Scaling theory for the zeros of the Mayer cluster sums for the Ising lattice-gas

~~~~p~r~r.9

ckn~.

VOI.

14, NO.

4. pp.

2955298,

1990

Printed in Great Britain. All rights reserved

Copyright

0097-8485190 $3.00 + 0.00 0 1990 Pergamon Press plc

SCALING THEORY FOR THE ZEROS OF THE MAYER CLUSTER SUMS FOR THE ISING LATTICE-GAS D. S. GAUNT Department

of Physics, King’s College, Strand, London WCZR 2LS. England (Received in revised form 20 December 1989)

Abstract-We study the zeros of the Mayer cluster sums (b,) for the Ising lattice-gas, or equivalently the high-field polynomials (L,) of the spin - : Ising ferromagnet with nearest-neighbour interactions, for two- and three-dimensional lattices. There are (n - 1) zeros in the temperature range T, < T c cc and their behaviour in the limits n + cg, T-+ T, and n + a, T-co is described by different scaling laws. Close to r,, “horizontal” and “vertical distances” are controlled by the standard critical exponents A and 6, respectively, while at high temperatures the corresponding exponents are dimensionally invariant constants. In the limit n+c~. the interval T, < T < co becomes dense with zeros and we use scaling arguments to determine the form of the limiting density function.

,CL and u are magnetic field and temperature variables, respectively, defined by

1. INTRODUCTION As is well-known, the pressure of an imperfect gas may be expanded in powers of the activity z as follows (Mayer & Mayer, 1940): p/k8 I- = 2 h,,(T)z”. fl= I

,u - exp( - 2mH/k,

interactions, the expansion

CAC

M/&-O

exp( - 4J/ks T). (3)

In n = (u’)~ + (2~’ - 2$&* f(6U4-

l6u’+

10$24~)~3+~~~ .

The expansion for the corresponding lattice-gas obtained by first re-writing equation (4) as

(4) is

In /i = (u*p) + (22.-I - ~+)(u’~C)~ + (6u-* - 16u-’ + I~#)(u~P)~ +. .

.

(5)

Using the lsing lattice-gas analogy (Lee & Yang, 1952), we now replace In/f by p/kBT and u2p (or more generally &,u) by I, to obtain

the configura(Sykes et al.,

1965) In.4 = 5 &(u)p”; I,= I

u -

[As a result of its odd coordination number (q = 3), the variable x (=u’/‘) is used instead of u for the honeycomb lattice.] The L,, in equation (2) are polynomials in u, called the high-field polynomials. They have been derived to quite high-order by the “code” method of partial generating functions (Sykes et al., 1965, 1973a, b, 1975; Sykes & Hunter, 1980). In two dimensions (d = 2), the number (N) of known polynomials are N = 25, 21 and 12 for the honeycomb, square and triangular lattices, respectively; and in three dimensions (d = 3), N = 17, 15, 1 I and 8 for the diamond, simple cubic, body-centred cubic and facecentred cubic lattices, respectively. It is these highfield polynomials (L,z) which are the analogues of the Mayer cluster sums or integrals (b,). As an example, consider the two-dimensional square lattice for which the expansion in equation (2) begins like (Sykes et of., 1965)

(1)

The coefficients b,, are functions of the temperature T, and their behaviour has been the subject of considerable speculation for many years (Mayer & Mayer, 1940; Katsura, 1954, i958, 1963; Kihara & Kaneko, 1957; Kihara & Okutari, 1971; Majumdar, 1974; Majumdar & Ramarao, 1976). Questions of interest include the radius of convergence of equation (l), the location of the closest singularity on the real positive z-axis (which determines the phase transition) and its relation to the corresponding singularity of the associated virial series. Unfortunately, for most interaction potentials, it has not proved possible to calculate more than the first few coefficients. For example, for the Lennard-Jones 124 potential, only the first five coefficients have been calculated (Barker et al., 1966). Considerably more progress has been made for the simple lsing lattice-gas, where up to 25 and 17 coefficients are available in two and three dimensions, respectively, depending on the lattice. We have made a detailed study of these coefficients in the hope of elucidating the more general features of the &,-coefficients and have shown how their behaviour determines the critical temperature and critical exponents of the corresponding Ising lattice-gas. For the spin -4 Ising model with nearestneighbour ferromagnetic tional ftee energy has

T),

p/kBT=z+(2um’-2$)z*

(2)

+(6u-‘295

16~‘+

l@)z3+...

