Condensed Matter

Z. P h y s i k B - C o n d e n s e d M a t t e r 40, 2 3 3 - 2 4 0 (1980)

Zeitschrift fQr Physik B

9 by Springer-Verlag 1980

Square Lattice Ising Antiferromagnet in an External Magnetic Field J. Zittartz Institut fiir Theoretische Physik der Universit~it, K61n, Federal Republic of Germany Received September l, 1980 We study the anisotropic square lattice Ising antiferromagnet in terms of three parameters: an external magnetic field B, an effective temperature 2 and an anisotropy parameter K. The model, i.e., partition function and free energy, is solved exactly in the anisotropic limit, K--* 0, for arbitrary temperature and field by using the transfer matrix method. We also calculate the first corrections beyond this limit. The limit is non trivial and the phase transition is completely preserved. It is of the expected Ising type. The transition temperature 2c(B, K) is determined exactly for both K--* 0 and K ~ oo and the results are used to check the validity of a recently conjectured formula by MiillerHartmann and the author.

I. Introduction The Ising antiferromagnet on a square lattice in an external magnetic field is defined by its Hamiltonian

fiW=K1 2 a.cr+K ~ a . a - B ~ a i v.n.n,

h.n.n,

(1)

i

with anisotropic couplings K and K~ and magnetic field fi-lB. The first summation is over all vertical nearest neighbour pairs, the second summation over all horizontal nearest neighbour pairs, and ai= +_1 denotes the spins at sites i. Its partition function

Z(K, g l , B)= ~ e-P~e

(2)

thus depends on three parameters. For a general set of parameter values (K, K1, B) this model so far has escaped an exact solution in closed form and presumably is not "soluble". At zero field (B=0) we have of course Onsager's solution of the Ising problem [1]. The model is expected to undergo a phase transition from a 2-fold degenerate antiferromagnetically ordered low-temperature phase to a unique (i.e., non-degenerate) disordered high-temperature phase, the two phases being seperated in parameter space by the critical surface. It is generally accepted that this transition is universally character-

ized by the Ising type singularity, i.e., a logarithmically divergent specific heat and critical exponent ~=0. In this paper the model is solved exactly in the anisotropic limit (K--*0, K1--*oo or vice versa) for arbitrary field B. It is then shown that one can systematically expand around this limit, and the lowest corrections are calculated. This limit has the advantage that the phase transition of the model is preserved completely, and the transition is indeed as expected. In order to explain our procedure and results it is convenient to introduce another parameter 2 instead of one of the couplings, say Kp The dual coupling L is defined by the well-known equivalent relations [1] t a n h L = e -2~:',

sinh 2L =(sinh 2K1) -1 .

(3)

We then define 2 by L = 2. K

(4)

and henceforth consider 2, K, B as basic parameters. While K is a measure for the anisotropy, 2 may be viewed as an effective temperature, as obviously 2 increases in (0, oo) as temperature increases (K and

0340-224X/80/0040/0233/$01.60

234

J. Zittartz: Square Lattice Ising Antiferromagnet

K 1 decrease from infinity to zero). The phase transition will then occur for some critical value L=L(K,B).

(5)

The well-known zero field result sinh 2K e= sinh 2L~

(6)

\{cosh B)-1

can then be conveniently expressed as 2~(K, O) =

K~

(7)

1

independent of the anisotropy K (see also Fig. 1). The meaning "anisotropic limit" can now be made precise. For fixed B and 2 we use K as a small parameter and calculate the partition function to the lowest nonvanishing order and then determine the first correction. For the free energy this means that we write f(2, B, K) =f(2, B, K) + O(K ~)

(8)

Fig. 1. Qualitative plot of the exact transition temperature 2c(K, B) for some B as compared with 2~(B=0)-=1 and 2c according to formula (13) (broken line)

work known at that time [3], we believed it to be exact generally. With (3) and (4) the formula can be written as cosh B = sinh 2 K/sinh 22cK

(13)

with the asymptotic expansions

and f is obtained in closed form. f shows the expected logarithmic Ising singularity (a=0) and the transition temperature is given by the formula

2~. coshB = 1 +~(K. tanh B) 2 + O(K4),

2 c - cosh B = 1 + e(K. tanh B) 2 + O(K*),

2c=1

(9)

correct to order K 2, with the exact coefficient 16 e..... 3~z

5 6

0.8643.

