Partition function zeros for the two-dimensional Ising model

Physica 129A (1984) 201-210 North-Holland, Amsterdam

PARTITION FUNCTION ZEROS FOR THE TWO-DIMENSIONAL ISING MODEL*

John STEPHENSON and Rodney COUZENS

Physics Department, University of Alberta, Edmonton, Alberta, Canada

Received 12 March 1984

The distribution of complex temperature zeros of the partition function of the two-dimensional Ising model in the absence of a magnetic field is investigated. For anisotropic square and triangular lattices the distribution function is two-dimensional and satisfies a partial differential equation derived from a generalized scaling theory. Corresponding results for the isotropic square, triangular and honeycomb lattices are also presented.

1. Introduction

As is well known the thermodynamic properties of a physical system can be derived from a knowledge of the partition function. Since a partition function is a sum of exponential Boltzmann factors, its mathematical features are com- pletely determined by the location and distribution of its zeros. For real temperatures, and magnetic fields, etc., the partition function is real and positive. Therefore all its zeros are complex. The occurrence of a phase transition is associated with the manner in which the zeros approach the real axis. Yang and Lee’) have shown how the “magnetic field zeros” of the Ising model lie on the unit circle in the plane of the complex “fugacity” variable p = exp(-2mH/kT), and approach the real axis as T+ T,, the critical temperature. When the magnetic field is absent, we know from Onsager’s solution of the two-dimensional Ising model that the phase transition occurs at a single critical temperature, where the specific heat exhibits a logarithmic singularity”‘). The analytical properties of the zero-field partition function are now determined by its zeros in the complex temperature plane. It is these “temperature zeros” which are the subject of this pape?13).

In what follows we will first show how thermodynamic quantities are extracted from a knowledge of the location and density distribution of zeros. Then we will display, and briefly indicate the derivation of, the explicit results for the Ising model, drawing attention to some novel and interesting features which appear in the anisotropic case when the horizontal and vertical inter-

* This work is supported in part by NSERC through grant No. A6595

037%4371/84/$03.00 @ Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

202 J. STEPHENSON AND R. COUZENS

actions are unequal. We will conclude by showing how these results fit into a

generalized scaling theory, which yields a class of partial differential equations,

one of which is satisfied by the density distribution function of the two-

dimensional Ising model.

2. Significance of zeros

From a factorization of the partition function

G(w) = GKO rI (1 - Z) i I

in terms of a suitable complex variable w, with zeros at wi = x + iy = w, + iy

where w, is the critical value of w, one obtains the internal energy as follows:

E 1 aw _-- I&- N-Nap i Wi_W’

p=&.

For a one-dimensional distribution of zeros on a curve in the w-plane,

YO

E aw yap dy I aw,- WI

(w, - w )’ + y* g(y) ’ 0

where g(y) is the density of zeros, i.e. the number of zeros in (y, y + dy) per

lattice site N. g(y) is even in y, and for a second order phase transition tends

to zero at the real axis. y. is a cut-off near the real axis. For the isotropic Ising

model, g(y) m 1 yl. For a two-dimensional distribution g(x, y)

YO

See refs. 9-12 for details.

3. Isotropic square lattice

(4)

Now let us look at the Ising mode16). For the isotropic two-dimensional

square lattice the temperature zeros lie on the unit circle in the complex plane

PARTITION FUNCTION ZEROS FOR ISING MODEL 203

of an appropriate w-variable, with a density g(y) - ly1/27r near the real axis ferromagnetic (antiferromagnetic) critical point at w = +1(-l). This result may be extracted-with a little handwaving-from the factors which make up the four multiple product expressions in the partition function2’3):

z,, = ; i P, ) P; = 22m” fi i {* . .} )

I=1 r=l. s=l

(5)

where

{*- *}= clc2-slcos~l”-s,cosc#J~‘. (6)

Here Si = sinh 2Ki, Ci = cash 2Ki, Ki = JJkT, i = 1,2 for horizontal, vertical. We select the fourth term with

~ ,

= or - lb ) ~ = (2s - lb s (7)

m n .

The range of 4, and 4, can be taken as (-7r, m) for convenience, instead of (0,27r), as desired. The retention of only one term is believed not to affect the distribution of zeros on a large lattice 1o,13). When the interactions are equal, J, = J* = J,

{- . .} = S[S + s-’ - (co@, + cosf$,)] . (8)

The running variables &, d3 combine into a single parameter:

cosa, = ;(cos+, + cos&) (9)

so the distribution of zeros is now one-dimensional, on the unit circle of the variable

w s s = ei@ . (10)

4. Anisotropic square lattice

The interactions Jr and J, are now different, and their ratio will be taken as fixed, so that the distribution of zeros may be studied in terms of a single appropriate complex variable, w. First we make the technical step of setting


