The Ising ferromagnet with impurities a series expansion …rapaport/papers/72b-jphc.pdf · 2003....

J. Phys. C : Solid State Phys., Vol. 5 , 1972. Printed in Great Britain. @ 1972

The Ising ferromagnet with impurities : a series expansion approach: I

D C RAPAPORT Wheatstone Physics Laboratory, King’s College, London

MS received 30 March 1972

Abstract. High temperature expansion techniques are used to study the changes in the critical behaviour of the susceptibility of the Ising ferromagnet arising from the introduction of nonmagnetic impurities. We consider two kinds of system in which the impurities are randomly distributed and fixed in position-in one the impurities are located on the lattice sites (the dilute magnet) while in the other they are allowed to occupy the bonds of the lattice. The mobile impurity version of the latter system is also studied, both analytically and numerically, and we find that the normal methods of series analysis are unable to predict its critical properties correctly. Analysing the expansions for the fixed impurity models also presents difficulties and, even though the two types of bond impurity model are fundamentally different, the analysis fails to detect any significant difference in their behaviour, except at very high impurity concentrations.

1. Introduction

The critical properties of ideal magnetic systems such as the Ising model are believed to be fairly well understood, thanks to the availability of powerful techniques for analysing series expansions, backed up by a number of exact analytical results (see eg reviews by Fisher 1967, Kadanoff et al 1967). The large body of information accumulated by these methods has confirmed the inadequacy of classical theories of second order phase transitions and led to the proposal of a number of new theories of the features underlying critical behaviour (notably scaling and universality). However, the models which are generally studied have translationally invariant local properties, and before making contact with experiment it is desirable to have some idea of the effects of factors which reduce this symmetry. Since the phase transition is ‘driven’ by the long wavelength fluctuations of a suitably defined order parameter, a change in critical behaviour is likely to result from any alteration of the local properties extending over a macroscopic region. Examples are a finite (but large) lattice size, grain boundaries, local variation of interactions (eg by coupling with the vibrations of a compressible lattice), and impurities. A number of these factors have been studied, especially in regard to their effects on the two dimensional Ising model. A finite system obviously cannot display any singular behaviour, but in a large, yet finite, system it is often difficult to distinguish between a true singularity and a sharply peaked analytic function (Ferdinand and Fisher 1969). Periodic grain boundaries-pictured as rows of spins coupled by interactions whose strength differs from that of the rest of the system-do not affect the critical exponents,

1830

The Ising ferromagnet with impurities: I 1831

but merely shift the critical point and change the amplitude values (Fisher 1969). The introduction of a compressible lattice can result in a renormalized second order or even a first order transition (Baker and Essam 1970, 1971).

A familiar impurity problem in the literature is the dilute magnet-a lattice in which a random selection of sites are occupied by Ising or Heisenberg spins, the remainder by nonmagnetic impurities. An equally simple, yet not so well known scheme of incorporat- ing impurities does not require the absence of any spins; instead the nonmagnetic impurities are allowed to occupy the bonds between neighbouring lattice sites-to simplify matters a maximum of one per bond-and their only effect is to prevent the interaction of the spins at either end of each occupied bond?. It requires little effort to envisage numerous extensions of these models but, as we shall see, they present sufficient difficulty as they are.

It has been pointed out (Brout 1959) that the impurities need not necessarily be in complete equilibrium with the spins. The time characteristic of any fluctuation in the position of an impurity might be expected to be considerably longer than for spin relaxa- tion; but the degree to which the critical spin fluctuations are affected by the mobility of the impurities is not clear. One begs this question by considering the limiting possi- bilities, namely that the spin fluctuations are either too rapid to be affected by the impurity mobi1ity;or sufficiently slow for the impurities to participate in the fluctuations so that a state of equilibrium exists. The two kinds of situation can be realized by intro- ducing the impurities at high temperature and then either cooling the system rapidly- quenching-which results in the impurities being ‘frozen’ randomly into the lattice, or slowly-annealing-to obtain the equilibrium state.

Determining the thermodynamics requires evaluation of two averages. In an annealed system the free energy is found by averaging simultaneouslv over all possible configurations of spins and impurities, generally with the aid of the grand canonical ensemble. The Syozi model-an Ising system with magnetic impurities on the lattice bonds-is one system which has been treated in this way (Syozi and Miyazima 1966, Essam and Garelick 1967). Systems which contain quenched impurities are not strictly equilibrium systems, and cannot be treated in the same way. Instead (Brout 1959, Mazo 1963), an ensemble of systems is considered, each one having fixed, randomly distributed impurities with a given overall concentration. Once the free energy of each member of this ensemble has been determined, the free energy (more precisely, the mean free energy) is found by averaging over the ensemble. In terms of the partition function, ln(Z)s,c is essentially the free energy in the annealed case, whereas for the quenched system it is (In (Z)J,, with s and c indicating spin and impurity configurational averages. A method of treating quenched site impurities by means of a special ensemble whose equilibrium properties are those of the quenched system has been proposed (Morita 1964), but actually using the method beyond the leading order approximation presents considerable difficulty.

Classical approximation techniques have been used to study the effects of various kinds of annealed and quenched impurities in Ising and Heisenberg systems (eg Bell 1958, Sato et al 1959, Bell and Fairbairn 1961, Bell and Lavis 1965). Such methods cannot predict critical properties correctly, but they frequently provide a useful guide to the behaviour away from the transition.

Quenched site impurity systems have also been investigated by means of series expansion methods, most of the effort having been directed to finding out how the

t We call this a ‘bond impurity’ system, and the dilute magnet a ‘site impurity’ system.

1832 D C Rapaport

temperature at which the susceptibilitv diverges varies with concentration for various lattices. Two types of expansion have been used : one in terms of concentration (Cooper- smith and Brout 1961, Rushbrooke and Morgan 1961, Elliott and Heap 1962, Heap 1963, Morgan and Rushbrooke 1963); the other a high temperature expansion (Behringer 1957, Morgan a n i Xushbrooke 1961, Rushbrooke 1971). Between them, these expansions allow, in principle, the study of the behaviour above the transition point at both high and low impurity concentrations. The consequences of extending these expansions for the king model on the face centred cubic (FCC) lattice are among the topics discussed in this and a subsequen, article (referred to as 11).

There are a number of rigorously established results for Ising and Heisenberg ferromagnets with quenched site impurities (Griffiths and Lebowitz 1968), among which are the existence and continuity of the free energy, and for the Ising model, its analyticity in nonzero field, and the existence of spontaneous magnetization when both the impurity concentration and temperature are sufficiently low. The problem of whether the formation of an infinite cluster of spins is possible is one of site percolation theory (see eg Frisch anc Hammersley 1963): if the spin concentration is less than a certain (lattice dependent) critical value p," the probability of an infinite cluster occurring is zero and any transition is impossible (Elliott et a1 1960, Domb and Sykes 1961). A further result is that the Ising magnetization is a nonanalytic function of field strength H at H = 0 for a range of temperatures extending at least as far as the critical temperature of the pure system (Griffiths 1969).

Similar results might be expected to apply to the quenched bond problem, but proofs have yet to appear. One result which can be demonstrated however (see 11) is that if the concentration of bonds not occupied by impurities is less than the critical concentration p," of bond percolation theory a transition is impossible.

The only soluble system with large scale random properties (excluding one dimensional systems which do not have a transition-Katsura and Tsujiyama 1966) is a square Ising system with constant nearest neighbour interactions between spins in different columns ; however the interaction between spins in different rows is a constant for any given pair of rows but its actual value is a random variable with a prescribed distribution. If this distribution is narrow, the divergence in the zero field specific heat disappears, to be replaced by a function which is analytic except at a certain temperature T,, where, to leading order in the distribution width, it is both singular and infinitely differentiable (McCoy and Wu 1968). Extension to the case of an arbitrary distribution (McCoy 1970) again results in the same kind of essential singularity at q, but it is not known whether the singular behaviour is confined to T, alone. Other results derived for this model indicate the possibility that the susceptibility is infinite for a range of temperatures above T,, and that T, itself is the temperature at which the spontaneous magnetization vanishes (eg McCoy 1971).