(10)

Anisotropic limit in (9) means K = 0 , while the K zterm is the first correction. The other limit of anisotropy, K ~ c ~ , can also be worked out with the result 1c=1

In coshB +(~+ 89 2K

e-4K

(11)

It should be remarked that both limiting formulas (9) and (11) at zero field, B = 0 , coincide with (7), valid for all K. This indicates that all corrections in (9) and (11) vanish at B =0. Figure 1 shows a qualitative plot of the exact transition curve for some B # 0 which should be below the zero field result 2c = 1 (7) for all K, as the presence of a magnetic field reduces the transition temperature. The above results can be used to check the recent conjecture by Miiller-Hartmann and myself [2] for the transition temperature cosh B = sinh 2 K . sinh 2K 1.

(12)

This result was derived by an approximate calculation of the interfacial tension. As formula (12), and also the interfacial tension result, checked with all exact results in limiting cases and also with numerical

K~0

(14a)

and in cosh B

2K

e-4K

+sinh2B.~_+O(e

6K).

(14b)

Obviously formula (12), or (13), reproduces the exact expansions (9) and (11) only to lowest order, it is incorrect in the next order corrections. As the exact coefficient s (10) is larger than the corresponding e = ~ of (14), formula (13) underestimates the transition temperature in both limits and presumably does so far all K. This is also plotted qualitatively in Fig. 1. That formula (12) might not be generally exact has also been concluded in recent numerical work [4, 5]. Baxter et al. [4] have determined the transition of the hard-square lattice gas very accurately from activity expansions. As the corresponding transition point is directly related to the slope of formula (12) at zero temperature in the isotropic case (K=K1), they concluded from their result that (12) is not exact, although a fairly accurate approximation. Other numerical work, based on Monte Carlo calculation [6, 71, seemed to support the validity of (12). It thus seems that it is a good working formula with fair accuracy. Nevertheless it is somewhat disappointing that the simple formula is not exact generally, but only in limiting cases. Lin and Wu [8] suggested that the special method of ref. [2] to determine transition temperatures leads to correct results if the problem is equivalent to a "free fermion" problem. This is supported by the investigation of this paper. As will be shown in the following, the anisotropic limit (K ~ 0), where also formula (13) is exact, is equivalent to a

J. Zittartz: Square Lattice Ising Antiferromagnet

235

'~

fermion" problem. In the next order the K 2corrections generate an "interaction" of the fermions and (13) is no longer exact. The plan of this paper is as follows. In section II we apply the transfer matrix formulation to the partition function. By using simple relations for Pauli matrices (Appendix A) the partition function is brought into a form that a systematic expansion in the anis0tropic limit, K--*0, can be performed. In section III the expansion is carried through. The partition function is thereby expressed in terms of an transfer operator which is correct in leading and next to leading order in the limit K-~ 0. In section IV and Appendix B the transfer operator (35) is diagonalized using the Jordan-Wigner fermion representation of Pauli spin matrices. A closed form expression for the free energy is then derived and the phase transition of the expected Ising type is discussed. In section V we finally determine the transition temperature 2~ in both limits K ~ 0, K---, oo for arbitrary magnetic field B.

