phase transitions of “extended” ising models on a square lattice

Z. Physik B - Condensed Matter 41, 65-74 (1981) Condensed Zeitschrift Matter f~r Physik B

�9 Springer-Verlag 1981

Phase Transitions of "Extended" Ising Models on a Square Lattice*

J. Zittartz

Institut ftir Theoretische Physik der Universit/it, KSln, Federal Republic of Germany

Received September 22, 1980

The ferromagnetic square lattice Ising spin system is dynamically coupled to another set of Potts variables r. We show that the usual Ising phase transition is universally preserved, but the transition temperature T~ is shifted upwards. To investigate the transition and to calculate T c we use both a method by Miiller-Hartmann-Zittartz as well as a systematic expansion about the soluble Ising limit. The comparison shows that the MHZ-formula for T~ is presumably a very accurate fit to the correct transition temperature. The results are relevant for special cases of more general q-state models, for instance the Ashkin-Teller model.

I. Introduction

In this paper we investigate a class of two-dimensional models on a square lattice which one may view as "Extended" Ising models. Their definition is given in Sect. Ii in terms of usual Ising variables a and Potts variables z coupled to them [1]. These z- variables only determine the excitation energies of the Ising system and do not order themselves, it is nevertheless interesting to investigate such a coupled system, as one is interested to know how the Ising phase transition is modified. Furthermore these models occur as special cases of more general q-state models, for instance the Ashkin-Teller model [2]. It will be shown that "Extended" Ising models have their transition temperature T~ always above the Ising value T O due to the fact that the z-variables are almost free in the ordered phase, but severely restrict each other in the disordered phase. This then leads to a drastic "relative entropy reduction" in the disordered phase with the tendency to shift T~ upwards. In special cases it is even possible that the basic Ising system remains ordered for all temperatures. The transition is universally of the Ising type, i.e., it is characterized by the specific heat exponent ~ = 0. To investigate the transition and to calculate T~ quanti- tatively we use two different methods. In Sect. III we

* Work performed within the research program of the Sonder- forschungsbereich 125 Aachen-Jfilich-KSln

apply the method of Miiller-Hartmann-Zittartz (MHZ) [3]. In this method one determines the interface tension by summing only a restricted class of configurations and then calculates T c from the vanishing of the interface tension. This method which also has been applied in other cases [4, 5] is particu- larly convenient in the present case leading to a neat simple formula for T~ (3, 5). In Sect. IV we consider a soluble limit of the "Exten- ded" Ising models which is just the Ising case. We then expand the free energy systematically around this limit to fourth order in a small parameter 2. The Ising type transition is thereby shown to be preserved order by order and the shift of the transition temperature is calculated. These results are then compared with the corresponding MHZ-formula in Sect. V. In turns out that this formula is correct in the Ising limit and three further orders in 2, it only fails slightly in 24. From this we conclude that the formula is presumably numerically very accurate. Finally we construct a few phase diagrams which also should be useful in the investi- gation of more general q-state models [6]. Two Ap- pendices A and B are included in this paper in which we briefly review the solution of the Ising model and the calculation of some near neighbour Ising correlations which are needed in Sect. IV.

0340-224X/81/0041/0065/$02.00

66 J. Zittartz: "Extended" Ising Models on a Square Lattice

II. Definition of the Models

We shall consider the square lattice with ~AF sites and periodic boundary conditions. With each site we associate 2 dynamical variables, an Ising variable a~ = _+ 1 and a Potts variable z i [1] which can take on # different values (/~>2). The # states should be ar- ranged on a circle such that the variable z can be represented conveniently by the discrete angles

27r z = - - v , v=0 . . . . . # - 1 (mod27z) (2.1)

/l

and the last state is again adjacent to the first state. The interaction in the z-system, d(z, r ' ) - fl- t . K(r, -c'), shall be confined to nearest neighbor pairs (i j) , it shall be strictly positive and depend only on the angular difference. Thus

K(z, z') > Kmi n > 0 (2.2 a)

K(z, z') = K(z + z", z' + r"), (2.2b)

such that the average

K=-# 1 .~K(z ,z ' ) (2.3) z'

is independent of z. "Extended" lsing models are then defined by the Hamiltonian with the special coupling

fl'-Y{~ = 2 K('ri' z j) (1 -- al a j) (2.4) (i j )

and its partition function

z = ~ e -p~. (2.5) {a, ~}

The general model (2.4) contains as the simplest case the usual ferromagnetic Ising model for a z-independent interaction, K(z,z')-=K in (2.3). Its well- known solution [7] is briefly summarized in Appen- dix A. The z-variables then play no dynamical role giving rise only to the multiplicity factor /~w in the partition function. The phase transition of the Ising model from a 2-fold degenerate, ferromagnetically ordered low temperature phase to the unique, i.e., non-degenerate, disordered high temperature phase occurs at the critical temperature T O = K o t (A.15)

sinh 2K o = 1, (2.6)

and the singular part of the free energy per site is given by (A.18)

fif~ t ~ O (2.7)

with

K = Ko + t. (2.8)

Later on we shall be interested in two special cases of the general model (2.4).