The relevance of these results to an Ising system containing quenched impurities is questionable for two reasons. First there is no translational invariance at all, whereas in the McCoy and Wu model invariance in one direction remains. This might be expected to further complicate the mathematical structure of the quenched models. Second, there is only one interaction strength involved rather than a distribution of values, and this would tend to simplify the behaviour. Nevertheless the results do establish a prece- dent for more complicated behaviour than the usually encountered exponent type singularities and, consequently, deductions based on series analysis must be made with great caution.

The principal purpose of this article is to describe the derivation (5 4) and analysis


(4.5 and 5 6) of high temperature susceptibility series for spin-+, nearest neighbour Ising systems containing quenched ’bond and site impurities. We have confined our attention to the FCC lattice and derived the first ten terms of the series for both kinds of impurity system-for the site problem this represents the addition of two extra terms to Rush- brooke’s (1971) results, and we believe that this is the first time a study has been made of the quenched bond problem. We found a particularly strong similarity between the quenched bond system and the corresponding annealed model, and for this reason, in addition to discussing the latter from an analytic point of view (4 2) , we treat 11 a b a problem in series analysis (4 3 and 4 6) in order to make the comparison of the two systems possible. A discussion of concentration expansions for the susceptibility is deferred to paper 11, as is a study of the fourth field derivative of the free energy. Preliminary accounts of parts of this work have already been published (Rapaport 1971a, b).

2. The annealed impurity model

We consider an assembly of N Ising spins situated on the sites of a regular lattice of co- ordination q. The spin on the ith site can take the values oi = -t 1. The spins are assumed to interact through an exchange coupling between nearest neighbours of the form - Joioj which, for J > 0, favours ferromagnetic ordering; in a uniform field H there are additional interactions - poHoi, where po is the spin magnetic moment. This system is just the usual Ising model. Nonmagnetic impurity atoms are now introduced into this system and are allowed to occupy the bonds between nearest neighbour sites. Not more than one impurity atom may occupy any bond, and the effect of an impurity is to prevent any interaction between the spins at either end of the bond. Thus, if pij = 0,l denote respectively the presence or absence of an impurity between the sites i and j , the interaction between the corresponding spins can be written as - Jpijoioj.

If { p } denotes a given arrangement of impurities the hamiltonian for configuration {PI is

ZN{P} = - J pijoioj - poH oi bonds sites

In this model the impurities are in complete equilibrium with the spins, and its behaviour at any temperature T can be determined from the grand partition function, which involves a sum over all configurations { p }

q N l 2 z”(K, L, 5) = 1 en( 1 E’ exp { K p..o CJ + L 1 oi} (2.1)

where K = P J , L = PpoH, 5 = P x chemical potential (P = l/k,IT: k, is Boltzmann’s constant) and the prime on the sum over p i j indicates that it is subject to the total number of bonds without impurities being equal to n, that is

n = O ( u , = * l } { p , J = o , l } bonds ” ’ sites

We can rearrange (2.1) to give

ZJK, L, 4 ) = C n exp (Loi) n Pij(ai, oj) {a, = i 1) sites bonds

where Pij is the contribution to ZN of the bond i-j. Now

P,, (o,, 0,) = 1 exp ((Koioj + 5) pi,} = 1 + exp (Koiaj + 5 ) p i j = o , 1

1834 D C Rapaport

If we are able to express Pi, in the form 4 exp (K’op,), with 4 and K‘ suitably chosen functions of < and K , then the problem reduces to the ordinary king model-the reference system-on the same lattice

Z,(K, L, 5 ) = C exp (Loi) n 4 exp (K’aiaj) {U, = f I ) siter bonds

(2.2) - - $*“2 2; (K’, L)

where Z i is the ordinary Ising partition function. Expressions for q5 and K’ are readily found, confirming the validity of (2.2). T h r j are

42 = (1 + ec+K)(l + (2.3)

This technique of eliminating ‘decorated’ bonds and reducing the model to a simplified reference system is well known (eg Fisher 1959).

The relation between the model and its reference system is simpler than in the Syozi model (Essam and Garelick 1967) for, unlike the Syozi model, the variable characterizing the field dependence is unchanged in the reference system. In the thermodynamic limit

In Z(K, L, 5 ) = lim N - ’ In ZN N-+Z

= $qln+ + l n Z R where

l n Z R = lim N - ’ 1nz; N-CC

We denote the concentration of impurities by j , where

and P = l - p

2 p = - lim N - ’ ( n )

N + z

with

2 a eR(K’,L) = ---1nZR q dK’

eR is just the pair correlation function (per bond) of the reference system. p is the concentration of bonds not containing impurities.

The next step is to determine how the reference system variable K‘ is related to K and p. Using equations 2.3-2.5 we can eliminate the quantities 4 and 5 with the result that

where


TC(K’,L) = :(I - e-2K‘ + ER) (2.8) (Equation 2.6 differs from the corresponding Syozi result in that exp (2K) is replaced by cosh (2K).) We know that in zero field the reference system has a critical point at some lattice dependent value K’ = Kb. If the impurity model is to exhibit a transition it can only occur at the corresponding value of K , namely Kc(p), given by

where

P o = %)(Kb,O) & = N q , O ) Both p o and pf’ are obviously positive, and pf’ < 1; thus a physical solution to (2.9) exists only if p 2 pf’. A phase transition will only occur for p between and 1, so that $ represents a critical concentration below which no transition is possible. It has the same value as in the Syozi model.

I . O I A

Figure 1. The curves T,@)/T,(l) against p for the annealed bond (AB), quenched bond (QB) and quenched site (QS) impurity systems. For QS, T,@) could not be determined below p E 0.6; for QB the curve extends down as far as p = 0~15-below this value there is too much scatter in the Pade results. The maximum separation between the QB and AB curves is less than the line thickness.

Tfie critical temperature 7 J p ) = J jk ,Kc(p ) is equal to that of the reference system when p = 1, and zero when p = &. The locus of critical points, or critical curve, can be

1836 D C Rapaport

determined numerically from (2.9), and figure 1 includes the curve for the FCC lattice?. F o r p 2 $

and the gradient of the critical curve is infinite at $. Near p = 1 we find

which for the FCC lattice becomes

that is for p near 1 the curve is very close to the straight line q ( p ) = pT(l)-see figure 1. The derivation of the thermodynamic properties follows the Syozi model. The internal

energy is

a u(K,L,p) = - J - l n 2 8K

which for p = 1 reduces to the expected -iJqcR. The specific heat is

and since

it is easily shown that near the critical point

(2.1 1)

where A(p) and B,(p) are positive for of the reference system, which in three dimensions has the form

< p < 1. dER/dK' is essentially the specific heat

A , 1 - - + A , + . . . ( E;)- K ' 6 Kb (2.12)

in zero field. The accepted value of c( is $, and the values of the constants A , and A , are known from series analysis (Sykes et al 1972b).

In order to determine the critical properties we must express 1 - K'/KL in terms of

+ Based on the latest estimates of the critical quantities for the FCC lattice (Sykes et ai 1972a, b), the values of p,and f f are 0,0852 and 0.1 152 respectively.