This formula is the starting point for our investigation. It is known that in the thermodynamic limit (M, N--* oo) one only needs the largest eigenvalue of the operator in brackets (or a symmetrized form of this operator) to determine the free energy. Now we combine the B-dependent part of (17) with Tt in (18) and use formula (A.3) of Appendix A to write e-~Ba3eLale_ 89

=e2~aa 2 e Lat e2~ a a

2

(19)

with new couplings (A.4) sinhL=sinhL.coshB

(L>__L)

tga=tanhL.sinhB

(0__
(20)

Using (19) in (18) and rearranging operators under the trace the partition function can then be written as 2 = Trace R M/2

(21)

where R = T,. T2(a) 9TI 9T 2 ( - a )

II. Transfer Matrix Formulation We consider a square lattice with M rows and N columns and assume that both M and N are even and that the lattice is periodically closed on a torus. In order to apply the transfer matrix method to the model (1,2) it is convenient to substitute a - - + - a for all spins in the even rows of the lattice. This changes the Hamiltonian (1) to flH=-K

1 ~ a-a+K v.tl.rt,

with ~ = T,(L) (16) and --

i

Tz(a)=e2 ~

2

e

_KEa3.a3

iaYa2

e 2

.

(23)

Formula (21) is of course equivalent to (18), but it is a more convenient starting point to investigate the anisotropic limit of our model and to expand around this limit.

~ a.a-B2(-1)~-*.a h,n.n.

(15)

where v denotes the rows. In the transfer matrix representation of the model [9] the transfer of interaction from any configuration of spins in one row to any other configuration of spins in the next row is described by the operator N

L~

I~ (eKl+e-~q.a~)=A 97-1,

cr~

TI(L)=e ,.~ ,

(16)

III. The Anisotropic Limit As mentioned in the introduction it is hopeless to study the model for general values of the parameters. For this reason we confine our investigation to the anis0tropic limit. This limit has already been defined in the introduction. Instead of the coupling L we introduce the "effective" temperature 2 by setting (4) L = 2. K,

with the constant A=(2sinh2K1) N/2. a 1 is the spin flip operator in standard notation and the dual coupling L has already been defined (3). The interaction in odd (even) rows is measured by the operator T2(_+B), Ta(B)=e . . . . . .

(22)

fin " ~+ 1

(i7)

where the Pauli matrix ~3 is diagonal in standard notation. The partition function (2) is then given by Z / A M - Z =Trace [T 1T2(B) T 1T2(-B)] M/2.

(18)

(24)

our basic parameters are now K, 2, and B. In the anisotropic limit K is considered to be a small parameter while 2 and B are kept fixed. It will turn out that this limit preserves the phase transition of the model. It is also the limit which one would study in a quantum field theory. In that case one would first perform the continuum limit along the "'time" axis (vertical direction) in order to obtain a quantum field model in one spatial dimension (horizontal direction). It is seen immediately from (20) and (24) that both quantities L and a are of order K. This implies that in

236

J. Zittartz: Square Lattice Ising Antiferromagnet and the validity of (30) can again be checked by direct expansion to order K 4. Substituting all terms into (21) we obtain the partition function

(22) and (23) all operators appearing in the various exponentials are of order K. To leading order K these operators commute, as commutators are of order K 2. Therefore we may combine them in one exponential whereupon R (22) reduces to

Z = Trace _~M/2

Ro = [eL~, e-XZ~3o~]2.

with

(25)

Replacing R in (21) by R 0 and comparing this expression with (18) at zero magnetic field, B = 0, we see immediately that both are identical in form. Only L has to be replaced by L which means that in the anisotropic limit the magnetic field merely renormalizes the coupling L--* L (20). Our model can therefore be solved in terms of "free fermions" (see below). We shall proceed by calculating the first correction terms in the exponentials which will turn out to be of order K 3, or of relative order K z when compared with the leading order. Using formula (A.2) of Appendix A the operator Tz(a) (23) is explicitly given by

~2(a)=e-l<~,~3,~3-Hl+c = e~C+O(K~)e_g~3 -n~ e~C+O(K~)

R=Tleae-X~'~-n~e-G~e-%-ez~'~3-me G

(33)

where all operators in the exponentials are correct to order K 3. Now the formidable expression (33) simplifies considerably. First we observe that G is of order K 3, as indicated in (31); as in the case of H t it can therefore be treated as if it would commute with all other operators in the exponentials. Obviously G then cancels out completely in (33). The remaining expression is a square which is substituted into (32). Rearranging again under the trace and observing that H 1 commutes we obtain the final expression Z = T r a c e S M,

(26)

(32)

S=So e-H~

(34)

with the symmetrized operator

with the new coupling /~ ~ K . cos2 a = 0 (K),

(27)

an "'interaction part N

H1 =3; ~ an1 G1+ I , .