Case a. The interaction is given by

{K~ z - r ' = 2 1 z (0 . . . . . v - l )

# (2.9) K(r, z') = otherwise

(1 <v<~t). For a = - a ' in (2.4) any r is thus restricted to just v of all # possibilities by the z' at a nearest neighbour site, while there is no restriction for a = a'.

Case b. The interaction is precisely that of the standard Potts model [1]

K(z, z')= ( K , - K 0 6~,~, + K I (2.10)

with two positive couplings K 1 and K u. The corresponding Hamiltonian (2.4) is then also a particular case of more general q-state models considered for instance in Ref. 6, 8. For /~=2 we obtain a special case of the Ashkin-Teller model [2,9] which can be defined by

fl~C~AW = Z I[(K1 + K 2 + K 3 ) - ( K 1 + K 2 -K3)ao- '

- ( K t + K a - K z ) z z ' - ( K 3 + K 2 - K I ) . a a ' z z ' ] (2.11)

with 3 positive couplings for two interacting Ising systems: a=___l, z=_+l . In the case K 3 = 0 (2.11) reduces to (2.4) with (2.10) and K,-~K2, as one may check. In the following we shall be mainly interested in the phase transition of the general "Extended" Ising model (2.2, 4) which is expected to be universally of the Ising type (2.7). The Ising variable a and the Potts variable dynamically play quite different roles. If a~ = a j for nearest neighbours, the corresponding z- variables are independent of each other; if a i=-c ry , however, then the z-variables mutually influence each other as they determine the structure of the a-excitation energies. One therefore can expect that only the Ising system orders at low temperatures with a spon- taneous breakdown of the global a--*-rr symmetry of the Hamiltonian. For in the a-ordered phase the typical configuration will mostly have a i = aj on nearest neighbour sites such that the z's are overwhelm- ingly independent of each other and thus disordered. In the a-disordered phase one has both al = aj and a~ =--Cry almost equally distributed in a typical configuration, which again implies that the r's are glo- bally disordered. However, now the mutual restrictions of the z-variables via K(z,z') are much more nu- merous. The multiplicity of a typical configuration which in the a-ordered phase is close to its possible maximum #x is then substantially reduced. This also means that the total entropy increases much slower

J. Zittartz: "Extended" Ising Models on a Square Lattice 67

with increasing temperature when compared with the case of the Ising model without z-variables. This effect of "relative entropy reduction" due to the decreasing entropy of the r-system will be seen to be most drastic in case-a-models where some excitation energies are infinite. Therefore, as we only expect the a-system to order at low temperatures, the phase transition should be universally the same as (2.7)

General: ]~flsi~-~ Ct2 ln[t], t - * 0 . (2.12)

Only the transition temperature To= T~({K(~, z')} will be shifted relative to the Ising case (2.6) and also the amplitude C in (2.12) will depend on the interaction function K(z,-c'), as will be shown below. We now shall argue that in fact T~ satisfies the inequality

T~__> T o = K o I. (2.13)

First we observe that the Hamiltonian (2.4) is bound- ed from below by

/~]~J'dt'Cmin~--- S Kmin(1--aiaj), (2.14) <ij>

using (2.2a). It means that the excitation energies of all a-configurations are larger or equal than the energies of the pure Ising system ~m~n" T~ then should be larger than the Ising T O because the destruction of order requires more energy, or equivalently T~ can only be shifted upwards because of the effect of "relative entropy reduction". In fact we shall find below by an approximate calculation which, however, we believe to be qualitatively correct, that for the case-a-models T~ can even be shifted to infinity. The general model (2.4) presumably cannot be solved exactly. Therefore we shall attack the general problem by two different methods. In the next section, we first shall derive a formula for T C by an approximate calculation of the interracial tension using the MH Z method [3] which also has been applied in other cases [4, 5]. In Sect. IV we shall assume that the interaction K(q {) deviates from its average (2.3) only by a small quantity. In that case a small parameter expansion around the soluble Ising limit is possible. By expanding to fourth order we shall in fact verify the universality assertion (2.12) to this order and also determine the corresponding T~ which is then compared with the result of the M H Z method.