1 - K/Kc(p), or T / q ( p ) - 1. This we do by integrating (2.10) with respect to K’, and the result is found to be

- T - 1 2: m,(P)( 1 - 6) + (1 - p ) m,@)( 1 - E)l-a (2.13)

T(P) KC Kb neglecting the terms of higher order. The smoothly varying functions q51 and q52 can be shown to be positive when p t < p < 1 (for 41 this must be done numerically). The solution of (2.13) is

(2.14)

to leading order, and substituting this into (2.12) we find that (2.11) becomes

c = 4 4 - q P ) T 2 T(P)

where B2(p) is always positive. The specific heat curve therefore has a finite cusp at the critical point rather than the normal divergent singularity, and its slope is infinite at that point. This behaviour is just another instance of exponent renormalization (Fisher 1968). Similar behaviour occurs on the low temperature side (a is replaced by U‘), In two dimensions the cusp again appears, but it involves a logarithmic term, just as in the Syozi model.

x initial susceptibility. (In future, susceptibility will be taken to mean reduced susceptibility.) Since

We now turn to a discussion of the reduced initial susceptibility 31, which is just (Bp:)-

a 2 x(K, L, p ) = -In 3 aC

the susceptibility is the same function of K as that of the reference system is of K’. Now

(2.15)

(eg Fisher 1967), so from (2.14) it follows that

The value of the exponent y (believed equal to in three dimensions) has changed to the renormalized value y/( l - a) (= y). Renormalization also occurs below the critical point, and in two dimensions a logarithmic term appears in the susceptibility both above and below 7Jp) (cf the Syozi results). The same renormalized exponents are observed when, instead of approaching ?Jp) at fixed p , the temperature is held fixed at a value below q(l) and p allowed to approach the value at which that temperature is critical.

It will become apparent in the following section that if the model is studied using the standard methods of series analysis it is extremely difficult to determine the value of the renormalized susceptibility exponent. What in fact happens is that as the critical point

1838 D C Rapaport

is approached the system starts by mimicking the unrenormalized behaviour, and it is only very near the critical point that true renormalization appears. The key to this behaviour is the way in which (2.13) is solved for K', for use in (2.15). Sufficiently close to 7 J p ) equation 2.16 is correct to leading order-with correction terms it can be written as

3c c o t - ~ / ( l - 4 + 1 t ( - Y + a ) l ( l - a ) + c2t ( - y + W / ( l - a ) + , . . (2.17)

where t = T/Tc(p) - 1 and the ci are functions of p . However, a little further from the critical point it is the relative magnitudes of the coefficients in (2.13) which is a more important factor in determining the apparent (as opposed to actual) critical behaviour, rather than the difference of a between the powers of 1 - K'KL. The first order approximation to the solution of (2.13) is then

1 - K'/K; = t/$l

and so on to higher order, with the result that

More generally

N cot-' + c , t - ' - a + c 2 t - y - 2 a + . . , (2.18)

Each term diverges more strongly than its predecessor, but even so it is (2.18) rather than (2.17) which represents the behaviour over most of the critical region, excluding a very small region surrounding the critical point itself.

An order of magnitude estimate of the temperature T* at which renormalization starts to be clearly visible may be obtained by determining the temperature for which the two terms on the right side of (2.13) are equal. The value of K' for which this occurs is given by

Hence

and since $JcPl is of order low2 for p not too near 6, it is clear that over most of the concentration range T* is extremely close to the critical point. The temperature interval in which renormalized behaviour occurs is, therefore, usually extremely narrow (see also Guttman 1970).

The annealed version of the site impurity problem is merely a magnetic lattice gas in which atoms situated on adjacent lattice sites interact via the normal Ising interaction. Its properties can be expressed in terms of a reference system which is a spin-1 Ising model with an extra term in the hamiltonian proportional to C 0:. Series methods have recently been used to study the behaviour of the reference system (Oitmaa 1972), but the series are of insufficient length to allow determination of the exponents with any degree of con- fidence. If longer series do succeed in revealing the critical properties of the reference system, then the properties of the annealed site model should follow by means of a calculation similar to that described in this section.


The annealed bond impurity model is the simplest member of a class of models which includes Ising systems with more than one possible type of interaction between spins (eg the Syozi model-also Kasai et a1 1969, James 1971), the mobile electron ferromagnet (Fisher 1968, Scesney 1970), and lattice gas models of multicomponent liquids (eg Widom 1967, Clark and Neece 1968). Each of these systems can be transformed to an Ising reference system, but since they sometimes undergo two or more phase transitions the transformation can be quite complicated. A different approach to this kind of problem was followed by Lushnikov (1968,1969) who employed the steepest descent method in a study of the two dimensional annealed bond system.

3. Numerical study of the annealed model

We saw in the previous section that the full effect of renormalization is visible in a very small temperature range which itself lies within a region where the system appears to behave as if it were unrenormalized. This, together with the fact that the critical form of the susceptibility consists of a sum of confluent singularities of slowly varying strength, suggests that a numerical study of the high temperature susceptibility series might yield results which differ from those expected theoretically. Since we already understand the behaviour of the annealed model, the results of such a study would be of little use per se; but when we later consider the quenched version we will see that the relative difference between the values ofthe coefficients in the annealed and quenched bond series is generally quite small. Whatever difficulties are encountered in analysing the annealed series might well be expected to arise also in the quenched model. Prompted by this we have studied the annealed bond x series, and in this section some of the results of this analysis are described.

Twelve terms of the x expansion for the FCC lattice are readily generated using available series for the partition function and susceptibility (Sykes et a1 1972a, b) of the regular Ising model. The series for the correlation function eR follows from the partition function by differentiation, and when used in (2.7) and (2.8) yields expansion for n(K',O) and no(K', 0). These are used in (2.6) to produce a series of the form

K' = 1 h,(p) K" "10

and hence a series for w' (= tanh K ' ) in terms of w (= tanh K ) , which is the variable commonly used for high temperature susceptibility expansions. Finally this series is substituted into the susceptibility expansion with the result

These series manipulations were carried out by computer. The series (3.1) was analysed for particular values of p to determine the critical beha-

viour of x. The theory predicts that at p = 1, where (3.1) reduces to the series for the regular Ising model the series has a critical singularity of the form {wc(l) - w}-5'4 whereas for e < p < 1 the singularity should be {wc(p) - w } - ~ ' / ' . However the coefficients X ; are smoothly varying functions ofp, and it is hardly likely that for a value ofp only slightly below unity the estimated value of the exponent will have jumped to the renormalized value. This is indeed found to be the case.

The Pad6 approximant method (eg Baker 1965) applied to the logarithmic derivative (Dlog) of (3.1) yielded estimates of both the critical point and the exponent (see table 1).

1840 D C Rapaport

Table 1. Estimates for w,@) (upper entry) and y @ ) (lower entry) for the AB model obtained from the [D, .Wl Pad6 approximants for various values ofp (cf the expected value y = 10/7).

p = 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 D N

3 2

3 3

3 4

4 3

4 4

4 5

5 4

5 5

5 6

6 5

0,11329 1.257 0.11319 1,249 0.11326 1.255 0.1 1324 1254 0,11324 1.254 0.11323 1.254

0.11325 1,254 0.11323 1,253 0.11324 1.254 Oil1321 1,249

0,12774 1,266 0,12757 1,254 0,12769 1,264 0,12767 1262 0.12766 1.261 0,12766 1.261

0.12773 1.257 0.12765 1,259 0.12772 1255 0.12763 1,256

0.14642 1,277 0,14609 1,254 0,14636 1,275 0.14631 1.271

0.14630 1,271 0.14630 1,270

0.14686 1,009 0 14628 1.268 0,14622 1254 0.14626 1.266

0,17150 1.292 0.1 7012 1.151 0.17144 1,291 0,17133 1285 0 17132 1,284 0.17 13 1 1.283

0.17103 1,218 0 17127 1.280 0.17125 1276 0.1 7126 1.278

0.20697 1.315 0,20762 1.323 0.20696 1,316 0.20669 1.304 0.20666 1,302 0,20663 1.300

0.20643 1.274 0.20658 1.296 0,20656 1.294 0.20657 1.295

0.26100 1,351 0,26120 1.355 0,26102 1,352 0,26045 1.333 0.26038 1,330 0,26032 1.326