7 = K .sin2a=O(K 3)

(28)

n~JL

and the operator

C(a)=K.sina.cosa~,(a3 al +ala3)=O(K2).

(29)

We have indicated to which order K the new operators contribute. While the first line in (26) is still exact, the result in the second line can be derived by a direct expansion to order K 4. As we are interested only in the first corrections of order K 3 in the exponentials, operators of order K 4 can be neglected as indicated in (26). The coupling ~ of the "'interaction" operator H I is already of order K3; therefore H 1 can be treated as if it would commute with all other operators in the exponentials because commutator corrections would be at least of order K 4. If (26) is substituted into R (22) we see that by rearranging operators in (21) under the trace the Cparts can be combined with T~ resulting in

e+~C~ e T-~c= e_+G~ e_+~

(30)

S O(L, K) = e~Lz~ e-K~~

e~LZ~,.

(35)

It is instructive to compare this expression with the corresponding one in the anisotropic limti R o (25) which is correct to order K only. The dominant term S o has the same form as RU z, apart from symmetrization, with the only change K ~ J ~ . Therefore S o can be diagonalized in terms of "free fermions" (Appendix B). The Hi-term which has appeared as the next to leading order correction will be seen to represent an interaction among the "fermions". It is induced in order K 3 by the presence of the magnetic field, and its coupling 7 (28) goes to zero with B-~ 0 (20). It should be clear from the treatment in this section that in principle one could go on with a systematic expansion and determine corrections to order K 5, K 7, and so on. One would generate more and more interaction terms H z, H 3 .... of increasing complexity. It is a reasonable guess that a diagonalization of all these terms and a resummation is impossible, the model therefore seems to be not soluble in closed form in the general case.

IV. Free Energy

with either the upper or the lower sign. Remark that C(a) (29) is odd in a. The new operator G is the commutator

In the thermodynamic limit (M, N ~ oe) we just need the largest eigenvalue of the transfer operator S (34)

G =88

Sl~m,~)=e N/9 I~a~ )

L ~ a ~] + O(K `~)= O(K3),

(31)

(36)

J. Zittartz: Square Lattice Ising Antiferromagnet

237

to obtain the free energy f per site (apart from an unimportant shift due to the constant A (18) which we drop). It is clear from the foregoing that f is of leading order K and should be correct to order K 3. We now use a little trick which will prove to be useful later on. We change the couplings L, K in S O (35) by setting

L=L+~, K=s

c] = o ( g 3 ) ,

(37)

where 6 will be determined later. The 6-terms which arise in (35) commute with all other terms, as they are of order K 3, and thus can be combined with H a in (34) to the new "interaction"

~ :H~ -~[~ ~a +~ ~ ~3.

(39)

To calculate the free energy in (36) correctly to order K 3 it is obviously sufficient to determine the contribution from /41 by first order perturbation theory. Thus we can write f as formula (8) of the introduction with two parts f(Z, B, K) =fo +fa-

(40)

The first part fo is given by the largest eigenvalue of So

So(L,~2). 17~o)= e -NI~ 17Jo)

(45)

Near its zero we have

[

[ R \2 211/2

with c=/s163 -1 and t = I - L / K , which when used in the integral (43) indeed leads to the expected logarithmic singularity (44) implying that the specific heat exponent c~= 0. The contributions of the correction term f l (42) near the transition point (45) are investigated in Appendix B. There arise various t ~In It[-terms (B.17). However, by a suitable choice of the parameter 6 (37), namely (B.18) 8 = ~ - . ~p,

