations with the corresponding weights is taken, then a recursion r e la t i o n fo r the functions t (Eq. 8) is obtained. This can be used to give a uniform bound on I t J a . I, v a l i d fo r any set of

3 1 - a . . . . - a . 1

tr ia n g le s and f o r any T. One can see from Eqs. (6 ) , (7) and (8 ) th a t th is bound is e i f the con dition ( i i i / b ) i s s a t is f ie d f o r e. Now x<. = 1 /3 ,

^2n+1 = ^ anc* ^2n 41 ^ П asy mP t o t i c a l l y . To fin d a s o lu tio n f o r ( i i i / b ) , i t is necessary to use the much smaller exact values o f N^n f o r several small values o f n. One then obtains th a t

c =

a .

i t J I < 0.1 (9)

1 - a . . . . - a . '

* J

f o r any a . , . . . , a . . To estim ate the p a ir c o r r e la t io n s , we s t i l l need 3

the bound

N l j < 2 n ( 1 0 )

n

where is the number o f sets of le n g th n o ccu rring in the summation (7 ). Applying the estimates ( 6 ), (9) and (10) in Eq. (7) we get the f i n a l re s u lt

|<S.S.>| < Z 2n ( т ^ т ) П <4 x 0.7411-J 1 (11) J n > | i - j I 1" U- 1

which is v a lid f o r any T > 0 .

Thinking over the p ro p e rtie s used to derive (11) one fin d s indeed

those which were enumerated in the In tro d u c tio n . This shows t h a t the proof can be tra n s fe rre d to any model s a t is f y in g ( i ) - ( i v ) . The s u b s titu tio n of the tr ia n g u la r plaquettes w ith squares is allowed because the diagonal p a ir - c o r r e la t io n s and the fo u r - p o in t c o r r e la t io n vanish on a s in g le

f r u s tr a t e d square while the absolute value of the bond c o r re la tio n s

saturates at 1/2. Hence, Eq. (7) remains v a lid and 1/3 is to be replaced by 1/2 in (6) and (11) whenever an edge o f a square is considered. The estimates of Nn and depend on the p a r t ic u la r model.

- 8 -

3. A n a ly tic a l Behaviour of the Free Energy o f Superfrustrated Models at T = 0 A s in g u l a r i t y in the fre e energy is usu ally connected to the

appearance o f a genuine cooperative behaviour among i n f i n i t e l y many degrees o f freedom. The s i n g u la r it y a t zero temperature may present a remarkable exception [ 9 ] . Indeed, the fre e energy o f a sin g le Ising spin in an external f i e l d h,

f h = - kßT Яn (2 cosh (h / kßT)) (12)

also shows an essential s in g u la r it y a t T = 0. In f a c t , t h is type of s i n g u la r it y appears in the fre e energy o f any Is in g model w ith i n t e r ­ actions { j ß} (b is a s in g le s it e , a p a ir of s ite s e t c . ) , since the p a r t i t i o n fu n c tio n can be w r itte n as

Z ^ ( П cosh (J. / kDT)) E П tanh (J. / kDT) (13)

b b B "closed Ь b B

graphs"

These remarks in d ic a te t h a t the sign o f an eventual cooperative behaviour a t T = 0 is to be looked f o r in the second fa c t o r o f (13). In our case, i t is convenient to normalize the fre e energy according to the number o f dominant plaquettes and to w rite

f = ( a f t + (1 - a) f ) + ip (14)

Here f ^ and f are the fre e energies of the is o la te d t r ia n g u la r and square pla q u e tte s, re s p e c tiv e ly , and ф represents the c o r re c tio n due to the in te r a c tio n s among the d i f f e r e n t plaquettes. For fr u s tr a t e d plaquettes f . and f are given by

t sq 3

J

f t,sq = " kBT [ m£n (cosh Ji j / kBT) + An (1 - ta nh (| Ji j | / k ßT)]

plaquette

(15) where m=3 in f ^ and m=4 in f ... As (13) and (15) show, ф depends

on T via t a n h ( J . . / kRT) o r , a f t e r separating o f f the signs o f the i n t e r - a c tio n s , via w = tanh( | | / k RT ) . The s in g u la r it y o f f at T = 0 is th e re -