.

(6)

D. S. GAUNT

296 This expansion with

is of the same form as in equation

(I)

may be written

as

L.(z4)~n~‘-“‘8)F(v), b,=l,

b,=6u-?-116~‘+lOf,

I..

.

(7)

Finally, we note that in Mayer theory it is more usual to re-write equations (7) in terms of a new temperature variable

in terms b,=l,

of which,

s=u-‘-

1,

the Mayer

cluster

b,=2f--_:,

b,=6f2-44f+:

(8) sums become ,.__

where F is some (unknown) variable v defined by

2. THE ZEROS In earlier work (Gaunt, 1978). we found that all the known L,,-polynomials have exactly (n - 1) zeros lying on the real u-axis in the physical interval U, < u < 1, where u, is the value of u at the critical temperature T,. This interval corresponds to T, < T -c co. There are no zeros in the interval 0 < u < U, (corresponding to 0 < T < T,), so that any remaining zeros (apart from the multiple zero at the origin) lie either on the negative real u-axis or in the complex u-plane. No pattern for the location of these non-physical singularities has been noticed. ’ The existence of (n - 1) zeros of L,l(u) in the interval (u,, 1) is an intriguing result. Furthermore, the zeros are found to exhibit the interlacing property possessed by the zeros of many orthogonal polynomials. We conjecture that both these features are true for all lattices in all dimensions (and not just d = 2 and 3). Since the (n - 1) zeros interlace in the interval (u,, I), the first zero (i.e. the one with the smallest value of u) will approach U, as n increases, as will the second, third etc. zeros. Alternatively, the last zero (i.e. the one with the largest value of U) will approach u = 1 as n increases, as will the next-to-last zero etc. To investigate this behaviour further, we have developed appropriate scaling forms for L,,(U), one valid close to u = u, and one close to u = 1. We study each of these in turn. A. Vicinity qf u, we have already presented this region, 1978) a scaling formulation of L,,(u). This

scaling

v = n”3jJ

U-+#,,

(10)

function

of the

- u,);

(11)

6 is the critical exponent describing the shape of the critical isotherm and A is the gap exponent (Domb, 1974). For the two-dimensional Ising model, the exact values of the critical exponents are

(9)

In this paper, we study the zeros of the b,,-polynomials for the Ising lattice-gas. It can be seen, by comparing equation (6) with equation (4), for example, that these zeros are identical to the zeros of the &-polynomials for the ferromagnetic Ising model. [In addition, the _&-polynomials have a trivial multiple zero at the origin (u = 0), which is of no interest.] In view of this precise correspondence, we couch our subsequent discussion in terms of the Ising model and the high-field polynomials, .!,,,(u).

For (Gaunt,

n4oo,

&=2u_‘--22f,

6 = 1.5,

A = 15/8.

(12)

In three dimensions, renormalization group estimates are (Le Guillou & Zinn-Justin, 1980) 6 = 4.82 + 0.025,

A = 1.566 +_ 0.0035.

(13)

Denoting the zeros of L,(U) by uln, I+~, ula,. . . (where U, < u,,, -C IA,,”< Us.”K . . . ), we see that the i th zero of L,,(u) implies a corresponding zero in F(u) at uj = n ““(U,, - UC)> 0, whereO
(14)

Hence,

ui.,,=uC+(vi/n”*),

i=l,2,3

n-+o~,

.. ..

(15)

i.e. as n increases, the ith zero of L,(u), counting from u,, approaches the critical point U, at an asymptotic rate characterized by the exponent l/A and with an amplitude determined by the ith positive, real zero of the scaling function F(u). Similarly, one can show that the location of the max-min points of L,(u) should scale like 1/n ‘IA. While “horizontal distances”, measured from u,, scale like 1/n ‘I*, it is not difficult to show that “vertical distances” such as the magnitude of L,, at the critical point or at a max-min point, should scale like i/nz+(“6). B. Viciniry ofu

= 1

Since L,,(U) is a polynomial, it may be expanded a Taylor series about u = 1, namely L,(u)

= 5

m= 0

(LX’“‘(k)/m !)(u -

I)“,

in

(16)

where L!:“)(l) is the mth derivative with respect to u of L,(u) evaluated at u = 1. It can be proved rigorously that ,~‘fl’(l),(_lyl+‘/4 ,”

I.,?