0,26010 1.308 0.26024 1.321 0,26022 1.319 0.26023 1,319

0,35347 1,419 0,35336 1,416 0,353 17 1,413 0,35207 1,382 0,35186 1.374 0,35171 1.367

0,35148 1,354 0.35158 1.360 0.35156 1.359 0,35156 1,359

0.54937 1.597 0.54805 1.575 0,54709 1,560 0.54322 1,474 0.54234 1,450 0,54206 1,441

0,54196 1,438 0.54170 1,427 0.54179 1.431 0,54178 1.431

Expected 0,11324 0,12767 0.14631 0,17131 0,20664 0,26031 0.35164 0,54169 W , ( P )

The estimates of w,(p) are clearly in very good agreement with the expected values. The degree of convergence decreases slightly as p approaches d-at p = 0.2 the difference is still only about 1 part in 5000, but this is to be expected because the gradient of the critical curve becomes infinite at d. In figure 1 we have plotted the reduced critical temperature T(p)/T,(l) against p ; the difference between the results of the Pade study and those obtained from (2.9) is too small to appear on the figure. The ratio method (Domb and Sykes 1957) has also been used, and it produces results of similar quality. However for p below about 0.3 there remains a noticeable amount of curvature in the ratio plots, and extrapolation slightly overestimates w, (at p = 0.2 the difference is about 1 part in 500).

The Padk estimates for the critical exponent y start off at $, for p = 1, and increase steadily asp is reduced-see table 1 and figure 2. This kind of behaviour was also noticed by Rushbrooke (1971) in the case of the Syozi model. It is only when p is reduced to about 0.2 that the y estimates are close to the expected value of y , and it appears that they continue to increase as p is reduced further. The ratio estimates of y behave in a similar manner, and are quite close to the Pade values.

It is possible to gain some understanding of this strange behaviour by studying series expansions of functions with (say) a pair of confluent singularities of similar strength. Consider what happens when the ratio method is used to analyse the series

The ising ferromagnet with impurities: I 1841

for which the behaviour near the closest real singularity zc (on the circle of convergence) is

I .5

1.4-

- Q 2. v

1 . 3 -

1 . 2

where p1 > p2 > 0 and the coefficients e, and e, are of similar magnitude. The ratios rn = d,/d,- , will be expected to have the form

AB (theory p < I IO/ 7

0 . 5 I . o

where A = pl - p, and g = e,l+J/e,T(p,). From (3.3)

and if A N 0 the ratios can be equally well represented by

rn - z, - I ( 1+- i4 ; 1) + o ( n - 2 )

where p1 is a new effective exponent arising out of the 'interference 3f the terms o :3.2).

It appears unlikely that the estimates the ratio method gives for the exponent will correspond to the value of pl, even though the estimated z, may be quite accurate. Analysis of test functions with singularities such as (3.2) by the Pade method indicated a similar kind of behaviour, with the estimated value of the dominant exponent strongly affected by slightly weaker confluent singularities. In the annealed model the structure of the critical singularity is considerably more complicated than (3.2), and there seems to be little chance of relating the estimated exponent obtained by either ratio or Pad6 analysis to the true exponent.

H I 4

1842 D C Rapaport

Having failed to obtain the correct exponent directly from (3.1) for p < 1, we tried examining the series for xliV. If the value y = is used the series should have a simple pole at the critical point wc(p), and this ought to be revealed by studying the Pade approximants of this series. But what happens in practice? For p above about 0.6 we found that the Pade estimates of wc(p) appear to converge better for the series z4l5 (corresponding to y = :)than for x " ~ ' ; and it is only when p falls below 0.4 that the x7/10 results start to look any better than those of x4", although the improvement is really only mar- ginal.

The results of this analysis confirm what might have been expected in advance. Though the early terms ofthe series are 'aware' of the position of the critical point they are incapable of 'seeing' the true renormalized behaviour which generally appears only very close to the critical point. The same conclusion can be drawn from Fisher and Scesney's (1970) numerical study of a related model which also exhibits renormalization. They found that the behaviour could be represented by an 'effective' exponent whose value varied smoothly as the critical point was approached, but even very close to the critical point it had failed to achieve the correct value. Whether some different approach to series analysis will succeed where the conventional methods have failed remains to be seen.

4. Series generation

In this section the methods employed in deriving expansions for the quenched impurity Ising systems are discussed. We denote the impurity concentration in both the bond and site impurity systems by p (0 < p < 1); then p = 1 - p is the concentration of spins in the site problem, and the concentration of bonds which permit interaction between spins in the bond problem. There are two variables in the problem (H = 0), w (or T - ')and p , and it should be possible to obtain expansions in either of them with coefficients which are functions of the other. Such expansions would be written as

in the case ofthe susceptibility, with superscripts B and S used to distinguish between bond and site impurity series where necessary. The hope is that expansions of type (4.1) will provide the means for determining what happens as the transition is approached from infinite Ta t constant p , whereas those of type (4.2) will be useful in studying the behaviour as the transition is approached from p = 0 at constant temperature. Both expansions are based on the disordered state-the connection between them has been discussed in some detail by Rushbrooke (1964) for the case of site impurities; a similar relation obviously exists in the bond problem as well.

In this paper the emphasis is on expansions of type (4.1). Some of the matters discussed in this section are of greater generality, and in paper I1 we will describe their application to series of type (4.2) and also to series for the fourth field derivative of the free energy. Expansions about the ordered state are also possible in principle (Domb 1971) and would be used for studying the low temperature side of the transition. The configurational problems involved in generating such expansions are considerably more complicated than those about to be described.


When p = 1 (no impurities present) equation 4.1 reduces to the usual Ising model series. For this particular case the coefficients are most readily obtained with the aid of a counting theorem (Sykes 1961, Nagle and Temperley 1968) which reduces the class of graphs? which must be considered to those with no vertices of unit degree, and either two or zero of odd degree. The power of this method is that a large number of graphs which might otherwise have contributed to the susceptibility expansion (including eg polygons with chains attached) can be ignored. But with the introduction of impurities the benefits of the counting theorem are outweighed by the difficulties in determining the contribution of individual graphs, and so a different approach is called for.

Themethodweuseis thefiniteclustertechnique(D0mb 1960), whichcan beimmediately generalized to include either type of quenched impurity. The weak cluster expansion of a quantity, say Y, defined for an arbitrary linear graph G, and possibly dependent on one or more parameters denoted by x is

The sum is over all connected subgraphs-or clusters-g of G, and L(g, x) is called the weight of g. The significant feature is that the subgraph weights are independent of the original graph G, and depend only on the subgraphs themselves-whence the method’s name. Provided Y is an additive function, in the sense that for any pair of nonoverlapping graphs G and G‘

Y(GuG’) = Y(G) + Y(G’)

it can be shown to have a cluster expansion (Sykes et a1 1966, Jasnow and Wortis 1967). (Cluster expansions may also be introduced via cumulants, eg Kubo 1962, Abe 1964.) The relation (4.3) can equally well be applied to any of the subgraphs of G: if the different subgraphs (unlabelled) of G are ordered so that for i > j , gi cannot be embedded in g j (4.3) is equivalent to

where (si; g j ) is the number of ways of weakly embedding gi in gj. The weights L(g,, x) may then be obtained either by computing the quantities Y ( g j , x) and the embedding constants, and solving (4.4) by recursion, or else directly from the formula (Essam 1967)

where ei is the number of edges in gi and (Si; gj )F is the number of weak embeddings of gi in g j subject to the condition that all edges of g j not in gi are incident with the vertices used by gi. The cluster expansion also applies when G is the infinite lattice 3

w9, x) = (gi; 9) U g , , x) (4.6) i 21

where both Y and (gi; 3), the lattice constant, are defined to be the values per lattice site-or per spin in the site impurity problem. Evaluating the terms of this expansion requires consideration of connected graphs (clusters) only and the weights again depend only on the details of the clusters themselves.

