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

Perturbation theory, scaling and the spherical model

D C RAPAPORT Wheatstone Physics Laboratory, King’s College, Strand, Londbn WCZR 2LS

MS received 22 December 1971

Abstract. The four dimensional spherical model is treated as a perturbation of the three dimensional model. An exact resummation of the expansion is possible and the perturbed critical properties are compared with the predictions of the scaling theoretical approach to the problem. Both methods are found to yield the correct dependence on the strength of the perturbation, which can also be found from the exact solution of the model. A more general method of resummation is discussed and applied to the expansions of both the four and five dimensional models. The effects of perturbations which do not alter the critical exponents are also investigated.

1. Introduction

Attention has recently been focussed on the problem of how a perturbation affects the behaviour of systems which experience a second order phase transition.? Even for a system such as the Ising model different kinds of effect are possible depending on the nature of the perturbation. When the perturbation is a uniform field the ferromagnetic transition disappears entirely, but the effect on the antiferromagnet is a shift in critical temperature and a change of critical amplitudes, both of which appear to be analytic functions of the field strength. Yet another possibility is provided by an interacting pair of infinite Ising layers: a scaling argument suggests that the change in critical temperature no longer depends analytically on the strength of the perturbation, which in this case is the interlayer interaction, as it is reduced to zero. The most general case in which the second order transition survives allows a complete change in the form of the free energy, with the result that even the critical exponents are altered. This kind of behaviour is expected, for example, when the perturbation leads to a change of dimension, as with an infinite set of Ising layers with a perturbation which couples neighbouring spins on adjacent layers.

Consider a spin system with hamiltonian

2 = xo + /w1 where is to be regarded as a perturbation, and i a parameter which controls its strength. Assume that in the absence of the perturbation (ie at 1 = 0) the system undergoes a second order transition at some critical temperature Tc(0). If we further assume that the transition remains of second order for A > 0, the free energy in an applied field H and

t For reviews of the field of critical phenomena, model systems and scaling, see, for example, Fisher (1967) and Kadanoff et ai (1967).

933

934 D C Rapaport

at a temperature T can be written in the form

F = Fo(T, H , A) + f{T - Tc( i ) , H , A} where Fo is an analytic function of its arguments and f is singular at a critical temperature T,(I). (In systems such as the Ising ferromagnet there is no singular term unless H = 0.) What is normally found when perturbation methods are used to obtain an expansion for F about I = 0 is that the terms of the series diverge as T approaches T,(O). The reason for this could be simply that the perturbation has shifted the critical point, but it is more likely that it signals the onset of behaviour which cannot be adequately represented by just the first few terms of an expansion based on the unperturbed system.

The situation sometimes arises in which the perturbation does not crucially affect the critical properties: though both the critical temperature and the amplitudes are altered the exponents remain unchanged. This suggests rewriting equation 1.1 as

here the function f does not explicitly depend on 3, and the effect of the perturbation is merely to change the measurement scales for the temperature and field strength, in addition to altering the critical temperature. The hypothesis of universality (Kadanoff 1970, see also Griffiths 1970) states essentially that the critical behaviour of a system having only finite range interactions depends only on the dimensionality and on the ‘symmetry’ of the ordered state; f can therefore be regarded as a universal function which applies to a whole class of systems. On this basis, a perturbation which affects neither the dimensionality nor the ordered state symmetry and does not introduce infinite range forces, will not change the form of the function f, and vice versa. If infinite range forces are allowed, f may well depend on the way the interactions decay at large distances, as in the case of the spherical model (Joyce 1966). Universality makes no attempt to say anything about the A dependence of a, b and T,, though they are, of course, bounded for finite I .

The evidence for universality is to some degree experimental (see Kadanoff 1970), but in the main relies on numerical results obtained by extrapolating seties for various model systems. A three dimensional Ising model which is of finite extent in one direction seems to behave two dimensionally (Ballentine 1964, Allan 1970); the precise degree of anisotropy in the classical Heisenberg model does not appear to affect the exponents so long as the symmetry of the ordered state is unaltered (Jasnow and Wortis 1968), nor, apparently, does changing the range of a short range interaction in the Ising model (Dalton and Wood 1969). One can go a step further and assume that the functions a, b and T, are analytic in A. This assumption is known as the smoothness postulate (Griffiths 1971) and there is evidence from numerical studies that it applies in the case of the Ising antiferromagnet in a uniform field (Rapaport and Domb 1971).