2,X- i

*

n-co, where

m =0,1,2

. ..)

(17)

for all lattices A,, = 1.

(18)

Zeros of the Mayer cluster sums for the Ising lattice-gas

Substituting gives

expression

(17)

into

equation

(16)

The limiting

function

p(u)

is now

defined

by

L,(u)

N f (m=O

l)“+‘(A,,/m!)n*-‘(u

- ((-- l)“+‘/n)

5 (m-0

P(U) = hm P,(U), n-m

- 1)”

l)“(A,,/m

w-

(- lY”

!)

p(u)--R(u

G(y)

U--rl,

?ldco,

2

n

(19)

where y = (1 - U)RIZ.

(20)

G(y) is some unknown scaling function ZJ= 1 and with the property

defined

G(0) = A,.o = 1.

near

n-+co,

i = 1,2,3..

.,

(22)

i.e. as n increases, the ith zero of L”(u), counting from u = 1, approaches u = 1 like l/n* with an amplitude determined by the ith positive, real zero of the scaling function G(y). By arguments similar to those used in the vicinity of u,, one can show that expression (19) implies all “horizontal distances” measured from u = 1 should scale as 1/n *, and all “vertical distances” as l/n. 3. DENSITY

We have seen that L,(U) has (n - 1) zeros in the interval (y. 1) and that as n increases the zeros interlace, approaching u = u, and u = 1 like l/n”4. and 1/n’, respectively. Presumably, in the limit n + cc, the interval (u,. 1) will become dense with zeros. In this section, we use scaling arguments to determine the form of the limiting density function p(u). First, we define a probability density function p,(u) such that the number of zeros of L,(u) in the interval u to (U + du) is equal to (n - l)p,(u) du. Clearly, p,(u) = 0 Vn if u c u,. The total number of zeros in the interval (u,, l), equal to (n - 1), is obtained by integrating over U, i.e.

This means

that p,(u)

=n

is normalized p,(u) du = 1.

-U))‘,

u Z-U,,

(25)

Uh(n - I)pJu) s UC

du = i.

Substituting for p,,(u) its limiting of y, namely p(u) w %(u - u,)‘,

form in the vicinity

R, = R(1 - u,)_‘,

gives (n - f)R,

UI,” (ti-uU,)5du s UC

=i.

To calculate the integral, substitute w = u - u,, and for the upper limit use the expression given in equation (15). Thus, we obtain (n - l)(s + l)-‘~r;+‘,-@+l~~* For the 1.h.s. to be independent (s + 1)/A = 1, or

= i.

of n, we must

s=A-1.

(26) have (27)

An extension of this argument gives the scaling behaviour of the ui. Thus, substituting equation (27) into equation (26) we get

OF ZEROS

‘(n - l)p,(u)du JQ

-UJ(l

where R is a constant amplitude, and s and t are both positive. We now use scaling arguments to derive expressions for the exponents s and 1, which are dominant in the vicinity of U, and u = 1, respectively. The number of zeros up to and including u,,~ is i and, from the definition of P,,, is given by

(21)

Clearly, zeros of G(_v) imply zeros of L,(u). If G(y) has zeros at _Y,._Y~,Y~,... (O-=~y,cy~ ZQ > u,,~ > . . . . Hence,

I -(y,,h*),

(24)

and is zero for u < u,. For u 2 u,, numerical and other evidence suggests that for the two- and three-dimensional Ising model

x [(I - u)dy

,+=

density

297

- 1. such that (23)

&=iA/up,

and since R, must

be independent 2$-z

*I/A

of i, this implies

.