III. Transition Temperature from the Interfacial Tension

In [-3] a simple method has been developed to determine interracial tension and transition temperature

~=+1 9 11

f " . . . . I

8 10 12 I ,, I

1 7 " 1 3 [ " ;]6 - ; . . . . .

l.

c~ =-I

o

GiG

| L - - J

I

Fig. 1. Typical interface configuration. The broken line denotes the interface separating the up-spin phase from the down-spin phase. Lattice sites along the interface are numbered and denoted by the dots

of square lattice Ising models. In this method one sums only over a restricted ensemble of interface configurations and one needs not consider the more complicated bulk problem. The application to the Ising antiferromagnet in an external magnetic field produced a very simple formula for T~ which, however, is not quite exact as is now known [10]. Numeri- cally the formula is presumably fairly precise, as Monte Carlo calculations [11, 12] as well as renor- malization group [13] confirmed the formula within their limits of accuracy, only recent series expansions found discrepancies [14, 15]. The "Extended" Ising models of this paper are very convenient for an application of the MH Z method. Consider Fig. 1. The two extremal low temperature phases of our model (all a-spins up or all down) are assumed to coexist with one another along an interface. The interface is allowed to pass any column at arbitrary height, h = 0, _+ 1, + 2, . . . , but only once. This means that back bending is not allowed. The tension, i.e. interface free energy, follows from the partition function Zi, t. One just needs the excitation energies across the interface bonds. In [3] we used the one-dimensional transfer matrix method. Howev- er, for ferromagnetic models where lattice symmetries are not broken, the situation is even simpler. As all columns in Fig. 1 are equivalent, they must contribute the same to the partition function. Thus we can write

/~f~t = - l n Z o , (3.1)

where Z 0 is the column partition function. For the Ising model with excitation energy 2K each column contributes a Boltzmann factor co = e - z K for each bond of that part of the interface which must be attributed to this particular column. Clearly we have one factor for the one horizontal bond and CO Ih[ if the height difference between the column and the pre- ceding one is h: h = 0, _ 1, + 2, .... Thus we must sum

68 J. Zinartz: "Extended" Ising Models on a Square Lattice

f i f i~ t=_ l n ~ col+lal=ln__l-co (3.2) h= - ~o co(1 + co)"

This is the exact result of the Ising model [7]. The transition temperature follows from f = 0 [3] which leads to

coo =e-2K~ = 1 / 2 - 1 , (3.3)

equivalent to (2.6). Now we turn to the "Extended" Ising models (2.4). As all a-spins are up or down in the bulk portions of the lattice (Fig. 1) the corresponding "c-variables are free, i.e., independent of each other, as we discussed before. Only the >variables along the interface in- teract across the bond with energy 2K(r, z') or Boltz- mann factor co(z ,Q=e -2K"'~'). The mutual restriction, however, can be removed along the whole interface, if we associate an average Boltzmann weight

co=(e 2K<*'*')) = # - 1 - ~ e - 2K(*'*') (3.4)

with each interface bond and treat all z-variables as being free, i.e. independent of each other. To show this we start with % at position 1 in Fig. 1. It interacts only with %, otherwise being free. Its contribution to the interface partition function would be #. co. However, associating the same factor co with the bond and treating z~ as if it would be completely free is clearly equivalent. The next z 2 is then free from z~ and interacts only with %. "c a is then treated as % before. Then we consider z 3, z~, and so on. As we go along the interface one z after another is then freed by associating co factors with the bonds. The method works because there is no backbending. The situation is then the same as in the above Ising case. We have co-bonds such that (3.4) replaces co = e - 2 ~ before and free z-variables as if there would have been no interface at all. The tension corre- sponds to (3.2) and its vanishing determines the transition temperature of our models:

coo - 1/~ - 1 = ( e - z K(~, ~'))~ = co~. (3.5)

We shall see later that this is presumably a very accurate approximation. Moreover equation (3.2) with co according to (3.4) shows that the interface tension varies linearly in co-co~, or (T-T~), near the transition. As scaling predicts in two dimensions that the critical exponent should be (2-~)/2, we recover the universal specific heat exponent e = 0. Therefore the above results are consistent with formula (2.12). Let us apply formula (3.5) to the particular case-a- models. With (2.9) we obtain

cG~-e 2K,~=~- ( ]~- 1) (3.6) V

where l < v < # is possible. As # / v > l we see that T~ =KTc I is always larger than the Ising T O which confirms (2.13). If, however, the right hand side in (3.6) is larger than unity, then (3.6) cannot be satisfied with any finite T~ as K u > 0 implies cou~ 1. In that

case, namely # / v > ] ~ + 1 = 2 . 4 1 .... the system remains ordered up to infinite temperature. The reason for this to happen has been discussed before. It is due to the drastic "relative entropy reduction" in the z- system when comparing a-ordered with a-disordered configurations. Discussion of the case-b-models is postponed to Sect. V.