t For a review of graph terminology see Essam and Fisher (1970)

1844 D C Rapaport

As we mentioned earlier, there are two distinct averaging procedures involved in determining the properties of systems with quenched impurities. The first is over all possible spin orientations for each configuration of impurities having a given overall concentration p; the second is over all such impurity arrangements. The expansion of Y in the presence of quenched impurities therefore requires averaging over a whole set of of series, one for each impurity configuration. The effective values of the lattice constants are different in each of these series because the presence of an impurity on a lattice bond (site) means that it cannot be occupied by a graph edge (vertex). However, because the impurities are randomly distributed (and therefore totally uncorrelated in position). the results of averaging over the impurity configurations is merely to multiply the lattice constant of each graph with e edges and v vertices by p“(p”- ’) for the bond (site) impurity model. This simple result is only valid when a one-to-one correspondence exists between the vertices and edges of the graphs and the actual spins and spin-spin interactions of the model; it does not apply when the counting theorem, renormalized expansions (eg Jasnow and Wortis 1968), or the inverse susceptibility star expansion (Domb and Hiley 1962) are used.

In the case of the spin-? Ising model Y can represent, among other things, the zero field free energy, initial susceptibility, and other higher order field derivatives of the free energy. Insofar as the free energy is concerned the cluster method has already been used sucessfully for the pure system (Hunter 1968), and it is a simple matter to extend the analysis to include impurities. This will be discussed elsewhere. In this article we confine our attention to SI, and in paper I1 d2X/dH2 will be discussed.

Three distinct tasks are involved in generating the susceptibility series: a list of graphs must be obtained: their weights determined, and the lattice constants of graphs having nonzero weights computed. We discuss each of these in turn.

4.1. The g r a p h

Though most of the graphs required in this work have been tabulated (Baker et a1 1967), future extension of this work will require graphs for which no such tabulation exists, and for this reason we have developed a scheme for computer generation of the required graphs. As will be shown shortly. if the expansion (4.1) is to include powers of w up to the Nth, we must consider graphs with as many as N edges. In the case of (4.2), if the highest power of p required is M then in the bond (site) problem a list of graphs with up to M edges (M + 1 vertices) is needed.

We start by generating all Cayley trees (connected graphs with no closed loops) with up to N edges: given a list of Cayley trees with n edges (for n = 1 there is just the single edge) the trees with n + 1 edges are obtained by taking the n-edge trees one at a time and attaching a single edge to each vertex in turn. A Cayley tree with n - 1 edges has n vertices; all connected graphs with n vertices and n edges can be generated from the n- vertex trees by joining each pair of vertices in turn with a single edge. Those with n vertices and n + 1 edges follow similarly from the n-edge graphs, and so on, up to the maximum required value N(or n(n - 1)/2, if it is less). This algorithm applied to all trees with as many as N vertices results in a list of all connected graphs with up to N edges.

The difficulty inherent in this method is that most of the graphs will be generated more than once. In order to produce a list in which each graph appears only once we require a way of assigning a unique label to each graph, so that duplicates can be recognized and discarded. Various systematic methods have been proposed (eg Sykes et al 1966, Nagle 1966); though originally designed with star graphs in mind, they are readily extended to


include more general graph types. The 'best' method of labelling a graph depends on how the graph is to be used and which of its features are important to the problem: we require a label which a computer can readily assign and recognize, which at the same time completely describes the structure of the graph.

Our somewhat arbitrary labelling scheme bears some similarity to the one proposed by Nagle, and is as follows. The graphs are first grouped according to the numbers of vertices and edges. Now consider the set of graphs with e edges and v vertices. Number the vertices of each graph 1 to v, in any manner consistent with the requirement that the degree of any vertex is not less than that of its successor (for graphs other than those with all vertices of equal degree, this condition speeds up the labelling process-see later), and construct the adjacency matrix A (A = ( A i j ) ; A i j = 1 if vertices iand j are joined by an edge, = O otherwise). The elements of A are only zeros and ones, and if they are written out, row by row, infa single row, with A , , in the rightmost position, the result is a binary number B. Clearly no two different graphs can have the same value of B: but a single graph will usually have a number of possible values depending on the order in which the vertices are numbered. If all permutations of the vertex numbering-always subject to the ordering by degree-are attempted, a lowest value, Bmin, of B will emerge. Bmin is thus a unique label of the graph which also contains a complete specification of the structure.

When a graph is generated Bmin is evaluated and compared with a list of the values for previously generated graphs. Duplicates are rapidly detected by this method. The final stage is to construct a graph dictionary by listing the graphs in order of (say) decreasing Bmin.

4.2. Vie weights The (reduced) susceptibility of a cluster G having v(G) vertices is (see Domb 1960)

x(G) = u(G) + 2 2 ~ " ' ~ ~ ) / ( l + w"( '~) ) 4 2 E G yo E G

(4.7)

Both numerator and denominator are polynomials in w ; the numerator contains contributions from all the subgraphs g2 (including multicomponent) of G with two odd vertices, and the denominator the subgraphs go with no odd vertices. The cluster weights can be obtained from (4.7) by using (4.4) or (4.5). In general the weights are complicated rational functions of w-we will return to this general case in I1 when we discuss expansions of type (4.2).

The weight calculation for the high temperature series is greatly simplified because both x(G) and L(G) can be expanded as power series in w. To explain this simplification we assume that the edges of G are labelled and that a different value of w is associated with each edge. If wl denotes the value for the Ith edge, the sums in (4.7) will then be over products of subsets of the wl corresponding to the edges used by each of the subgraphs of G. x(G) can now be expanded as a multivariable power series in the wl. It has been pointed out by Domb (1970) that the subgraph subtraction process which results in L(G) (ie equation 4.5) can be accomplished by simply discarding all those terms in the expansion of (4.7) which do not have every wl appearing at least once. The truth of this statement is readily established for the quantity Y(G) which may represent not only z(G), but the free energy and all its even field derivatives at H = 0. First, the expanded form of the weight of any subgraph of G cannot, by definition, involve all the w,, so that the subgraph subtraction will not affect those terms in the expansion of Y(G) which include all the wI at least once. What we must now show is that the subtraction eliminates every term which does not involve all the wl. This is done by induction.

1846 D C Rapaport

Assume that for clusters with up to e edges L(g) = 0 if any w I = 0. We now show that the same is true for a cluster of e + 1 edges and, since it is trivially true for e = 1, it is true in general. By definition

L(G) = Y ' ( G ) - c -U) Y C G

for a graph G with e + 1 edges. Divide the connected subgraphs of G into two classes: the subgraphs g(') which contain a particular edge I , and the subgraphs g(') which do not. Then

L(G) = Y ( G ) - 1 L(g"') - L(go) 9'11 c G g( )cG

Since none of the subgraphs of G have more than e edges, w 1 = 0 implies L(g(") = 0 by hypothesis, and so

W ) , " * = O = Y G ) , " l = o - 2 L(9"') (4.8) all' c G

Setting w1 = 0 is equivalent to removing the edge 1 from G ; thus if G(') denotes G without the edge I , (4.8) becomes

a result which holds even if G"' is not connected. There are therefore no terms in the expansion of L(G) which do not include every w I at least once.

Once we have obtained the expansion of the cluster weight we can drop the edge labels and write

L(G) = u,(G) we(') + u,(G) + u,(G) w ~ ' ' ) ' ~ + . . . (4.9) where the coefficients ai(G) are numbers whose values depend on the structure of G. It sometimes happens that the first one or more ai(G) are zero: if a,(G) is the first nonzero coefficient 2 is called the entry parameter-for further discussion of this topic we refer to Domb (1970). Various theorems can be established which provide the values of the first one or more nonzero ai(G) for whole classes of graphs, but because the weights are easily obtained by computer we have not explored this matter in any depth.

