The specific heat of a general two-dimensional Ising model

Physica A 178 (1991) 551-560 North-Holland

The specific heat of a general two-dimensional Ising model A. Aguilar and E. Braun’ Departamento de Fisica, Universidad Authoma Metropolitana Iztapalapa, Apartado Postal 55-534, Mexico, D. F., C. P. 09340, Mexico

Received 2 March 1991 We analyze the specific heat for the class of models treated in a previous paper. It is shown that if a certain condition is fulfilled (eq. (3)), then the specific heat diverges logarithmically. This fact is established without recourse to the details of the interactions between the spins; thus, we show that the universality of the singularity holds for this class of models. The equations for the transition temperatures are presented for the models treated specifically in a previous paper.

1. Introduction In a previous paper [l], to be referred to as I, we treated a generalization of the two-dimensional Ising model which consists of a unit cell (UC) that is repeated throughout the plane. We were able to solve the general problem of obtaining the partition function per spin in the infinite limit, as a quadrature for any arbitrary cell UC, i.e., for any number of spins with nearest neighbours interactions and for arbitrary values of these interactions. Some of the previously reported models were the cases of two [2], three [3] and four [4] different energy parameters; these models are particular cases of the general model under study. Extensive reviews of the works presented in the literature are surveyed, for example, in refs [5,6]. Onsager obtained for the first time the exact specific heat, in the infinite limit, for an UC that contains only two different energy parameters. He showed that there is a particular temperature, in the neighborhood of which there is a logarithmic divergence of its value. The other authors mentioned also obtained this logarithmic divergence for the specific heat, in the models they explicitly solved. Since then, it has been assumed, following perhaps the universality of the phase transition hypothesis, that any two-dimensional model would exhibit a ’ Also at Facultad de Ciencias, UNAM, 04510 Mexico, D.F., Mexico. 0378-4371/91/$03.50 (Q 1991- Elsevier Science Publishers B.V. (North-Holland)

552

A. Aguilar,

E. Braun

i Specific heat of a general two-dimensional Ising model

logarithmic divergence in its specific heat. However, as far as both of us are aware, neither the universality of the phase transition hypothesis nor the divergence for other models in two dimensions have been explicitly demonstrated in general, in an exact way. It is the purpose of this paper to show that for the class of general Ising models treated in I if a certain condition is fulfilled (see eq. (3)), then the specific heat diverges logarithmically. Furthermore, this fact is obtained independently of the details of the interactions, which means that this is valid for any UC without regard for the particular value of the energy interactions, which in I were called eij and e:,. Therefore, in this way we can conclude that the universality of the logarithmic singularity for the specific heat indeed holds. In section 2 we start from (1.21), which gives the general partition function per spin, and show that for any set of the 9 values (defined in I), i.e., for any UC, if the function F(x, y, p) (see eq. (3) below) has a zero value, then necessarily the specific heat has a logarithmic divergence. This is done by examining the analytic properties of the function F(x, y, j3). In section 3 we exhibit the equations that the transition temperature T,, the temperature for which condition (3) is fulfilled, has to satisfy for some models treated in section 4 of I. In section 4 we make some final comments.

2. The behaviour

of c(T) near the critical temperature

In this section we study the behavior of the specific heat corresponding to the general case of the UC model. We will start from the general expression obtained in I for the partition function power spin L(T). We recall that L(T) is given in terms of certain !& quantities defined in I. Let us combine the two terms in eq. (1.21) to obtain the partition function L(T) in the form

L(T) = n+m lim -g?r’,,, :I

iln

F/AY? P) dY

’

(1)

0

The function F,(y, /3) can be read off from eq. (1.21) and we will not write it explicitly. It is easy to see that Fk( y, /3) has the following general properties. (a) From inspection of the procedure followed in I to obtain the q, k quantities one can assert that each q,k is a linear combination df exp[in(2k - 1) ln], exp[-im(2k - 1) ln] and powers of these exponentials. (b) The function F,(y, p) is a real positive definite quantity that is a

A. Aguilar,

E. Braun

I Specific heat of a general two-dimensional Ising model

5.53

function of sinh(2/3e,,), cosh(2pe,,), sinh(2Pe&,), cosh(2pe&), its products and its powers. (c) For any finite value of n, F,(y, p) never attains the zero value. A necessary condition for Fk( y, p) to become zero is that n+a. Taking this limit we get for L(T) the following expression:

L(T)

=

2.x

2a

0

0

dy In F(x,

+

167r qt

with Fk

Y,

Y, P)

P) = ,@e F,(Y, P>.