IV. Expansion around a Soluble Limit

If a problem cannot be solved exactly, it has always been a good principle to look for a small parameter in which one can expand. We thus assume that the interaction (2.2) can be written as

K(~, z') = K + W(z, ~'), W= 0(2) (4.1)

such that the deviation W from the average (2.3) is small of order 2 which will be our expansion parameter. The Hamiltonian (2.4) is then

f iYf = 2 Y K - K ~ ai aj + ~ W(z i, z ) (1 - a~ a). (i j ) ~ij)

(4.2)

To lowest order we have the Ising model whose solution is summarized in Appendix A with the free energy f o given by (A.13). As will be seen, the corrections to f o resulting from the expansion in W all contain nearest neighbour a-correlations (Appendix B) which at the transition point T o develop logarith- mic singularities. When combined with the zeroth order result (2.7), the transition temperature is shifted in every order 2. In order to avoid such a somewhat cumbersome procedure we can use a little trick which also has been applied recently in a similar problem [-10]. We add and subtract a small term 3a~ , j in (4.2) and set

t~=_K + 3 = K o +t (4.3)

such that 24 ~ (apart from the unimportant constant 2 Y K which we drop) is given by

P~=Wo+~, ~o = - R ~ a~a~ 

Y = ~ [W(ri, r ) (1 - al a ) + 6a i aj] (4.4) (i j )

with the new perturbation ~ . 3 is then adjusted in such a way that the corrections due to ~ no longer

J. Zittartz: "Extended" Ising Models on a Square Lattice 69

cause a shift of T,.. This means that Jfo already has its transition at the corrected T. which is then determined from (4.3) as t ~ 0 :

K~ = K o - "5. (4.5)

(3 will turn out to be of order 22 while the first part of the perturbation W is of order 2. The expansion of the free energy per site follows from the partition function and is given in terms of cu- mulant averages [16]

n f = n s o + ; 1 , _ ~ v! <Yf )~,m (4.6)

where f ~ is (A.13) with K replaced by K (4.3) and the averages in (4.6) are performed with respect to the Ising Hamiltonian 24Po. In order not to indulge in formidable diagrammatic calculations, we shall investigate the expansion only to fourth order in ). at which order a clarifying and decisive comparison with the Tcformula of the last section can be made. To the first order we have

~A#- t < ~ > = 2 [< W~j>~~ + (3 @,j>~]. (4.7)

Because of translational invariance we may take the average for some fixed nearest neighbour pair <i j}, which henceforth will be denoted by <01}, the sum over bonds gives a factor 2~2. We shall also use the obvious abbreviations W~j and cr~j~ a~ c;j. The averages in (4.7) and in following formulas all split into products of a- and z-averages. As d,Y 0 is independent of ~, the z-average in (4.7) is simply

~ 1 - <W01>~=/~- 1 ~ W(%, r 0 = 0 (4.8)

by using (4.1) and (2.3). Similarly we define the quantities

G -~ < WePt} = 0(2") (4.9)

which will appear below, and we have indicated the k-order. The a-averages which we shall need are just Ising correlation functions for products of nearby spins which are calculated in Appendix B. We shall need the nearest neighbour correlation (B.5)

g-= @o al> -= @o1> (4.10)

(for an illustration see Fig. 2), the next-nearest neighbour correlation (B.7)

g 2 = < a o a 2 } (4.11)

and the plaquette average (B.9)

p = <ao al a2 a3>. (4.12)

Fig. 2. Plaquette with its four spins

Furthermore the quantity

6g 0 K = ~ <a~ g)}, (4.13)

 =26g. (4.14)

As the term of order 2 from <W> vanishes, the quantity (3 which is still to be determined must be of order 22 .

2. order:

<~2}cum = <~2> __ <~r 2. (4.15)

The first part is then given by

y 1<~25 =2 }2 <Wo~ ~j(1-aoO(1-%) . (4.16)

The second term in (4.16) gives zero (4.8). The first term also gives zero unless <ij} coincides with <01} because of (4.8) again. The last term is combined with the second term in (4.15) and we get finally

zV I (J-f2>cum = 4cq(1 - g) + 2(32 ~ _ (4.17)

by using (4.13), (4.9), and the simple relation

(1 - aol)" =2"- 1(1 - a01). (4.18)

3. order:

<d~3>oum = <2~3) -- 3 < ~ 2 ) <24~) + 2 <2~)3. (4.19)

As we are interested only in terms up to order 24 and (3 = 0(22), the last term is not needed. The first term is then

Y - I < Y f 3 > =2 ~ < Wo 1W/j VVk,(1 - aol) , <kl>

�9 (1 -- ai) (1 -- akz ) + 3 (3 W o 1 Wij(1 - aol ) (1 - ~ij) crkt>.