When applied to the susceptibility of the bond and site impurity systems the cluster expansion (4.6) becomes

(4.10)

where the lattice constants are those of the regular lattice (without impurities). If in (4.10) we use the weight expansions (4.9) and regroup the terms according to powers of w we obtain expansions of the form (4,1), with coefficients X,, which are finite polynomials in p .

4.3. Lattice constants

Most of the lattice constants required for the first ten terms of (4.1) on the FCC lattice were computed with little effort. To reduce the computing time however, we used the counts given by Baker et a1 (1967) for a few of the larger, loosely connected graphs. There was no need to consider two classes of graph which might otherwise have presented


computing problems : graphs with more than two vertices of unit degree do not contribute at all, and graphs with e = U are easily seen to have entry parameter A , 2 1, which means that graphs having e = U = 10 will not contribute to Xlo. The only U = 11 graph which is required is the ten step self-avoiding walk-its count has been given by Martin et a1 (1967).

There are 36 graphs with e = lQ and U = 9 which contribute to Xlo. Their lattice constants are in general fairly large and to avoid any lengthy graph counting we determined their total contribution indirectly: the value of Xlo(l) is known, and if we subtract from it the contributions of all other graphs at p = 1 what remains is the contribution of the e = 10, U = 9 graphs to the coeficient of p" in X:o, or p 8 in Xso. Aside from this group of graphs a total of 367 were found to contribute to the FCC series.

5. Analysis of the quenched impurity series

In studying critical behaviour by means of series expansions one generally assumes (expli- citly or otherwise) the existence of a sharp singularity of simple form corresponding to the critical point. The justification for this assumption is provided both by available exact results which establish its plausibility, together with the apparent internal con- sistency of the numerical results. Systems which have randomly varying microscopic properties (eg impurities) could fall into a different category since, as discussed in the Introduction, it seems likely that the critical behaviour is much more complicated. There is in fact no evidence that a simple critical point exists at all, and there may well be an extended temperature region over which some kind of nonanalytic behaviour (not necessarily detectable however) occurs.

There have been no suggestions as to how one would go about identifying this kind of behaviour using series analysis. The only available course of action is to assume that the behaviour is of the type commonly encountered and analyse the series using standard

Table 2. Pade estimates for wJp) (upper entry) and y ( p ) (lower entry) for the QB model. Note p = 0.1 results which suggest w,(p) > 1 -that is no transition. Compare with the corresponding AB results in table 1.

D N p = 0.9 0.8 0.7 0.6 0.5 0 4 0.3 0.2 0.1

0,11329 1,257 0,11319 1,250 0,11326 1.255 0.11324 1.254

3 2

3 3

3 4

4 3

0,11324 1.254 4 4

011324 1.254 0,11326 1.254

4 5

012775 1.266 0.12761 1,257 0.12770 1.263 0,12768 1.262 0.12768 1.262 0.1 2767 1.26 1 0.12713 0.770

0.14645 1.278 0.14629 1.268 0.14637 1.274 0.14635 1.273 0.14635 1.272 0.14634 1.271 014645 1.266

0.1 71 59 1,294

0.17143 1,286 0.17 144 1.287 0.1 7144 1,287 0.17144 1.287 0.17144 1.287 0.1 7 144 1,287

0,20722 1.320

0.207 11 1,315 0.20688 1.302 0.20827 1,280 0.20698 1,309 0.20697 1,308 0.20697 1.308

0.26172 1.362 0.26178 1.364 0,2641 8 1.321 0.26 173 1.362 0,26121 1.340 0,26125 1.342 0.26124 1.342

0,35561 1,441 0.35665 1.465 0.35696 1472 0,35620 1.454 0,35242 1,276 0.35458 1,398 035438 1.388

~~

0.55384 1.574 0.56677 1.792 0.56693 1,795 0.56240 1.706 053630 1.037 0.55116 1,456 0.54982 1417

1.1248 1,417 1,3557 2.641 0.4754 1,385 1,2038 1,667 1,1411 1.343 1.1893 1,654 1.1685 1,500

1848 D C Rapaport

Table 3. Pad6 estimates for w,@) and y @ ) for the QS model. The absence of an entry means that the approximant does not have a pole corresponding to w,(p). (p = 1.0 results included for reference.) Contrast with the QB results of table 2.

p = 1.0 0.9 0.8 0.7 0.6 D N

010178 1.250 010171 1,245 0,10175 1,249 0.1 0 174 1,248 0,10174 1,248 0.101 74 1.248 0.10174 1.248

3 2

3 3

3 4

4 3

4 4

4 5

5 4

0.11374 1.275 0.11372 1,274 0.11374 1.276 0.11374 1.275 0,11376 1.277 0.11317 1.534 0.11381 1.283

0,129 13 1.319 0,12928 1.331 0,12909 1.317 0,12904 1,315

0.12952 1.360 0.12947 1,351

0,14992 1.405 0.1 5046 1.448 0.15008 1.419

0.1 5 127 1.555 0.1 5 105 1,517 0.15098 1306

0,18067 1.639 0.1 83 13 1,881 0 18353 1,935 0,18715 2,853 0,18550 2,310 0,18492 2.171 0,18373 1,923

Table 4. Extrapolated ratio estimates of w,@) and ;I@) for the QB model

p = 1.0 0.8 0.6 0.4 0.2 n

0.10178 0,12771 0,17146 0.26127 0,55771 1,249 1,261 1.285 1.341 1.595 0.10175 0,12768 0.17143 0,26126 055815 1,247 1.260 1,284 1.341 1.600 0.10172 0,12765 0.17140 0.26134 0,55985 1,245 1,258 1,283 1,343 1,620 0.10172 0,12765 0,17139 0,26128 0.55743 1.245 1,259 1,283 1.341 1.587 0,10173 0.12766 0.17140 0.26127 0.55610 1,246 1,259 1,283 1.341 1.566 0.10173 0.12766 0.17140 0,26125 0.55418 1.246 1.259 1.283 1.340 1.533

5

6

7

8

9

10

techniques. If the change in the nature of the phase transition as a result of the random- ness is felt over a wide temperature range, the series analysis might be expected to reflect this change; but if the effect is only over a very narrow range which itself lies within a larger region where the system looks as if it has the usual simple kind of singularity, then, as we saw in Q 3, there is little chance of the series studies correctly identifying the true behaviour. Bearing these remarks in mind we proceed with the analysis.

The procedures described in the previous section enabled us to compute the terms of (4.1) as far as XI, for both bond and site impurities on the FCC lattice. The Appendix


contains a selection of these series. (The same length series have been derived for the body centred cubic and triangular lattices, but they will not be discussed here.) The series were analysed by the usual ratio and Pade methods for a range of p values-the assumption being that we are looking for a singularity of the form {wc(p) - w ) - ~ ( ~ ) . Selections of Pade estimates for w,(p) and y ( p ) in the bond and site problems are given in tables 2 and 3; the resulting estimates of T J p ) and y ( p ) are plotted in figures 1 and 2. The ratio plots are presented in figures 3 and 4, and in table 4 a few of the results of ratio extrapolation in the bond impurity problem are given.

I I

0.4

4! 0.3 I - - - - - - - I

i ' 0.2 A - - _

21

Figure 3. Ratios of successive coefficients in the QB expansions for various p . Extrapolation to l l n = 0 yields an estimate for we@)-', and r@) is found from the limiting gradient. Note the p = 0.1 case where no transition is expected.