The concepts of smoothness and universality enable us to identify three types of perturbation. First there are those which change the class of universality: we have already cited a case where the dimensionality is changed. An example of the change in symmetry is the introduction of anisotropy into the isotropic classical Heisenberg model (the Ising ferromagnet in a field also falls into this category, but there is no longer any transition) a change in the form of the interaction at large distances is brought about by the addition of an infinite range interaction (eg molecular field) to the king model. These and other examples have been studied via a scaling approach (Riedel and Wegner 1969, Abe 1970, Suzuki 1971, Coniglio 1971). The second kind of perturbation leaves the universality class unchanged but does not satisfy smoothness-the pair of interacting Ising layers is

Perturbation theory, scaling and the spherical model 935

an example (Abe 1970). The third kind involves those systems which obey the smoothness postulate. The Ising antiferromagnet is one example, while another is the Ising model with both nearest and next-nearest neighbour interactions (Herman and Dorfman 1968).

There exist perturbations which, though they do not destroy the second order transition, affect the system in a manner which presents difficulties when looked at from the point of view of universality. We mention two examples. The first is a three dimen- sional Ising system regarded as an infinite set of layers such that the intralayer interaction alternates between the two values J , and Jb, and the interlayer interaction is J . Such a system is an extension of one discussed by Fisher (1969) and it would be expected to exhibit a transition characteristic of the ordinary three dimensional Ising model. For J = 0 the system splits into two sets of identical Ising layers and therefore has two critical temperatures. A perturbation expansion in terms of J will, of necessity, have a complicated structure. The second example is the exactly soluble eight vertex model whose exponents can vary continuously with a quantity which depends on the parameters of the hamiltonian (Baxter 1971). This model can also be regarded as a pair of planar Ising systems coupled by a particular kind of four spin interaction; if the four spin term is treated as a perturbation,% follows that the exponent values can depenc! on the perturba- tion strength. This problem has been discussed by Kadanoff and Wegner (1971); it is an example of a special class of system in which universality does not apply.

In this article we describe how the concepts of universality and smoothness arise in a study of the effects of perturbations on the spherical model. Exact solutions can be obtained for spherical models on lattices of different dimension and type and for various kinds of interaction (Berlin and Kac 1952, Joyce 1966), so the model seems an ideal vehicle for testing techniques for handling thermodynamic perturbation expansions. In $2 we derive such an expansion for the four dimensional spherical model in which the perturbation is the interaction which couples an infinite set of three dimensional systems into a four dimensional one. In $3 we examine the way scaling theory can be used to treat problems of this kind, and in &I we show how the spherical model series can be resummed to produce the correct four dimensional critical behaviour. In $5 we outline a more general technique for resumming the series and apply it to the four and five dimensional spherical models. In $6 we look at two examples of perturbations which do not alter the exponents but affect the critical temperature and the critical amplitudes in different ways. In the first example, a spherical model with certain next-nearest neighbour interactions, the smoothness postulate is found to apply, but not in the second example, which involves a suitably coupled pair of finite three dimensional spherical models.

2. Perturbation expansion for the spherical model

The hamiltonian of an array ofN spins located at the sites ofa lattice in a uniform magnetic field H is

N X N = - J(ri j )s iSj - p 0 H 1 si

l < i C j < N i = l

where J(v i j ) is the spin exchange interaction, p o the magnetic moment of each spin, and the spin variable si can take any real value subject to the overall spherical tonstraint

N

936 D C Rapaport

We now consider the case where the lattice is the four dimensional hypercube, and the interaction is ferromagnetic and confined to pairs of spins which are nearest neighbours; then if ai (1 6 i 6 4) are the basis vectors of the lattice, J ( r ) is nonzero only when r = a,. An extra parameter, A, is introduced into the problem by the requirement that the inter- action strength in one of the directions differs from that of the other three, that is

The technique for solving the spherical model (Berlin and Kac 1952) is immediately J ( q ) = J(l 6 i 6 3), J(a,) = U.

generalized to this case, with the result that the free energy per spin is given by

- 3.m (2.1) (PI-10H)2 -PF = -4 -31n2K + K(3fA)z +

4K(3 + A ) ( z - 1)

where 3

f(z) = i ~ - ~ so j d4w In ((3 + A) z - i = l cos w i - i cos w4

and z , the ‘saddle point variable’, is determined from the saddle point equation

Here p = l/k,T, K = Q J . In zero field a second order phase transition occurs at z = 1; the critical temperature-obviously a function of A-follows from equations 2.2 and 2.3,

In zero field and for T > T@) the specific heat and susceptibility are

dz dK

C = 4kB + kB(3 + A) K 2 -

I-16 x = 2J(3 + A ) ( z - 1)

The magnetization is

A H M = 2J(3 + A)(z - 1)

The behaviour near the critical point is expected to vary as the value of 2 changes. For A = 0 and ,i = 1 we have the isotropic three and four dimensional models respectively; the behaviour reduces to the linear chain (or ring) when A is infinite. For any finite value of 3, other than zero the properties are four dimensional in character. with both the location of the transition and the amplitudes of the thermodynamic quantities depending on A.