(4.20)

To be non zero its first part requires ( 0 1 ) = ( i j ) =(kl) because of (4.8). The second part requires ( 0 1 ) = ( i j ) for the same reason. This is then combined with the 2. term in (4.19) which follows from (4.14, 16). Adding together and using listed formulas one gets

ag (4.21) ~/ ' - l ( ~ 3 > c u m = 8~ --g) -- 12a0(2 c3i~'

4. order:

( j ~ 4 ) e u m = ( j ~ 4 > __ 4 ( j / ? 3 > ( j ~ > __ 3 (j.~2 > 2

+ 12 (~;r (,/~)2 _ 6 ( ~ ) 4. (4.22)

The 2., 4., 5. terms are not needed being of order 25 , 26, 2 s, respectively. The 3. term is from (4.17)

- 3 Jr ' -1 ( R e ) 2 = _ 24 0(e z ~. (1 - g)2, (4.23) (i j )

while the first term reduces to

~/ ' - 1 ( ~ 4 ) =2 ~ (WolWijWkt Wren (i j ) , (M) , (rnn)

�9 (1 - a o 1) (1 - oij ) (1 - %t)(1 - am,)) (4.24)

The W-average gives three different contributions. First, if all 4 bonds coincide, the contribution to (4.24) is

16(z4(1 --g). (4.25)

Secondly, if the four bonds coincide in pairs but are not all the same we get

24 0(~ ~, ( ( 1 - a o 1 ) (1 - ao . ) ) . ( 4 . 2 6 ) 

Combined with (4.23) this is equal to

2 ~g 2 24~2 ~-~-- 48~2(1 --g). (4.27)

The only remaining possibility for a non vanishing r- average in (4.25) is the case where the four bonds form a plaquette (Fig. 2). There are 12 possibilities with one fixed bond (01). Defining

(4.28)

for an average around the plaquette and calculating

( (1 - a o l ) ( 1 - ~12)(1 - a23)(1 - % o ) )

= 2 1 1 - 4 g + 2 g 2+p] (4.29)

in terms of g, g2, and p (4.10, 11, 12), we then obtain the third contribution to (4.24):

4864(1 -- 4g + 2g z +p). (4.30)

Adding (4.25, 27, 30) we have altogether

~/~-1 (z474)oum = 8 [(1 -- g)(2 ~4 -- 6 0(2) I -

+ 30(~ ~ - -+ 6 ~4(1-4g + 2g 2 + p)]. (4.31)

In order to obtain the expansion (4.6), correct to order Z 4, we now must add (4.14, 17, 21, 31) with the appropriate prefactors leading to

f = 2 [(6 + 0(2-- 20(3 + ~0(r ~2) g-- ~4(p + 2g2 -- 4g)]

2 ~g 2 1 - ( 6 + % ) ~ + 2 [ - % + 5 0 ( 3 -50(~+0(~ - ~ J +0(25).

(4.32) The coefficients 0(,(K) (4.9) arc via (4.3) analytic in t. However, the correlations g, g2, and p (B.5, 7, 9) all would contribute a tlnltl-term via b(t) (A.22), and Og/OI~ even produces a lnltl near the transition�9 Such terms when combined with f o (A.18) would shift the transition temperature away from t = 0. However, we now choose the parameter 6 in such a way that the prefactor of b(t) in the first bracket of (4.32) vanishes which then also eliminates the c?g/~?k-term in (4.32) to the required order. All remaining terms in f at most have a t 2 lnlt]-singularity (see Appendix B) which can be combined with f o in (4.6) leading to (2.12) with some correction of the amplitude. The condition fixing 6 is easily obtained:

6 : __ 0(2 ~_2(~3 __ 10(4_t_ 2 1 ~2 - g ~ ~ + 0(25) (4.33)

with the number

7 = 8 I-1 - 1/~(1 - ~z- 1)] = 0.288 .... (4.34)

We have thus seen that the "Extended" Ising models (2.4) preserve the transition of the Ising model (2.7) universally. This has been checked by the foregoing calculation to fourth order in an expansion around the soluble Ising limit. Presumably the result could be extended to every order of perturbation theory. The transition temperatute follows from (4.5) and (4.33). A comparison with the result of Sect. III will be done in the next section.