(2)

Thus, in the infinite limit, F(x, y, j3) is a positive semi-definite function. (d) In view of the properties (a) and (b) we can state that the function F(x, y, /3) is analytic in all its three variables. We will now show that if there exists some point (x,, y,, p,) in the domain of F(x, y, p) in the (x, y, p) space, such that

FCC,Y,, Pc)=O

(3)

7

then necessarily we obtain a logarithmic dependence in the specific heat c(T) near the “critical temperature” T, = 1 /k&. As a matter of fact, let us consider the following domain for the variables: (4) where all the E quantities are very small. Because F(x, y, p) is analytic one can expand it as a double Taylor series around x = x, and y = y,,

F(x, Y, P)=P(P)+

t~,,(PHx-x,)~+ ~P,,@)(Y

-Y,)'+

.a.

(5)

No linear terms in x and y appear due to the fact that F(x, y, p) is evaluated near a minimum. Here the indices x and y mean partial derivatives with respect to x and y, respectively. We have written p(p) = F(xc, y,, p), and

PII(P)

= (

a2F(xY, P) a’,2

),=,,,

y=y, 9

We have chosen the values of the different E’S in eq. (4) such that the first three terms in eq. (5) are the dominant ones.

554

The

A. Aguilar,

E. Braun

fact

F(x, y, p)

that

I Specific heat of a general two-dimensional king model

is a positive

semi-definite

function

implies

the

following: (i) The minimum p = p, where

of F(x, y, /!I) occurs

& is given

as the solution

necessarily

at x = x,, y = y, and at

to eq. (3). Therefore

A prime indicates a derivative with respect to p. (ii) In the neighborhood of th e minimum the function p( fl) has, necessarily, the form shown in fig. 1, i.e., it has an upward curvature, which means that

By the same

token

and all the second

derivatives

Let us now calculate gets

of p are positive,

the internal

energy

aP The specific

heat c(T)

1

16rr2qt is then

Fig. 1. In the space (x, y, p), if condition and upper curvature near this point.

F’(x, 0 given

Using

eq. (1) one

277

2T

u(T) = - -aL(T) = - ~

per spin u(T).

0

y, P)

F(x, Y, P)

(9)

by

(3) holds,

the function

F(x,.

y,. @) has a minimum

at p,

A. Aguilar,

E. Braun

I Specific heat of a general two-dimensional Ising model 277

c(T)

au(T) ap

= -k/3’

2n

kP2 dx

Y, P),

dyW

I 0

=16p2qr

555

I 0

(10)

where z(x

> > y

p)

m7

=

Y? P>fYx7 YT PI -

[F’k

YY PM’

Y7 PII’

1%

(11)

In the neighborhood of x = x, and y = y, the function Z(x, y, Z3) becomes, with the aid of eq. (5), and its derivatives

Z(x,Y, P) =

$ + %(X

- xJ2

a( f?) =

pp” - (p’)’

)

+ Uz(Y

- Yc)’

(12)

PD

with a =

a1 = a,(P)

= i(PP’:,

a2 = a,(P)

=

I(PP’b,

D = D(P) = P

+ P,,(x

+ Px,P”4P’P:,)

>

(13b)

+ P,,Pf’-2P’P’yy)

3

(13c)

- xJ2

+ P,,(Y

-Y,)’

.

(134

Let us now explicitly separate in the double integral (10) the domain (4), which we call R, c(T) =

2

kP2 (IIz(x,Y.P)dxdy+a(8))

167~qt

R

(14) Here zr(p) is the contribution from the term in eq. (12) that contains a; E2( p) is the contribution from the term in eq. (12) that contains ui( p) and CY~( p), and E(p) is the contribution of the rest of the domain of integration. Substituting eq. (12) into eq. (14) one finds that zr:“,(~3) is a R

P2 + PP&

dx dy

- xJ2

Making the change of variables a

+ PPJY

- Yc)’

.

(x - xc) = r cos 8 and 6

(15) ( y - y,) =

A. Aguilar,

556

E. Braun

I Specific heat of a general two-dimensional Ising model

r sin 8 we obtain

This integral is immediate.

%(P)-

Pm

a

Considering

explicitly the value at r = 0 we have

In p( /3) + rest

(16)

and using eq. (13a) we get

K(P) -

PP”_ (P’)’ In p(p) + rest

(17)

PVE&

Now let us to expand p(p), which is the first term in eq. (5), as a Taylor series around p,. From eqs. (5) and (3) we notice that p, is the root of F(x,, y,, p) = p(p) = 0. The required expansion is

:_p”(p,)~‘+...=~p”(p,)1412+...

p(p)=p(p,)+p’(p,)d+

(18)

with a = j3 - p,. Here we have used eqs. (3) and (6). Substituting eq. (18) into (17) we see that in the neighborhood of p,, the dominant behavior of the quantity 3, (p) is given by

%:“,(P> - v

P’YPC) Px,(PJ

In (Al + rest,

PJPJ

P-PC.