(42)

V. Transition Temperature The transition temperature in terms of our basic parameters 2, K, B is now determined from (45) with (37) and (47) 0 = ( I ~ - L + 1~67) 9K-1

dqE(q)

(48)

(43)

where E(q) is the solution of (B.Sa). The correction term f~ is given by (B.16) and represents a nearest neighbour interaction of the fermions (see (B.13)). Both expressions could be evaluated in terms of elliptic integrals which is omitted here. We are rather interested in the phase transition of the model and the corresponding singularity in the free energy. Of course, we expect just the Ising type singularity

f~-ct 2 In Itl+reg.

(47)

(41)

Diagonalization of S o is performed by the standard technique [9] in terms of free fermions and is described in Appendix B. The leading part fo is given by (B.10) fo = -

(46)

we can eliminate the leading singularity t In Itl in fa. The remaining contributions merely change the amplitude c in (44) in order K 3 while the location of the transition point given by (45) is unaffected.

and fl is the K3-correction fl = (%1/411%).

/~=L.

(38)

S in (34) is therefore changed to

S ~ So(L, fs e - ~ .

erate antiferromagnetically ordered low-temperature phase to the unique (i.e. non-degenerate) disordered high-temperature phase. A singularity in (43) can only arise if E(q) vanishes for suitable values of q and the other parameters, because cosh E in (B.8) is analytic in both parameters /(, L and thus the inverted function E(q) is analytic provided E # 0 . It is obvious from (B.8) that E = 0 only for q = 0 and

(44)

near the transition point ( t ~ 0 ) which universally should describe the transition from a 2-fold degen-

which must be evaluated to order K z. Using the various definitions (28), (27), (24), (20) and solving for ,%up to order K 2 we obtain the critical temperature 2 c 9cosh B = 1 + e (K. tanh B) 2 + O(K 4)

(49)

with the coefficient 16 ~-3~

5 6=0"8643 ....

(50)

This formula includes the anisotropic limit ( K ~ 0 ) where )~cO= (coshB) -1 ,

(51)

238

J. Zittartz: Square Lattice Ising Antiferromagnet

which decreases with increasing magnetic field as one would expect. The other limit of anisotropy, K ~ o o , can also be worked out very easily. It is obvious from the Hamiltonian (1) that the partition function (2) has the symmetry Z(K, K 0 = Z ( K > K).

(52)

Thus we can also introduce the coupling dual to K and a corresponding 21 by setting sinh 2Lx = (sinh 2K) -I,

L1=21 . K 1 .

(53)

K--.oo implies La ~ 0 and thus K 1 -~0 at fixed 21. In this limit we would obtain the same formula (49) for 2~c as a function of K~: 2 ~ . cosh B = 1 + e (K ~. tanh B) 2 + O (K4).

(49 a)

Then solving (53) for 2 , . K t in terms of K to third order in K~ and using (49a) we obtain the expansion for K ~ = K I ( K ) ( K ~ o v ) . Substituting this in (3) and using (4) we solve (3) to get 2 = 2 ( K ) for K ~ o o . After some algebra this leads to 2c=1

in coshB +(~+ 89 2K

for any values of 2 and the external magnetic field B in the anisotropic limit K ~ 0 such that the essential physics of the model, namely its phase transition, is completely preserved. The problem could be solved exactly in this limit and the leading order corrections could be determined. We have obtained the following results: a) the free energy f ( 2 , B , K ) (40) as a closed form expression, correct to orders K and K 3, b) the singular part (44) of f near the phase transition, which is of the expected Ising type, c) the transition temperature 2c(K,B) in both limits K---, 0 (49), correct to order K 2, and K ~ oo (54). These results have been Compared with the wellknown zero field formula, 2~(K)=1, and have been used to check a conjecture of Miiller-Hartmann and myself for 2c(B, K) which turns out to be only correct asymptotically, but otherwise presumably a fairly accurate approximation to the true formula which therefore still remains in "darkness".