(28)

A completely analogous treatment can be used in the vicinity of u = I and readily yields t =f. To summarize scaling arguments

yi * i*.

our main result, to show that

p(u) _ R(u - u,)A-‘(1

- u)-“2,

(29) we have

used

u 2 y.

(30)

The behaviour close to U, is controlled by the exponent A, as were the positions of the corresponding zeros [see equation (IS)], while close to u = 1 we find square-root behaviour for p(u), independent of dimensionality and reflecting the l/n’-scaling observed for the positions of the zeros [see equation (22)l.

D. S.

298

4. DISCUSSJON In this paper we have outlined a scaling theory for the zeros of the Mayer cluster sums (b, or, equivalently, L,,) for the Ising lattice-gas. There are two distinct scaling regions, one close to the critical temperature (corresponding to u > u,) and the other at high temperatures (corresponding to u < 1). Our main predictions are as follows. The rates at which the zeros approach u = U, and u = 1 are I/H”~ and l/n’, respectively, for n large. (A is a standard critical exponent called the gap exponent.) In the limit n + cc, the interval U, c u < 1 becomes dense with zeros. Close to u,, the limiting density function, p(u), behaves like (u - u,)~-’ while, in the vcinity of u = I, p(u) has the form (1 - u)-“~. Many interesting aspects of this problem remain to bc explored. Numerical results consistent with the l/n”A behaviour have been presented by Gaunt (1978) for the two- and three-dimensional Ising models. How well does the numerical evidence support the other scaling predictions? Clearly, l/n”* and I/n’ are only the leading terms in asymptotic expansions about u = U, and u = 1, respectively. The form of the higher-order correction terms is of particular interest. We have formulated our scaling theory in terms of a simple Ising lattice-gas, but preliminary work suggests that many of the rest&s may be far more general (e.g. Gaunt, 1978). Finally, we would dearly love to have a proof-rigorous, in the case of the lsing model-for the existence of precisely (n - 1) zeros on the real, positive u-axis between u, and 1, and an understanding of the extent of its apparently wide generality.

GAUNT

Acknowledaements-I have benefited from stimulatine discussions with all my colleagues in the King’s Coll& Statistical Physics Group, but especiallyG. S. Joyce, on all aspects of this work.

REFERENCES Barker J. A., Leonard P. J. & Pompe A. (1966) J. Chem. Phvs. 44. 4206. Domb C. (i974) Phase Transitions

and Critical

Phenomena,

Vol. 3 (Edited by Domb C. & Green M. S.), _ D. 357. Academic Press, New York. Gaunt D. S. (1978) J. Phys. A: Math. Gem 11. 1991. Katsura S. (1954) Proa. Theor. Phvs. 11. 476. Katsura S. i1958) Proi. Theor. P&S. 20; 192. Katsura S. (1963) Adv. Phvs. 12, 391. Kihara T & K&eko S. -(1957) J. Phys. SOC. Jpn. 12, 994.

Kihara T. & Okutari J. (1971) Cllem. Phys. 63. Lee T. D. & Yang C. N. (1952) Phys. Ret>. 87, Le Guillou J. C. & Zinn-Justin J. (1980) Phys. 3976. Majumdar C. K. (1974) Phys. Rev. B 10, 2857. Majumdar C. K. & Ramarao I. (1976) Phys. 1542. Mayer J. E. & Mayer M. G. (1940) Stu~isticaf

L&t.

8,

410. Rev. B 21,

Rev. A 14, Mechanics.

Wiley, New York. Sykes M. F. & Hunter D. L. (1980) J. Phys. A: Math. Gen. 13, 1121. Sykes M. F., Essam J. W. & Gaunt D. S. (1965) J. Math. Phys. 6, 283.

Sykes M. F.. Gaunt D. S., Mattingly S. R.. Essam J. W. & Elliott C. I. (1973a) J. Math. Phys. 14, 1066. Sykes M. F., Gaunt D. S., Essam J. W., Heap B. R., Elliott C. J. & Mattingly S. R. (1973b) J. Phys. A: Math. Gen. 6, 1498. Sykes M. F., McKenzie S., Watts M. G. & Gaunt D. S. (1975) J. Phys. A: Marh. Gem 8, 1461.