V. Trans i t ion T e m p e r a t u r e s and E x a m p l e s

We shall now compare the MHZ formula (3.5) for the transition temperature with the correct expansion to order 2 4 of the last section. If W is small of order 2, we use (4.1) in (3.4) and expand to order 24 . Compar- ing this expansion with the expansion of the quantity e -2(K+a) up to order 24 we can identify

e- 2(K+a) = CO- e~4 + 0(25) = co [1 + 7 ~ + O(2S)]. (5.1)

J. zittartz: "Extended" Ising Models on a Square Lattice 71

The transition is then given by (3.5) and (4.5), respectively:

O)o=] /2_1 =/'co ~ MHZ (5.2) co~(1 +7 ~)~+ O(;P) exact "

The general result found by the M H Z interface method thus reproduces the expansion remarkably well, it fits all orders 2 ~ 21, 22, 2 3 correctly and only fails in fourth order by the small term 7 ~ . ~4 is given by (4.28) and can also be written as

g4 = (K01' Kt2" K23" K30) -K4- (5.3)

This plaquette average is in fact the first contribution of a closed lattice polygon besides tree-like diagrams. This throws some light on what terms are missing when one applies the interface method of Sect. III. From (5.2) we obtain a second (upper) bound on the exact T C as 2--*0 by observing that 3,~r (5.3) is positive which, together with the lower bound (2.13), implies

T o <= T~ < T Mnz. (5.4)

We suggest that also the upper bound holds general- ly, not only for 2 ~ 0, Let us now consider the special case-b-models (2.10) with their standard Potts-type excitation energies. The average of (2.10) is then

K =(K.- K,). #-t + G (5.5)

which with 2 = K , - K 1 leads to

W(~, ~')= Z(6~,~,-# ~), (5.6)

# - 1 ~ = 7 s - . 2or (5.7)

The average Boltzmann weight (3.4) becomes

co = # - t [CO,-F ( # - 1) call, co~=-e -2K~ . (5.8)

The transition according to (5.2) is now given by

(#_l)cot~+co,c=#coo[ 1_T~4~24+#-1 O(2S)] �9 (5.9)

In Figs. 3 and 4 we show phase diagrams in the co D c%-plane (0< co~ < 1). The thermodynamic path (from T = 0 to T = oo) of the soluble Ising case (2 = 0) is just the diagonal cot=co . and the point S denotes the transition at c o o = l / 2 - 1 . In Fig. 3 we have plotted the transition lines according to the MHZ-formula, i.e. without 24-corrections in (5.9). These are all straight lines. Explicitly given are the cases # = 2, 3, 5, 10 which for # -* oo would accumulate at the broken line. The exact curves would presumably be very close to these straight lines smoothly joining them at

COl "=CO

'1 I I I I I ' I I I /

/ 2 /

disorderT, phase

5 S / lo . . . . . . } / _ . . . . . . .

10 / / / " 5

/ , \ , ,. , , , \ , T=0 co o 2 uJ!a

Fig. 3. co t, (G-phase diagram. The ordered phases are in the lower left part; the straight lines are the transition lines. Numbers on the lines denote the number/~ of possible states for the z-variable

I I I I I I I

";QQ

/ \ /MHZ /

dis.~d, ered phase

I I I I ~ I I I ~ h . I

T=0 coo 0.5 Fig. 4. o01, co2-phase diagram

03 2

the point S with a 4. order slope (as the correction in (5.9) is of order 24). Our results predict that at c o l = 0 ( K l = o o ) no transition takes place as temperature (or cG) increases, i.e. the a-system remains ordered, for all # > 3. Thus only # = 2, i.e. the special AT-model, is an exception. The latter case is again illustrated in Fig. 4 which also contains a qualitative plot of the exact curve. De- viations of the two curves should always be only a few percent, as experience has shown in the case of the Ising antiferromagnet in an external field [6, 12, 13].

Part of this work has been done while I visited the Institute for Theoretical Physics, Santa Barbara, Ca. I would like to thank its Director, Prof. W. Kohn, for his kind hospitallity. I also would like to thank E. Miiller-Hartmann and D. Stauffer for very helpful discussions.