- 9 -

fore determined by the behaviour of f t (and f ) around T = 0 and the behaviour of ф around w = 1. As mentioned e a r l i e r , the former is not s i g n if ic a n t because i t r e f l e c t s the p ro p e rtie s o f a f i n i t e system. Hence, one concludes t h a t any fe a tu re o f the fre e energy which is p a r t ic u la r to s u p e r fr u s tr a tio n , must appear in the w = 1 behaviour o f the fu n c tio n ф.

In Ref. [ 8] , a comparative study of ф f o r the KFF and SFF models was performed by l o c a liz in g the zeros of the corresponding (reduced) p a r t i ­ tio n functions on the complex w-plane. I t was found th a t f o r the KFF model the 0 < w á 1 segment l i e s in the i n t e r i o r o f a domain fre e of zeroes while w = 1 is on the border o f the domain o f zeroes, f o r the SFF model. The proof uses nothing p a r t ic u la r to the KFF model except the p ro p e rtie s ( i ) - ( i v ) . Hence, one can draw the fo llo w in g conclusion:

In an Ising model s a t is f y in g the c o n d itio n s ( i ) - ( i v ) , the p a rt o f the free energy which describes the in te ra c tio n o f d i f f e r e n t plaquettes is an a n a ly tic fun ction o f w = t a n h ( | J . . | / k RT) f o r 0 ^ w ^ 1 .

This property a lso shows t h a t the name " s u p e r f r u s tr a tio n " is w ell j u s t i f i e d the f r u s t r a t io n is so strong t h a t i t makes the i n f i n i t e system to behave e s s e n t ia lly l i k e the union o f independent small subsystems.

4. S e n s it iv it y to the Boundary Condition in the Ground State and the Decay of the C orre la tio n a t T = 0

As [ 7 ] and the present Sect. 2 i l l u s t r a t e , a c e r t a in amount o f mathe­

matical work is needed to f i n d the rate o f decay o f the p a ir c o r r e la t io n fu n c tio n in d i f f e r e n t f r u s t r a t io n models. However, to convince ourselves th a t there must be a d iffe re n c e between the SFF and KFF models, i t is s u f f i c i e n t to look a t some general p ro p e rtie s of the ground s ta te con­

f ig u r a t io n s in the two cases.

I t is well known that the c o r r e la t io n s usually depend on the choice of the boundary co n d itio n which is maintained while the thermodynamic l i m i t i n g process is performed. Instead o f p a ir c o r r e la t io n s , one may con­

s id e r the magnetization a t the o r ig in , < S >v . so th a t the spins are 0 V , s ,

fix e d in the c o n fig u ra tio n S outside the volume V. (S may eve n tu a lly be zero in some o r a l l o f the s it e s . )

I f

:S o S = 1 im

\J-HX>

depends on S, t h i s ind ica te s th a t So remains c o rre la te d with the

boundary spins w h ile the boundary disappears in the i n f i n i t y . In such a case, the c o r r e la t io n length must be i n f i n i t e which implies e i t h e r a long range order or a power-law decay. I f <Sq>^ does not depend on the boun­

dary condition S then one expects a f i n i t e c o r r e la t io n length, i . e , an exponential decay of the c o r r e la t io n .

Let us look a t the TAF and SFF models. I f a ground state c o n fig u ra ­ t i o n is singled out at random, i t w i l l almost s u re ly be a c o n fig u ra tio n in which every spin can be fr e e ly flip p e d e ith e r by i t s e l f or together w ith a few o th e r spins. I f any of these n o n-iso la ted ground s t ates is taken as the boundary c o n d itio n S, the in fin ite - v o lu m e c o r re la tio n s w i l l not depend on the p a r t i c u l a r choice and w i l l c e r t a i n l y give <S > ^ = 0.