Appendix A

d -4K

(54)

which is formula (11) if the introduction. Both limits (49) and (54) reduce to the well-known zero field result 2 c ( B = 0 ) = 1 (7), valid for all K. In Fig. 1 we have plotted qualitatively the exact transition curve as a smooth interpolation between the two limits (49) and (54). Of course, this curve should always be below 2~= 1 as a magnetic field reduces the transition temperature. Also plotted is the curve corresponding to formula (13) which has been conjectured by Mtiller-Hartmann and myself [2]. As already discussed in the introduction this formula checks with the exact limits (49) and (54) only to leading order. For finite values of K it is presumably a very accurate approximation and also a useful lower bound for the exact curve.

We list some formulas for Pauli spin matrices ai,

airyj + as a i = 26ij

(A.1)

which are frequently used in this paper: i ----a~

e 2

~Xff3

e~

3

~-a~

o e2 ,~-J.

-- !XO-3

ey e ~

3

=o- - c o s a + a l - s i n a , i 2 --aO-

=e2

e

~al

(A.2)

~ a o -2

e2

(A.3)

with sinh y = sinh y. cosh x tg a = tanh y. sinh x,

(A.4)

and e89

eY(a 3 c o s q + a ~ sinq)e89

i

2

i

~ e ---aa 2, e ~a3 e2--aa

2

(A.5) with:

VI. Summary cosh y = cosh y. cosh x + sinh y- sinh x. cos q Instead of using the anisotropic couplings K 1 and K to describe the square lattice Ising antiferromagnet we have introduced a more convenient parameter 2 (3, 4), representing some effective temperature, instead of the one coupling K,. This choice made it possible to use the transfer matrix method* effectively * Similar methods have also been used to solve some inhomogeneous Ising models [10]

sinh y. sin a = sinh y. sin q

(A.6)

sinh y. cos a = sinh x. cosh y + cosh x. sinh y- cos q. These relations (and cyclic variants, of course) hold by using

ex~' = cosh x + erz' sinh x.

J. Zittartz: Square Lattice Ising Antiferromagnet

239

Appendix B

[1~0) = U IO),

To diagonalize the operator S O (35) we use the standard Jordan-Wigner representation of spins in terms of fermion operators G, c,+'

(10) is the state of full occupancy) and f0 in (41) is given by

(Tn1

fo = - N -1 ~ E(q)= - ~ S d q E ( q ) ' o

2c,+c, - 1,

all q,

(B.9)

(B.10)

q>0

~C+C

1)-,=1 '....

- - f i n3 -- -- ( - -

"~q10> = I0>,

Expectation values of z-operators within this state are

" ( C : 4- C.),

9 3 O'n. 1 (7n2 = --lO'n

(B.1) ('C~)=O,

Using (B.1) for the operators in (35) and Fourier transforming we obtain

(z~)=sinaq

(z3q)=cOSaq,

(B.I1)

leading to

(c~ cq) = (C+qC_q) =~(1 + cos aq)

N 3

~=Z(2C+Cq - 1 ) -=2 Z vq n=l

q

(B.2)

i (Cq c+q) = (cqc_q) = - ~ sin aq.

(B.3)

The expectation value of H z (28) is in terms of fermion operators

q>0

(B.12)

N

3 Crn3 O-n+ n=l

= 2 ~ , ( ' c ~ . c o s q + z q1 sinq) q>0

with

( H 1 ) = ? ~ ((2c + c , - 1)(2cL 1c,+, - 1)) .

3 zq=c~cq+c+qc ~- 1 ,

"Cq ~=i(c+

C+q+C,C_q),

3 zq2 - i,.cl Zq.