Appendix A. Ising-Model

We briefly summarize the solution of the Ising-model by the transfer matrix method and using the Jordan- Wigner representation of spins by "free fermions" which is well-known from the literature [17] (see also [18] for a recent application). We consider a square lattice of M rows and N columns with JU = N - M sites closed on a torus. For the Ising Hamiltonian (K =fiJ)

13~gf=-K ~ aiaj (A.1) 

the partition function can be written as

Z =~, e # ~ =(2x) w/2 Trace(T 1 �9 T2) M (A.2) {~}

with

N

T~ =e ~=, , T e e, ....... = e n = z (A.3)

3 and the diagonal % where the spin flip operator o-, are Pauli matrices in standard notation. The operator

. t = l f ~ - . Ta,,~ (A.4) e l ( q - e - l( (r n

with

x -= sinh 2K = (sinh 2L)- 1 (A.5)

transfers "interaction" from any configuration of spins ( ~ = • in one row to any other spin configuration in the next row, it thus takes care for the interaction along vertical bonds. The operator T 2 measures the interaction within a row along the horizontal bonds. Using the Fermion representation

1 = 2 C + C n _ _ l ' 3 ~. c+c + G - G = ( - 1 ) ~ < 2 "(c, +G) (A.6)

one obtains for the exponential operators in (A.3) and Fourier transforming

2cr~ = 2 2r3, Z~2a3+l =2 2 (z3c~ sinq) n q>0 n q > 0

(A.7)

with

z 2 = c + c . + c + _ ~ c , - 1 ,

= iz~ ~ (A.8) =i(c + c+~+qc_~), z~ ~ , "Cq

and q ranges in ( - ~ , ~z) with spacing 2~z/N. The symmetrized operator T= T(/2. T z. T 1/2 (instead of T 1 . T 2 in (A.2)) can then be written as

T= ~ T(q)= U. ~[ eE(q)'~ . U , (A.9) q > 0 q>0

with the unitary operator

U = l~ e-~ a~3 (A.1 O) q>O

and

coshE(q)=x+x-l+cosq, E>O

sinhE.~sinaq ~x sin q �9 (A.11) (cos aq = [ 1]/ i~x ~ + ] ~ + x 2 cos q

To derive (A.9) from (A.3, 7) one observes that T(q) gives 1 when applied to the two states of single occupancy (either q or - q occupied) where zq3 =0. This checks with the right hand side in (A.9). In the subspace of zero plus double occupancy for q and -q( 'c3= • 1) the >matrices (A.8) are Pauli matrices and (A.9) is derived by using e~=cosh~+'c.sinhc~ (for each "c I, z 2, ~3). In the thermodynamic limit M, N ~oo the partition function (A.2) is determined by the largest eigenvalue of T I �9 T 2 or equivalently of T

TlO0) = e- uy i~to ) (a.12)

such that the free energy per site is

1 ~ fif~189 f = - � 8 9 dqE(q). (A.13)

The eigenstate of largest eigenvalue for (A.9) is obviously

I~o}=uI0}, ~t0}=10) (allq), (A.14)

the eigenvalue then leads to the right hand side expression in (A.13). At the phase transition the free energy develops a singularity in temperature, i.e., in K (or x). As cosh E (A.11) is both analytic in x and q, a singularity can only result from the inversion of coshE to obtain E, which determines the transition point: E( G, Ko)=0. Obviously (A.1t) this point ist:

q~ = ~, x o = sinh2K o = 1. (A.15)

Expanding E near the transition point with

K=Ko+t , t--,O (A.16)

we have

E(q) ~- ]fl~t 2 + (q - g)2 , (a.17)

Using this in (A.13) the singular part o f f follows at o n c e

fifols~g_4 t2 in IG (A.18)

J. Zittartz: "Extended" Ising Models on a Square Lattice

Expectation values of >operators within I~o} are:

<z2} = 0, @qa} = cos aq, <zq ~ } = sin aq, (1.19)

leading to

<c~ cq}=<c+qc q>=�89 +cosaq),

i <Cq C+q)=(CqC q)= --~sinaq. (A.20)

Near t =0 these quantities are (from (A.11))

sin ; = [16 t 2 q- (1 + cos q) (3 + cos q)]- 1/2

aq

COS aq)

"I sin q [1 + 2 l / 2 t +O(t2)]

[ - 4 t + ( 1 +cosq)(]/~+2t)+O(t2)" (A.21)

Denoting q-averages over (--~,~) by a bar, we have

(t o is const.)