One may say t h a t any of the no n-iso la ted ground s ta te s (which form the overwhelming m a jo rity o f the whole set o f ground s t a t e s ) , taken as boun­

dary c o n d itio n , determines one and the same phase o f the system: a phase w ith a f i n i t e entropy per sp in . There a re, however, isolated ground states

in the TAF and SFF models. Figure 1 shows a gauge-invariant representation o f such a ground state in both cases: the fr u s tr a t e d bonds are crossed w ith a lin e segment. To f l i p the spins in a f i n i t e domain, the cost of energy is p ropo rtion al to the length o f the border o f th is domain minus the number o f the f r u s tr a t e d bonds along the border. S ta rtin g from the ground states o f Fig. 1, the energy co st of any lo c a l transformation is p o s itiv e which means th a t the re s u ltin g c o n fig u ra tio n is not a ground

I

s ta te and th e re fo re is not a va ila b le a t T = 0. Hence, taking an isolated ground state S as the boundary c o n d itio n , one ob tains that

11

f o r any set В o f s it e s . This shows the existence o f a zero temperature e q u ilib riu m s ta te which is concentrated to the s in g le co n fig u ra tio n S.

I t is found th e re fo re th a t the in fin ite -v o lu m e c o r re la tio n s are sensi­

t i v e to the boundary c o n d itio n in the TAF and SFF models. According to our former speculation, t h i s means the i n f i n i t y o f the c o r r e la tio n length: long range order i f the boundary co n d itio n is an is o la te d ground s ta te and power-law decay i f i t is n o n -iso la te d .

Unfortunately, I cannot give a rigorous proof to the claim th a t there is no is o la te d ground state in the KFF or g e n e ra lly , in the su p erfrustra ted models. At le a s t, my attempt to f i n d such a co n fig u ra ­ t io n was unsuccessful. Figure 2 presents a ground s ta te of the KFF model.

I t is to be noted th a t the model is o v e rfru s tra te d which means th a t in any co n fig u ra tio n there are plaquettes w ith more than one f r u s tr a t e d bond. Indeed, the r a t io o f the number o f tr ia n g u la r and hexagonal plaquettes is 2 : 1, hence one cannot cover the l a t t i c e with t r ia n g le - hexagon p a irs . I f in a ground state every t r ia n g le has one fr u s tr a t e d bond and a p o rtio n у o f the hexagons has three o f them then

2 = 3 у + (1 - y)

which gives у = 1/2. The ground s ta te shown on F ig . 2 is o f t h i s kind.

The absence o f isolated ground states stro n g ly suggests th a t the i n f i ­ nite-volume c o r re la tio n s do not depend on the boundary c o n d itio n which

is in agreement with the form erly found exponential law o f decay.

In f a c t , as i t was mentioned already in the In tro d u c tio n , the s t a b i l i t y o f the set o f the ground states against boundary p e rturbatio ns is l i k e l y to be a s u f f i c i e n t and necessary c o n d itio n f o r s u p e r fr u s tr a tio n . This condition can be formulated s t i l l in another way. We say th a t the set o f the ground states is connected i f f o r any two ground states S and

1 2 n

S' there e x is ts a sequence o f ground states S ,S , . . . , S , . . . so tha t s ' E S , Sn is d if f e r e n t from Sn+' on ly at a f i n i t e number o f s ite s

( f o r any n) and Sn tends to S' pointwise ( i . e . , f o r any s i t e i there e x is ts a number n. such th a t s" = S', f o r any n > n . ) . Now I summarize a l l the speculations o f t h i s section in the fo llo w in g conjecture:

12

A model is s u p e rfru stra te d i f and only i f i t has more than one ground state and the set o f the ground states is connec­

ted.

I f the Hamiltonian is in v a ria n t with respect to the global transform ation S -+ -S then the above c o n d itio n f o r the s u p e rfru s t­

ra tio n is id e n tic a l w ith th a t conjectured by Hoever et a 1 - C10] fo r the absence o f the breakdown o f t h i s symmetry. Hence, there is a strong in d ic a tio n th a t models w ith purely p a ir in te r a c tio n s are sup erfru stra te d i f and only i f the global s p i n - f l i p symmetry is not v io la te d in the T=0 e q u ilib riu m s ta te .