The Pade method was used to obtain estimates of wc(p) and y ( p ) for p values right down to the critical concentration p: in the bond impurity model, and down as far as about 0.6 in the site model. In the bond case the Pade results are apparently well con- verged down to about p = 0.4. Below this value the scatter in the Pad6 estimates starts to increase, but even at p = 0.2 (and perhaps even lower) it is possible to get a reasonably well defined estimate of wc(p), though y ( p ) is less well behaved. The increased scatter probably just reflects the higher sensitivity of wc(p) to a small variation in p near $, Included in table 2 are results of the Pad6 analysis for p = 0.1; the estimated wc(p) is now greater than unity, consistent with the fact that for p < p," (known from studies of the concentration expansion to be slightly below 0.12-see 11) the probability of having an

1850 D C Rapaport

infinite cluster of interacting spins is zero and a transition is impossible. The series is still sufficiently well behaved close to $’ to enable us to obtain a rough estimate for the critical concentration: a search of the Pade results to find the value of p which corresponds to w,(p) = 1 (ie 7 J p ) = 0) suggests that it lies just below 0.12-in agreement with the value from the concentration expansion. The results of the ratio analysis of the bond impurity series (figure 3 and table 4) are practically the same as the Pade results. At low p there is a slight tendency to overestimate w,(p), just as in the annealed model.

- - _ - - - I

2t d 01 112 I /4 1/10 0

I ln

Figure 4. Ratios of coefficients in the QS expansions for variousp. Contrast with the behaviour of the QB ratios in figure 3.

The site impurity series show avery different kind ofbehaviour. Though the coefficients of the quenched bond and site series are identical as far as X,, some of the higher order X,” become negative at values of p (eg 0.25) near to, but still above the critical value p: ( - 0 . 2 k s e e 11). Down to about p = 0.6 the Pad6 approximants yield fair estimates of w,(p), the scatter again increasing with decreasing p and always greater than for the bond impurity results at the same value of p . For lower values of p (starting somewhere between 0.6 and 0.55) most of the approximants cease to have a singularity on the positive real axis capable of corresponding to w,(p). Correspondingly, the ratio extrapolations (figure 4), which for p > 0.6 yield results in fair agreement with those of the Pade analysis, show a strong downward curvature for lower values of p . Whether this represents some weird aspect of the model’s behaviour, or is merely the analysis techniques showing their lack of versatility at probing singularities of complex structure, is something we are unable to say.


The y ( p ) estimates obtained from both the ratio and PadC analyses increase steadily with decreasingp, with the site impurity values climbing more rapidly than the bond values (figure 2). On the basis of these results alone it does not appear possible to decide whether this occurs for the same reason as in the annealed model, or whether it means that the singular behaviour cannot be characterized by an exponent at all.

Over the range 0.15 6 p 6 1.0 the difference between the critical temperature of the annealed bond model and the q ( p ) estimates for the quenched bond model is too small to be shown in figure 1. Clearly there must be some separation of the two curves because 8 slightly exceeds dt. At p = 0.8, if there is any difference, it is undetectable. At p = 0.4, the difference is less than 1 part in 200, and at p = 0.2 the quenched w,(p) is about 1 part in 50 greater than its annealed counterpart-the difference is in the right direction. At first sight this is rather surprising, because there is no a priori reason to expect any similarity in the behaviour of the two systems. A close inspection of the actual values of the series coefficients prompted by this result revealed that, over most of the concentration range, the fractional difference between X: and X f is quite small (up to n = 4 they are identical) and, for example, at p = 0.8 the difference between X:, and X:o is only 1 part in 4000. This by no means implies any similarity in critical properties, merely that unless the temperature is very close to the (annealed) critical point there is very little difference in the apparent behaviour over most of the concentration range as a result of fixing the impurities at random instead of allowing them the freedom to achieve complete equilibrium. The similarity of the two curves may be something of a red herring, because at p = 1 they must coincide, and the concentrations at which they go to zero are very close together, so perhaps it is not surprising that they stick together over most of the range.

It would be interesting to compare the quenched site impurity model with the corresponding annealed model mentioned in § 2. Ifthe two models were to show the same degree of similarity as the bond impurity models, then some clue may emerge as to why the series analysis breaks down at p ‘v 0.6. But such a comparison must wait until the properties of the annealed version of the model are understood.

We have also studied the Pad6 approximants of x4/5 and x7/lo for the quenched bond model in the hope of determining which, if either, of the values of y (: or 9) is more appropriate. The results are inconclusive. The scatter in the w&) estimates with the assumed y = appears to be less than for 9 down to about p = 0.5, but below this there is little obvious difference.

The analysis we have presented has failed to answer the question of what kind of transition the quenched systems experience. The analysis suggests that the susceptibility diverges at some temperature x ( p ) , and if a unique critical temperature does exist then it would seem likely that it has this value. As for the exponent y(p) , it appears to behave in a similar fashion to that of the annealed model. But in the annealed model such behaviour is anomalofs, and is attributed to the existence of confluent singularities ; any number of reasons can be put forward to explain the behaviour in the quenched models-confluence, essential singularities, etc-and there seems to be no way of identifying the correct one. The question of the existence of a unique critical temperature could perhaps be resolved by studying the behaviour of other derivatives of the free energy, both above and below Q).

t Though small, the existence of this difference is well established: cf the triangular lattice, where the values can be determined exactly (Essam and Garelick 1967).

1852 D C Rapaport

6. Expansions about the pure state

A different way ofapproaching the impurity problem is to regard the impurities as arising from a perturbation applied to the pure p = 1 state. Though not strictly a perturbation problem in the sense that a small change is produced in the hamiltonian, it turns out that quantities such as the susceptibility can be expressed as expansions in j j (= 1 - p ) about the pure state

For j j @ 1 the susceptibility is generally well represented by the sum of the first few terms of (6.1): the greater the number of terms included, the better the description of the effect of the impurities. In this sense we use the term 'perturbation expansion'.

Elsewhere (Rapaport 1972), we have discussed the possibility of determining the effect of a perturbation on critical behaviour by means of an expansion such as (6.1). We found that as the critical point of the unperturbed system is approached each term of the expansion diverges, the strength of the divergence increasing with the order of the term. What is wanted-and it can be realized in certain simple examples-is the ability to resum the most singular parts of the terms of the expansion and obtain the critical properties of the perturbed system. To this end we re-examine the x expansions of $ 3 and 4 5; not so much in the hope of actually finding that resummation is possible, but rather to see if any trends can be found in the singular behaviour of the terms of (6.1).

We start by considering the annealed system. The coefficients X:(p) in (3.1) are just polynomials in p which can equally well be expressed as polynomials of the same degree in j j . Then (3.1) can be written

where x'") is ( - l)"/m! times the mth p-derivative of x evaluated at p = 1. susceptibility of the pure system. The early coefficients of the expansions

itself is the

can be computed with little difficulty. The critical behaviour of (6.3) follows from (2.15): the dominant singularity is

(6.4)

y + m (6.5)

%(m) Iv c ( m ) wcm(wc - w)-~'"'

where y(m) =

and

= - 1*013( y + m - 1 )


C'O' is the usual susceptibility amplitude, that is CO in equation 2.15. In deriving (6.6) we used the result

which follows from (2.6).