- (sinh E(q))- * = 1 in t_ _= b(t), (A.22) ro

cos aq = ] / 2 -1 + 4t b(t) + t + O(t z lnt)

(1 + cos q). cos aq = 2 ] / 2 (1 +]/2 t) + O(t 3 in t) 7C

s i n q - s i n a q = ( 1 - - 2 ) ( l + 2 ] / 2 t ) + O ( t 2 1 n t ) . (1.23)

Appendix B. Near Neighbour Correlations

Near neighbours are shown in Fig. 2 of Sect. IV. To obtain the near neighbour correlations we observe that in the transfer matrix formulation a vertical pair (such as a o 0- 3 in Fig. 2) changes the transfer (A.4) at column index n between the two rows to the quantity

eK--e-K 17nI=(eK§ ' ( ~ - ' } - X - 2 - - X - i 171). (B . I )

In 17-averages we thus have to insert

17o % ~ 1 / 1 + x - z - x -1 171 (B.2)

at the appropriate place between two symmetrized operators T (A.9). Similarly a horizontal pair 17o 17, is replaced by

~-1/2 3 3 .T)/= (B.3) 170 a l ~ 11 Gn an+ 1

at column index n by observing that 0 .3 does not commute with T~.

73

Using (B.2) the nearest neighbour correlation g = <17o 171} = <% %} is expressed as

g = < ~ , o 1 r x - 2 - x - 1 a.'l~,o>. (B.4)

In terms of fermion operators (A.6) this is easily calculated

g = ] / l q - x - 2 - - x -1 c o s a q

~_ l ~ - 1 _ 4t b(t) - t + O(t 2 In t), (B.5)

where (A.6, 8, 19) and translational invariance have been used. The small t-expansion follow from (A.5, 16) and (A.23). The next nearest neighbour correlation gz=<a l 173} =@o17317oa1} (Fig. 2) is with (B.2, 3) expressed as

-1 /2 3 3 . [ 1 / l + x - 2 g2=<~'01L a~ o-.+~ - x - 1 o~]. r11/=10o> =(1--e -4~) l<00l[a~_ie_2Ka2]

.[a L ._2K 2 l+{e o'.+ i ] l@o}. (B.6)

Again using (A.6, 8, 19) and translational invariance this leads to

g2 = l / i + x - 2 cos q. cos aq + sin q- sin aq (B.7 a)

with its small t-expansion following from (A.23)

2 g 2 ~ - - - 4 ] / 2 t . b ( t ) + 2 1 / 2 1-- t+O(t~lnt).

7~

(B.7b)

To calculate the plaquette average p =<(a0a3)(%a2)} (Fig. 2) one uses (B.2) twice at column number n and n+ 1, but between the same r o w s :

_ _ X - 1 1 ~n+ ,] I~'O>" (B.s)

In terms of Fermion operators (A.6), then calculating all Fermion contractions and using (A.20), one obtains:

p = l + x - Z - 2 x - l l f f l + x - 2 cosaq

X - 2 [-(1 -- COS q) COS aq" (1 + COS q) COS aq

+sin q. sin a,t2]. (B.9a)

Its small t-expansion follows from (A.23):

4 p = l - - - - - 8 1 - t[b(t)+rt ~]+O(tZlnt). T~ 2

(B.9b)


References

1. Potts, P.B.: Proc. Camb. Phil. Soc. 48, 106 (1952) 2. Ashkin, J., Teller, E.: Phys, Rev. 64, 178 (1943) 3. Miiller-Hartmann, E., Zittartz, J.: Z. Phys. B27, 261 (1977) 4. Burkhardt, T.W.: Z. Phys. B29, 129 (1978) 5. Lin, K.Y., Wu, F.Y.: Z. Phys. B33, 181 (1979) 6. Zittartz, J.: (to be published) 7. Onsager, L.: Phys. Rev. 65, 117 (1944) 8. Domany, E., Riedel, E.K.: Phys. Rev. BI9, 5817 (1979) 9. Wu, F.Y.: In: Statphysics 13. Annals of the Israel Phys. Soc. 2

(1), 1978 page 371 10. Zittartz, J.: Z. Phys. B40, 233 (1980) 11. Rapaport, D.C.: Phys. Lett. 65A, 147 (1978) 12. Binder, K., Landau, D.P.: Phys. Rev. B21, 1941 (1980) 13. Schuh, B.: Z. Phys. B31, 55 (1978)

14. Baxter, R.J., Enting, I.G., Tsang, S.K.: J. Stat. Phys. 22, 465 (1980)

15. Racz, Z.: (to be published) 16. Kubo, R.: J. Phys. Soc. Jpn. 17, 1100 (1962) 17. Schultz, T.D., Mattis, D.C., Lieb, E.H.: Rev. Mod. Phys. 36,

856 (1964); Mattis, D.C.: The theory of magnetism. New York: Harper & Row 1965

18. Hoever, P., Wolff, W.F., Zittartz, J.: Z. Phys. B41, 43 (1981)

J. Zittartz Institut ftir Theoretische Physik Universitgt zu K51n ZiJlpicher Strage 77 D-5000 K61n 41 Federal Republic of Germany

phase transitions of “extended” ising models on a square lattice

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