However, our conjecture re fe rs also to models the Hamiltonian of which has no in te rn a l symmetry. C le a r ly , in t h i s l a t t e r case the con dition ( i v ) , given in the In tro d u c tio n , is not s a t i s f i e d . The conjecture can be used to get examples f o r models with such i n t e r ­ a c tio n s , which are l i k e l y to be s u p e rfru s tra te d . Let me remind the reader th a t the general d e f i n i t i o n o f the c o r r e la t io n le n g th y comes from the asymptotic (la rg e r ) behaviour of the "tru nca ted" c o r re la ­ tio n fu n c tio n :

<S S > - <S > <S > ^ exp ( - r / £ ) (16)

o r o r

In the example treated in S ect.2, <S > = <S > = 0 because o f the purely p a ir in te ra c tio n s and the vacuum boundary c o n d itio n ; there­

fo r e , the fin ite n e s s o f F, fo llo w s from the exponential decay of

<S S >. In the case o f a f r u s t r a t io n model w ith a Hamiltonian con- o r

ta in in g , f o r instance, an external f i e l d , the f a c t of the eventual s u p e rfr u s tra tio n can be in fe rre d from the exponential decay o f the truncated c o r r e la t io n fu n c tio n a t T=0.

Now, l e t us consider a tw o - s u b la ttic e antiferrom agnet in a c r i ­ t i c a l external f i e l d . The Hamiltonian o f such a model can be w r i t ­ ten as

13 -

H = £ S.S. -z Z S. (17)

<1J> J 1

where the f i r s t sum runs over nearest neighbour p a irs and z is the coordination number. The purely a n tiferrom a gne tic model is not f r u s ­ tra te d on the same l a t t i c e but the competition between the external f i e l d and the a n tiferrom a gne tic in te r a c tio n gives r is e to f r u s t r a ­ t i o n . Any c o n fig u ra tio n not containing negative nearest neighbour spins is a ground s ta te . The two an tife rro m a g n e tic and the a l l - p i u s ferromagnetic ground states are connected to each other by lo ca l transform ations in the sense defined above and i t is ra ther obvious th a t the whole set of the ground states is also connected. Provided t h a t our conjecture is c o r r e c t, we o b tain the fo llo w in g r e s u lt :

Any two-subl a t t i c e Ising antiferrom agnet in a c r i t i c a l exte rna l f i e l d is a model f o r t he s u p e r f r u s t r a t io n .

There i s , a t le a s t, one case in which a rigorous proof is p o ssible:

the antiferrom agnetic chain in a c r i t i c a l f i e l d . Consider the model described by the Hamiltonian

H* (1) =

Z

Z

i i

The e ff e c t o f the external f i e l d can be simulated by a ferromagnetic coupling to two neighbouring chains in which the spins are fro z e n -in a t the value +1. Let a ( i ) and ß ( i ) be the neighbours o f the s ite

i perpe ndicu la rly to the chain ; then

H(1) =

Z

Z

provided th a t S ^ . j = ^ ( i ) = 1 f o r a l l i . The magnetization in the model (18b) was studied in an e a r l i e r paper [ 11] dealing w ith the f r u s t r a t io n models o f Longa and Oles' [ 1 2 ] . I t was found t h a t f o r (18b) and, hence, f o r (18a) the magnetization a t T = 0 in the i n f i n i t e chain is

<S. > = 1//5 (19)

f o r any s i t e i . Using the re s u lts o f [ 1 1 ] , i t is easy to c a lc u la te

- 14 -

<S S > . For a f i n i t e chain containing к spins above S and £ spins

o r r

below Sq , one obtains the fo llo w in g p r o b a b ilit ie s a t T = 0:

P rob (V V 1) - nfc Пц V l / nk+ttr+1 P r o b ( S0= S r = - l ) = n k _, V ) nr _3 / nkti+ r+ 1 Prob (S0= - S r = 1) - nk_, n,, nr . 2 / nk u + r t ,