(B.4)

2zt Generally q ranges in ( - ~ , re) with spacing - - . so(L , R.) is thus represented as N So =

[ 1 ez~a "e - 2k(~{

oosq+q sine), e~.~a.

(B.5)

q>O

S O=

1-] eE(q)~" U - 1

U

and obviously represents a nearest-neighbour interaction of the fermion system. Fourier transforming and forming all Fermion contractions we obtain using the averages (B.12) N - 1( H 1) = 3~[(1 - cos q) cos aq -(1 4- cos q) cos aq 4- sin q. sin aq-2]

Within the subspace of single occupancy (z 3 =0) for each q S o is already diagonal with eigenvalues 1. In the subspace of zero plus double occupancy (z 3= + 1) the z-matrices have the properties of Pauli matrices. Formula (A.5) then diagonalizes So: (B.6)

q>O

with the unitary operator

(B.13)

(B. 14)

where 1 g(q) =~-= _~dq g(q).

(B.15)

The expectation value of the 5-part in /tz (38) is calculated by using (B.2, 3, 11) and combined with (B.14) leads to (41) fl =[7(1 - cos q) cOS a q -- ~5]

(B.7) q>O

9(1 + cos q) cos aq + [~ sin q. sin aq - -

~]-

sin q. sin aq.

(B.16)

and (from (A.6)): Near the transition (45) the quantities sinai, (B.8) go like

cosh E = cosh 2/(. cosh 2L - sinh 2/s 9sinh 2L. cos q, sinh E. sinh aq = sinh E.

cos

--

cOS aq

E>0

sinh 2/s sin q,

E(q). sin aq -~ - q sinh 2/(,

-rc
E(q). cos aq ~- -- 2/~ t + 89

aq = sinh 2L. cosh 2/(

- sinh 2/s cosh 2L. cos q. The eigenstate of largest eigenvalue is

(B.8)

sinh 2/(.

This implies (using (46)) that sin q- sin aq ' ~ C1 4- ca 1 2 In Itl

(B. 17 a)

240

J. Zittartz: Square Lattice Ising Antiferromagnet

(1 - cos q). cos aq ~-c 3 + c4t 3 In

It[

(1 +COS@ cosaq ~ c 5 + c o t In ttl.

(B.17b) (B.17c)

The second part in f , (B.16) thus produces a t21nltl term which when added to fo slightly corrects the amplitude c in (44) but does not change the location of the transition. The t . l n l t l - t e r m in (B.17c) would shift the transition point. However, by a suitable choice of a which enters the prefactor in the first part in (B.16) we can eliminate this shift. The condition is 6 = 7 ( 1 - cos q) cos aq],= o = 8 = ~ 7 7.

~f~dq(1-cosq) ~-]/2 0

~ (B.18)

Then the first part in (B.16) does not even correct the amplitude of the singularity (44).

References i. Onsager, L.: Phys. Rev. 65, 117 (1944) 2. Miiller-Hartmann, E., Zittartz, J.: Z. Physik B27, 26i (t977) 3. Rapaport, D.C., Domb, C,: J. Phys. C4, 2684 (1971) 4. Baxter, R.J., Enting, I.G., Tsang, S.K.: J. Stat. Phys. 22, 465 (1980) 5. Racz, Z.: (to be published) 6. Rapaport, D.C.: Phys. Lett. 65A, 147 (1978) 7. Binder, K., Landau, D.P.: Phys. Rev. B21, 1941 (1980) 8. Lin, K.Y., Wu, F.Y.: Z. Physik B33, 181 (1979) 9. Schultz, T.D., Mattis, D.C., Lieb, E.H.: Rev. Mod. Phys. 36, 856 (1964) Mattis, D.C.: The Theory of Magnetism. New York: HarperRow 1965 10. Hoever, P., Wolff, W.F., Zittartz, J.: Z. Physik B (to be published) J. Zittartz Institut fiir Theoretische Physik Universit~it zu K61n Ztilpicher StraBe 77 D-5000 K/31n41 Federal Republic of Germany