K=i(K,iK,), k=i(K,-K,), with k = aK, 0 < CY < 1, (11)

so the factors {* . .} become

(...)=S,C, (~+$-c+-gc_). k K K k

(12)

where C, = cos.4, ?cos&, SK = sinh 2K and C, = cash 2K, etc. The appropriate choice of complex variable is

w = SK/c,, (13)

so the critical points are at w = tl when &,&, + 0 or +7r. Inspection of the

factors { . . . } reveals that the transformation from the angle variables +,, 4, to the real and imaginary parts of w = x + iy is now non-singular, so the zeros will be distributed over a two-dimensional region in the complex w-plane. Moreover, this region has natural boundaries located by setting 4, and 4, equal to 0 or T. The boundaries pinch together in a cusp-like fashion at the real-axis critical points w = ?l. The unit circle is contained within the distribution where

‘I 6 +I

Fig. 1. The locations of zeros in the complex plane of the variable w, in the special case KI= 3Kz soK=2k=2Kzandw=2sinhK.


4, = ‘4,. Zeros pinch the imaginary axis at w = ?i where 4, = 24, = ?5-12, and return again whenever #J, = ‘4, and SK = C, = 0 at w = +i/a. [In this last case, one must have 2K = ?im’rr and 2k = *(n’+ $T with (Y = (n’+ :)/ml for some integers m’ and n’.] (Fig. 1.)

5. Distribution of zeros

To display the symmetry breaking explicitly we write the vanishing factors in the form

(1 + w2 - WC+ - RC_) = 0, with R = &C,/C: . (14)

To obtain the distribution of zeros near the real axis ferromagnetic critical point where 4, = 0, +s = 0, expand R as a function of (w - 1) to obtain real and imaginary parts

[’ denotes w-derivative, and c subscript denotes a value at the critical point]. The boundaries (4, and 4, zero) of the cusp-like region can be extracted easily. The unit-circle locus of the symmetric case, R = 0, is contained within the cusp. The two-dimensional density of zeros g(x, y) now follows immediately. Since zeros occur in cells of size 2rlm x 2~In, we obtain

bd, AAl = 2n2g(x, Y) Ax AY = Y Ax AY

2[{d(l - x) - by2}{ay2 - c(1 - x)}]“’ . w

Let us summarize the important features:

9 ii)

iii) iv)

v)

existence of boundaries, cusps at wc= +l, singular density at the boundaries, numerator proportional to y, sum of zeros (integral over x) between boundaries for fixed y gives

x2

g(y) = j- dxg(x, y> = “’ 4%-(cd)1’2 ’

(17)

XI

independent of the ay2 and by2 terms. Note that (cd)-l” = 2(sinh 2KJ2.


6. Triangular lattice

We sketch briefly how to extend our calculations to the anisotropic triangular

lattice with a third (diagonal) interaction J3. The factors {. . .} in (6) generalize

to3-5)

{* . *} = c,c*c, + s,sp, - s, cos4, - s, cos4, - s, d4, + A> 9 (18)

which, with the same notation for K and k as before, may be written

(...)=sK~k(~+~-c+-~c_+--$l-cos(m,+b,,l). k K K k

(19)

Now the appropriate complex variable is

w = SK e+2K31Ck. (20)

At the ferromagnetic critical point w = +l with 4, = 4, = 0. For an antifer-

romagnetic lattice with J, < J2< J3< 0 the range of w is (-03, m), and the

antiferromagnetic critical point is at w = -1, with 4, = 4, = ?7r. [With a

different ordering of the interactions the antiferromagnetic critical point would

be at one of the points w = *&Sk/C&,]. Here we choose the ratios of the

interactions to be fixed so J3 is the weakest interaction and

K, K, K3 -= -=A=

1+cw 1-a p K=k, withcu+p<l.

(Y (21)

When 4, = -&,, the zeros lie on the unit circle in the w-plane. Zeros pinch the

imaginary axis at w = -+i where 4, = -4, = +rr/2, and return again whenever

4, = -4s and S, = C, = 0 at w = *i/a, provided that e2K3 = ?l. [We now have

2K, = 2pK = +Iim ‘71. for some integer m’, as in the square lattice case.] Again

we write the vanishing factors in terms of w:

1+ w*- WC, - R,C_ + R,[l - cos(& + +,)I = 0, (22)

where

R, = SkCK e2K3/Ci and R, = S, e2K31C: . (23)

In principle the calculation of the density of zeros is as before, but is technically


complicated by the appearance of terms involving the product 4&s in the expansion of (22) near the ferromagnetic critical point:

The

X~l-x=~<~~+~:)+~R;,(~~-~~)+~R;,(~,+~,)’,

Y=Y*= ~(~5+~5)+~R,,(~~-~5)+~R,(~,+ 4,)‘. (24)

distribution of zeros has the same sort of analytical properties as before.