(19)

P+&.

WV

The coefficient of In 12 1 is a finite quantity. In a similar way we obtain P’Y PC) =2(P)

-

VPX.Y(PJ

P&t>

In Id I + rest ,

Using eqs. (14), (19) and (20) we see that the specific heat c(T) is a function such that, near the critical temperature,

c(T) -

P'Y PC> VP,,(Pc)

P&u

In

IAl ,

(21)

where

(22)

A. Aguilar,

E. Braun

I Specific heat of a general two-dimensional Ising model

557

Thus from eq. (21) we can say that if condition (3) holds then the specific heat has a logarithmic singularity in the vicinity of the temperature T, = 1/ kp,, with /3, given as the root of eq. (3). It is important to mention that the other terms in c(T), i.e., the sum of 8(p) and the rest of the terms in eqs. (19) and (20) are finite, well behaved functions of p in the neighborhood of p,. As a matter of fact, they have these properties for all values of the temperature. Therefore, we have been able to show that for all the two-dimensional models encompassed in the UC model if the condition (3) is fulfilled then the specific heat diverges logarithmically, independently of the particular details of the interactions between the spins. In this way we have explicitly obtained that for the class of models which obey condition (3) there is a universal singularity for the specific heat. Following an analogous procedure as the one just given, we can find that the internal energy per spin behaves, in the neighborhood of T,, as u(T) - Ap’ In (p) + Bp In (p) + rest .

(23)

The first two terms vanish as p -+ /3, (see eq. (18)). Thus the internal energy per spin is the finite quantity, as it should. Summing up we have shown that if there exists a point (x,, y,, p,) such that eq. (3) is satisfied, then the specific heat diverges logarithmically. It is possible, at least in principle, that there are several such points (x,, y,, /3,) thus giving rise to several transition temperatures. We have not been able to decide, in the general case, how many such critical temperatures there are. If there were at least two of these critical points, the results we have just presented are valid only if the corresponding critical temperatures are sufficiently separated. Otherwise, it is not justified to keep only the second order terms in the expansions such as eq. (18). In a forthcoming publication we will analyze in more detail this point. In order to obtain the explicit value of the transition temperature T, for a specific model, the procedure to follow is the following. Obtain the corresponding y,k quantities; substitute them in eq. (1.21); write down F,(y, p); take the limit rt+ w; finally, with the resulting function solve eq. (3). For a UC with a given set of interaction energies, it might be possible that eq. (3) will have no solution for any positive temperature. For this case one cannot conclude that there is a logarithmic singularity for the specific heat. Therefore, there could be other types of behaviors for that quantity. We have not been able, so far, to ascertain for which combinations of the values of the energy interactions between the spins eq. (3) is in general fulfilled. This is a question we are presently looking at. We have not been able to perform analytically, for the general UC model, the integral given by eq. (10) for any value of the temperature. However, we

558

A. Aguilar,

E. Braun

I Specific heat of a general two-dimensional Ising model

have performed the integrals numerically for the set of particular cases of the UC model that were discussed in I, and expressed the equation that the transition temperature T, satisfies for each of them. In the following section we will present the equations that the temperature T, has to satisfy for the various particular models treated in I.

3. Equations

for the transition

temperatures

Let us consider the 2 x 2 model. The corresponding F(z, w, T) function was presented in eq. (1.31). If in eq. (1.31) one puts z, = 0 and w, = 0, one gets the following equation (cf. eq. (3)): 0 = (1 + c, ,c,*cz,c22 + S,,S,*%,%*)(I + ~

(b,,b,*

+

-

(Ql2%1

+

~ll%)(~L2~21

+

(bl,b,,

+

~12~*2)(C*lCI2

-

(%I%

+

%~**)(c11~12~21c22

-

(~,I~22

+

~1*~2*)(~,,~,2

-

(S11%

-

(bl,a,,b2l%,

-

(Qll%L

+

(b,,b,,a,,a,,

+

~2lM(C,lCl2w**

+

+

+

+

b,,b,A,M

~ll~,ZC214

+

C2lC22)

+

+

SI IC12CZIS22)

s2,%2)

+

~,1~,2~2,%2)

~1,~,2~2lM(C,,C2l

~dd(CllC12

+

SllS22)

~,*~**)(~,,~,2%,~22

+

alla12%1%2

+

+

c12c2,

cd&)

>

~,,~,2~2,~22)(~,lc,2~2,~22

+

(241

c,,s,zc2,~22)

The solution of this equation is p, = llkT,, where T, is the transition temperature. We now write down the forms that eq. (24) acquires for the cases treated in section 4 of I. (A)

The Utiyumu model U (four parameters): sLls,*ul,

(B)

+

s,,sj2u,2

General hexagonal

+

S,,alla,2

+

S12~,,~l?