Table 5. Estimates of w c and y'"' for the AB model-cf the expected wc = 0,10174, j""' = 1.25 - m.

m = l 2 D N

0,10176 0.10179 2,257 3.266

0,10176 0,10181 2,257 3,270 0.10177 0,10180 2.258 3,268

0,10176 0.10180 2,257 3,269

0,10177 0,10182 2,258 3.270 0,10177 0.10173 2,258 3,252 0.10177 0.10177 2.257 3.263 0,10176 2,257

3 2

3 3

3 4

4 3

4 4

4 5

5 4

5 5

3 4

~~

0.10186 0.10178 4,284 5.271 0.10185 0.10178 4.282 5,272 0.10175 0.10175 4.258 5,260 0.10180 0,10178 4.272 5,271

0.10174 4.255

Table 6. Estimates of the amplitudes C'"' for the AB model.

m = l 2 3 4 D N

3 2 3 3 3 4 4 3 4 4 4 5 5 4 5 5 5 6 6 5

- 0'069947 -0.070107 - 0.070079 - 0.070079 - 0.070047 -0.070057 - 0.070057 -0,070038 - 0.07007 1 -0.070071

0,080482 -0'087683 0'094322 0,079910 -0.087576 0'094683 0,079842 -0'087639 0'094583 0,079843 -0.087641 0.094583 0.079847 - 0'087720 0.094650 0,079840 -0.088170 0,079843 - 0.088229 0.079942

Estim,ites of w, and y ( m ) (1 < m < 4) obtained by Pade analysis of Dlog x(m) appear in table 5. The values of w, are quite close to the expected w, = 0.10174, and the y ( m ) obey the linear relation (6.5) fairly well. In table 6 we give the amplitude estimates found by evaluating the Pad6 approximants of (wc - w ) ~ + ~ x ( ~ ) at w = wc. The C(O)

1854 D C Rapaport

Table 7. Estimates of wcand y'"' for t h e ~ ~ m o d e l . Compare with the AB results of table 5.

m = l 2 3 4 D N

0.10176 0.10178 0.10188 0,10173 2,257 3.265 4.288 5,257

0.10176 0.10181 0.10184 2,257 3.270 4,280 010177 0,10180 2.258 3,268 0.10176 0.10180 2,257 3,268 0.10 177 2,258

3 2

3 3

3 4

4 3

4 4

et a2 (1972a). When the ratios C""'/C(m- ') are compared with (6.7) we find that the numerical value 1.013 is estimated as 1.012, 1.013, 1.02, 1.02 for m = 1 to 4 respectively, and (not shown) 1.01,l.O for m = 5,6. These estimates are very sensitive to the choice of value for wc, and a 1 in lo4 change in the value of w, used in the analysis produces a change in

of about 1 part in 200.

Table 8. Estimates of C'"' for the QB model. Compare with the AB results of table 6.

m = 1 2 3 4 D N

3 2 -0'069947 0'080211 - 0'087909 0'094322 3 3 -0.070107 0.079929 -0'087574 0.093214 3 4 -0'070079 0.079832 - 0'087653 4 3 -0.070079 0'079834 - 0'087656 4 4 -0'070047 0.079848 4 5 - 0'070057 5 4 - 0.070057

Expansions of type (6.2) whose coefficients are series of the form (6.3) can also be derived for the quenched models. In the course of deriving the series we noticed that the terms of f" for the quenched bond model were identical to those of the annealed ~ ( l )

series. For other m values the %("') series of both bond impurity models were, of course, identical up to the term in w'. The similarity of the series is an important factor in the explanation of why the series analysis fails to detect any significant difference between the annealed and quenched systems.


The results of a Pade analysis of the quenched x ( ~ ) series are displayed in tables 7-10. A comparison of tables 5 and 6 with 7 and 8 does not reveal any obvious divergence of the annealed and quenched bond results. On the other hand the site results appear to indicate a different kind of behaviour, and it is possible that the sites x("') are not even singular at the same point (which would mean that the corresponding C'") estimates are invalid).

Table 9. Estimates of wcand y"' for the QS model. Contrast with table 7

m= 1 2 3 4 D N

010172 2.270 0.10 168 2,266 0,10177 2,276 0,10189 2.298 0.10183 2,285

3 2

3 3

3 4

4 3

4 4

0.10190 0,10243 0.10193 3.329 4.469 5.312 010223 0.10193 3.400 4,350 0.10191 3.322 0.10204 3,352

The numerical results for the annealed model are inadequate to allow the prediction of the renormalized critical behaviour The reason is that (6.4) et seq may also be obtained by evaluating the p-derivatives of (2.18) at p = 1, at which point all the fine structure of the singularity is lost. All that can be derived from the analysis is a value for the initial gradient of the ?Jp) curve, that is dx(p)/dp at p = 1, but this is already known from (2.9).

Table 10. Estimates of C'"' for the QS model. Contrast with table 8.

m= 1 2 3 4 D N

3 2 - 0'072439 0.085123 -0.093778 010735 3 3 -0.072596 0.085819 -0'095642 0.10830 3 4 -0.072418 0.084256 - 0'096526 4 3 -0'072470 0'084314 -0,096586 4 4 - 0'072057 0'084806 4 5 - 0'072129 5 4 - 0.0721 3 1

The positive side of the analysis as a whole is that it emphasizes the similarity of the series in the two bond impurity problems, and with it the difficulty of determining any difference in behaviour on the basis of just ten terms. It remains to be seen whether the addition of a few more terms will clarify the situation.

1856 D C Rapaport

7. Closing remarks

It goes without saying that the problems encountered in analysing the impurity series remain unresolved. Our estimates of q ( p ) will be used in further work on other kinds of expansions, but the question of the critical behaviour of the susceptibility (and what x ( p ) really represents) is still open. One thing which is clear however is that bond impurity series appear to be more amenable to analysis than the site series, and a study of the Heisenberg bond impurity system might therefore prove to be of value.

We close with a mention of two approximate treatments of the configurational problems involved in the quenched site impurity model. In one (Mikulinskii 1968) it is claimed that for the two dimensional Ising model the ‘critical temperature’ is independent of impurity concentration, and the specific heat is finite at that point. It is difficult to say whether this result bears any relation to the nonanalyticity of the magnetization as a function of H at H = 0 for temperatures at least as high as q(1) (Griffiths 1969). Possibly the so-called critical temperature of the analysis is really just the temperature at which the magnetization becomes nonanalytic at H = 0, with the actual transition occurring at some lower, concentration dependent temperature. The other treatment is due to Domb (1972) who claims, on the basis of a study of the asymptotic contributions of various graphs to the series, that the critical exponents should be unaffected by the impurities. It ought to be possible to test these approximation schemes by numerically examining some of their assumptions, but until this is done any judgment of their bearing on the problem is premature.

Acknowledgments

The author wishes to thank Professor C Domb for his interest in this work, and to acknowledge the financial support of a Commonwealth Scholarship.

Appendix

In tables A1 and A2 we list the values (to 10 figure accuracy) of the coefficients X:(p) and X,S(p) (0 < n < 10) for various values of p .

Table A l . The coefficients A’:@) (0 4 n < 10) for various p values

p = 0.9 p = 0.7 p = 0.5 p = 0.3

1 .o 10.8

106.92 1023.516 9609,678

89145.53172 820 339,1773

7505 665356 68375532 79

620 780 983.5 5620 724 269

1 .o 1 .o 8.4 6.0

64.68 33.0 481.572 175.5

351 3,006 912.75 25301,50476 4679,625

180670,9186 23 762,0625 1282224.554 119824,2188 9057 942,635 601 085.2031

63 756253.52 3002987.398 447 458 973.0 14953 647.18

1 .o 3.6

11.88 37.908

117.774 359,09244

1081.022 436 3224,033442 9548.340643

28 124.91945 82484.61773


Table A2. The coefficients x(p) (0 G n < 10) for variousp values.

p = 0.9 p = 0.7 p = 0.5 p = 0.3

1 .o 10.8

106.92 1023.516 9605.79

89 032.468 68 818239,3424

7473980,789 67951 971.56

615 547 231.3 5559 507 563.0

1 .o 8,4

64.68 481,572

3505.95 25 143.309 24 178 385,9953

1255369,966 8778 357,068

61 068 206.43 423024391.2

1.0 6.0

33.0 175.5 906.75

4583,625 22765,3125

11 1 406,4063 538255,8281

2571 360.945 12 160521.81

1 .o 3.6

11.88 37,908

114.75 329.39676 891.408 564

2246,633 248 5 149.1 72 833

10 185,90069 14902,955 52

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