7. Differential equation for g(x, y)

To conclude this discussion of two-dimensional (area) zero distributions, let us relate our new results for g(x, y) to scaling theory. Abe has shown that if we perform linear scaling

X’=aX, y’=ay, witha=w,-w, (25)

in the integrand of the specific-heat integral, and take into account the IT- TJ” or a-” divergence of the specific heat, then g(x, y) satisfies the homogeneous Euler partial differential equation

ag+X-$+y~=O. ay (26)

The general solution is

g = x-xylx). (27)

Clearly the two-dimensional Ising-model distribution function with (Y = 0 does not have this form. The remedy is to allow for the parabolic cusp-like nature of the region containing the zeros by scaling the real and imaginary parts of w separately. (We are indebted to Dr. N. Rivier for this suggestion). If we perform instead a non-linear scaling

X’= a"X, y’= ay, (28)

then we obtain a modified differential equation

(29)


This new equation is now satisfied by the Ising distribution function in the case

(Y = 0, corresponding to a logarithmic specific heat singularity, and it = 2 for a

parabolic cusp.

8. Isotropic lattices

We conclude this paper with a summary of our results for the isotropic

square, triangular and honeycomb (hexagonal) lattices. In every case the zeros

lie on lines in the complex plane of the variable u = tanh K = x’+ iy’, which

intersect the real axis at the critical point(s), where the density of zeros in u has

the form6,‘)

g,(y’)= g,(y’)=$: c

so that the energy behaves like

and the specific heat has the well-known logarithmic divergence:

c 8hK2 -- -Llnlv,- ~1. Nk n-S2

(31)

(32)*

(* There is an exception -see below.) Values of A, and S = sinh 2K, are in table

TABLE I

Critical data, density of zeros g”(y), and the specific-heat critical behaviour for the square, triangular

and honeycomb lattices.

Lattice

s,

Square, Triangular,

ferromagnet ferromagnet

+1 11x0

Triangular,

antiferromagnet

--3o

Honeycomb,

ferromagnet

V/3

g”(Y) $ d3Y I djlY I 3V/3lY I

274 47-l 47rvf

-~lnlv,- U, 12x4 32d\/3

-

CINk --K~InIv,-vl -- K’ e4K 57 7r -+K:InJv,-vi


I, where we also list the densities of zeros in the vicinity of the real axis critical points, and the corresponding specific heats. The form (30) in the variable ZJ holds even for the isotropic antiferromagnetic triangular lattice near its “critical” point, u = -1, T = 0, but (N.B.) an extra factor of 2 is required in (32)*, since both factors (sinh K)-’ - 4 eK and u, - u - 2 eX are important in (31) near T = 0. However if the w-variable, introduced previously, is employed then for the isotropic triangular lattice

W = S(C + S) = ~(f’ - 1) , with z = eezK, (33)

where 0 < z < 1 and 0 < w < ~0 for the ferromagnet, and 1 < z < CQ and -k < w < 0 for the antiferromagnet. Under the inversion transformation w + w+ = l/~.~*~) In the w-plane the triangular lattice zeros lie not only on the unit circle, but also along the negative real axis in the unphysical interval (-2, -3. Moreover near the antiferromagnetic zero-temperature point, w = -i, the density of zeros is constant: g(y) - X6/r. These results are readily derived from (18) (19) or (22) on setting

cos@ = ;<w + w-l) = $S(C + S) + l/S(C + S)]

= $cos& + cos+, + cos( 4, + 4,) - l] ) (34)

in which the range of values (of the r.h.s.) is such that -gc cod c 1. The antiferromagnetic critical point is now at 4, = 4, = +2rr/3, where w = -i.

The results for the honeycomb lattice may be deduced from those for the (ferromagnetic) triangular lattice by means of the star-triangle transformation.3,4)

9. Summary

We have obtained the two-dimensional distribution function for temperature zeros of the anisotropic two-dimensional Ising model, and have shown its relation to a generalized scaling theory. More details of the present calculations, with examples on the anisotropic triangular lattice, and a discussion of some of the remaining complications and puzzles, will be presented else- where. Some of the results contained in this paper have previously been announced at conferences: 3rd Open University Statistical Mechanics Con- ference, Milton Keynes, England (December 1979); 14th International Con- ference on Thermodynamics and Statistical Mechanics, Edmonton, Alberta, Canada (August 1980); 18th Solid State Physics Conference, York, England


(January 1981). (Note: Whilst the present work was being prepared for pub- lication (February 1984) the authors received a preprint of a paper by van Saarloos and Kurtze14), who have also examined the problem of temperature zeros for anisotropic Ising lattices.)

References

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Partition function zeros for the two-dimensional Ising model