From eq. (1.34) we get =

Sl1

+

Sl2

+

all

+

aI2

.

(25)

gh (six parameters):

@II + sds21 + S22)(b,,b + 1) = ~,,~2*(s,,s,z- l)(s2,s2z- 1) +a

11 a 22

ccII

12 c 21

c 22

(26)

A. Aguilar, E. Braun I Specific heat of a general two-dimensionalIsing model

559

(C) General triangular gt (six parameters): ( a11az2 +

W,,)(l

= (M,, +

SllSl2

c11c12c21c22

(II)

bll

sllszl - s12s22)

+ a11b22)(cllc21 + c12f22 - sllc12s21c22 - clls12c21s22) +

s21s22

In the ferromagnetic

=

+ cllc12c21c22 + slls12s21s22 -

+

+

+s12s21

(274

+s11s22.

case this equation can be simplified to

(~11~21-

s21h2

l)(s12s22

+

S22Wllb22

-

1)

+

alla221

*

(27b)

Tetragonal triangular tt (seven parameters):

a21(slls22 + s12s21) + all(clls12s21c22 + sllc12c21s22) +

bll(cllcl2s2ls22

+

~21(~11~12

+

(b11b21a22

+

alla2lb22)(cllc21

x

(~11a21~22

+

%1~2la22)(~11~21+

=

(b

11a21a22

+

allb2lb22)(sllcl2s2lc22

+

b22(cll%2

x

(1

+

+

+

s2922)

+

w12c21c22)

+

c21c22)

c11c12c21c22

+

a22tc11c22

+

+

+

(a11a21a22

~11~12~21~22)*

c12c21)

c12c22)

s12322)

+

+

Wl2C21~22)

bllb2lb22)

(28)

Unfortunately we have not been able to solve analytically the highly transcendental equations (24) to (28). However, once a set of values of the interaction energies eji and e; for each model are given, we are in position to solve numerically these equations. As was mentioned at the end of section 3, it is possible that for a certain model if a set of interaction parameters is given, the corresponding equation, either one of (24) to (28), is not satisfied for any temperature. In that case the specific heat for that model does not have a logarithmic singularity.

4. Final comments

In this paper we have shown that for the whole class of general Ising models treated in I, if condition (3) holds then the specific heat exhibits a logarithmic

560

A. Aguilar,

E. Braun

I Specific heat of a general two-dimensional Ising model

divergence. It was possible to demonstrate that statement in a general way without taking into account the specific details of the interaction between the spins of the system. In that sense there is a universality in the behaviour of the specific heat for the class of models treated in I. There is still an outstanding problem left, namely, the calculation of the spontaneous magnetization as function of the temperature for the general model. We emphasize that the function L(T) we have obtained in paper I is valid only for a system with no external magnetic field. Therefore we ought to start by obtaining the partition function per spin in the presence of that magnetic field. One would then like to obtain, from that partition function, that at the transition temperature the magnetization behaves as Ali’, for any model, independently of the details of the spin interactions. If one could work out this program then, hopefully, one could show that the temperature at which the spontaneous magnetization is set on is precisely the temperature at which the specific heat has the logarithmic singularity (if condition (3) is fulfilled, of course), as occurs for the q = t = 1 model solved by Onsager, and whose magnetization dependence was first obtained by Yang [7]. If that is the case one can then say that the temperature T, is indeed a transition temperature in the sense that there is a phase transition. This problem is under current research.

References [l] A. Aguilar and E. Brdun, Physica A 170 (1991) 643, to be referred to as I. [2] L. Onsager, Phys. Rev. 65 (1944) 117. B. Kaufmann, Phys. Rev. 76 (1949) 1232. [3] R.M.F. Houtappel, Physica 16 (1050) 425. H.N.V. Temperley, Proc. R. Sot. A 202 (1950) 202. G.F. Newell, Phys. Rev. 79 (1950) 876. G.H. Wannier, Phys. Rev. 79 (1950) 357. K. Husimi and I. Syozi, Prog. Theor. Phys. 5 (1950) 177. I. Syozi, Prog. Theor. Phys. 5 (1950) 341. (41 T. Utiyama, Prog. Theor. Phys. 6 (1951) 907. [5] C. Domb and M.S. Green, eds., Phase Transitions and Critical Phenomena. vol. 1 (Academic Press, New York, 1972). [6] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press. New York, 1982). [7] C.N. Yang. Phys. Rev. 85 (1952) 808.