Prob(S0= - S r = - 1 ) = nk n j . , nr _2 / nk t l + r + , (20) Here n is the number o f ground sta te s in a chain of к spins i n t e r -

k ( 1)

acting via H . This number was c a lc u la te d in [1 1 ]:

2/5+4 , 1 t / 5 vk 2/5-4 , 1 - / 5 , k , , , ,

" l ' í w S "2 ' F 7 T (“ 5 -> (21) Taking the l i m i t s к Д °° in Eqs.(20) we f in d from (1 9 ), (20) and (21) that

p

<SS > - <S ><S > = ! (i11®) (22)

o r o r 5 1+/5

According to (16) t h is means th a t the c o r r e la t io n length

Z

a t T = 0, f o r the one dimensional a n tife rro m a g n e tic chain in a c r i t i c a l external f i e l d .

5. Summary

In t h is work I discussed a fa m ily o f Ising models in which the e f f e c t o f f r u s t r a t io n reaches i t s extremes. I found t h a t the systems described by these s o -c a lle d s u p e rfru stra te d models show no cooperative behaviour a t any T >_ 0 : the c o r r e la t io n length remains f i n i t e and the free energy is e s s e n tia lly t h a t of a f i n i t e system. Such models can be con­

structed in any dimension: the c o n d itio n s ( i ) - ( i v ) have to be f u l ­ f i l l e d . In p a r t i c u l a r , the example o f the KFF model shows th a t not a l l the two dimensional f u l l y fr u s tr a te d models belong to the same u n iv e r­

s a l i t y c la s s , in c o n tra s t to what was expected. This fo llo w s from the d e ta ile d c a lc u la tio n s and is r e fle c te d also by the existence of

15 -

is o la te d ground s ta te s in the TAF and SFF models and t h e i r absence in the KFF model. A q u a li t a t i v e argument suggests th a t tw o - s u b la ttic e antiferromagnets in c r i t i c a l external f i e l d s are s u p e rfru s tra te d . This has been j u s t i f i e d rig o ro u s ly f o r the one dimensional model.

Acknowledgement

I am indebted to P. Erdős f o r his c r i t i c a l reading o f the manuscript.

I have much p r o f it e d from several remarks o f G. Toulouse and from a correspondence w ith P. Fazekas and W.F. W o lff.

16

References

1. G. Toulouse, Commun. Phys.

2

3. K. Kano and S. Naya, Prog. Theor. Phys. H) 158 (1953) 4. J. V i l l a i n , J. Phys. C: Solid S t. Phys. H) 1717 (1977) 5. J. Stephenson, J. Math. Phys.

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6 . B.W. Southern, S. T. Chui and G. Forgács, J. Phys. C: S olid St. Phys.

13

7. G. Forgács, Phys. Rev. В 22 4473 (1980) 8 . A. Sütő, Helv. Phys. Acta 54 (1981) in press

9. Another exception is the G r i f f i t h s s i n g u l a r i t y (R.B. G r i f f i t h s , Phys. Rev. L e tte rs 23 17 (1969)).in random ferromagnets below the p e rc o la tio n th resho ld.

10. P. Hoever, W.F. W o lff and J. Z i t t a r t z , Z. Phys. В 4J[ 43 (1981) 11. A. Sütő, J. Phys. A: Math. Gen. T4 (1981) in press

12. L. Longa and A.M. OleS, J. Phys. A: Math. Gen.

13

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Figure Captions

Figure 1 The fru s tra te d bonds in an is o la te d ground state o f the f u l l y fr u s tr a t e d model

a) on the square l a t t i c e b) on the t r ia n g u la r l a t t i c e .

Figure 2 The fr u s tr a te d bonds in a ground sta te o f the f u l l y f r u s tr a t e d model on the Kagomé l a t t i c e .

18

Fig. la.

Fig, l b .

*

(n

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Kroó Norbert

Szakmai lektor: Forgács Gábor Nyelvi lektor: Forgács Gábor

Példányszála: 470 Törzsszám: 81-485 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1981. augusztus hó

*