If the interaction in the a4 direction is set to zero (ie 2 = 0) what remains is an infinite set of noninteracting three dimensional spherical models. We regard this as our un- perturbed system and introduce a perturbation which corresponds to that part of Z N arising from the interactions in the a4 direction. The perturbation expansion which can be derived for the free energy takes the form of a power series in A; in zero field it is

Perturbation theory, scaling and the spherical model 937

where F(0) is the unperturbed free energy, and

If the exact solution were known only for A = 0, the terms of the series (2.8) would have to be found by considering the multi-spin correlation functions of the unperturbed system, since +,(O) is just a sum of the 2n-spin cumulants over all possible sets of spins with the restriction that the spins in any given set can be divided into n pairs, the spins in each pair separated by a4. Fortunately the exact solution for the spherical model is available for arbitrary A, so that the terms of equation 2.8 may be obtained without going to this effort. The remainder of this section will be devoted to a description of the steps leading to such an expansion.

Letting x(") denote B"x/aA" evaluated at constant K , we find that

= K { z + (3 + A) z(1)} - + f " ) ( Z )

and since 3

71-4 1: d4\v cos b ~ v 4 ( U - i& cos \vi - A cos w4

= jOm dtZ,(t)3 eXp ( - U t ) I b ( A t )

where Zb(x) is a modified Bessel function of the first kind,

41(A) = 3 jOm d~Z,( t )~ exp { -(3 + A) z t } Il(At)

(2.10)

(2.1 1)

Clearly q51(0) = 0. In terms of two new quantities (and dropping the arguments for clarity)

P = exp {-(3 + A)zt} Q = I&) The higher derivatives of -PF can be wdtten L I S

4n+ = 3 J: dtZ:t-'(PQ('))(")

and thc result

follows. In deriving equation 2.12 we have made use of the fact that for A = 0 Q ( z t - 1 ) = 0

The 3, derivatives of P are conveniently expressed in terms of the quantity

R m - - --z(m-') - (3 + A)Z'"

(2.12)

F9

938 D C Rapaport

The resulting expressions are

PC1) = RI tP

P(2) = (R2 + R f t ) t P (2.13)

P(3) = (R3 + 3R2Rl t + R:t2) tP

and so on. Combination of equations 2.12 and 2.13 leads to an expression for &+l(0) in terms of various R, and integrals of the form

M , = 1: dtZo(t)3 exp (- 3zt) tk (2.14)

The behaviour of the R, is determined by differentiating equation 2.3 with respect to

2K = 1: dtl;PQ (2.15)

,? at constant K . Equation 2.3 for H = 0 can be rewritten as

and since the left side is independent of 2,

I = o

Setting n = 1 we find that because l o ( t ) is positive over the entire integration interval, P ( l ) and hence RI are both identically zero. For n = 2 equation 2.16 becomes

(2.17)

Before proceeding any further we must investigate the behaviour of the integrals Mk as TJO) is approached from above, that is, as z -+ 1 + 0. The integrand of equation 2.14 is positive and bounded above over the whole intewal, so if M , diverges in the limit z = 1 it will be due to the contribution from large values of t.t We therefore replace the modified Bessel function by the leading term in its asymptotic expansion

(2.18)

R 2 M l + i M 2 = 0

I o ( t ) - (2nt)-'I2 ef{i + O(t- l )}

and study the integral

= (2n)-3'2 11: dt eXp {-3(Z - l)t} tk-3'2 (2.19)

the constant to is chosen large enough for equation 2.18 to be a reasonable approxima- tion. Then if A, diverges at z = 1, M , will diverge in an identical manner to leading order. Equation 2.19 can be expressed in terms of the exponential integral,

and since

7 An abelian theorem exists to this effect (Widder 1946)

Perturbation theory, scaling and the spherical model 939

we find that for k 2 1

Substituting equation 2.20 into 2.17 we see that for A = 0

R2 - -&(z - l)-’ (2.21)

Further R, are found by evaluating equation 2.16 for larger values of n. From equations 2.12 and 2.21 we obtain

&(o) = &vl - 32n)- 3/2 3- r(+) (z - 1)-

+4(o) = ~ R , M , + + M ~ - &2n)-3/23-3/2r(3 (z - 1)-5/2

to first order. Also from equation 2.12, q53(0) = 0, and the fourth derivative is

Higher derivatives may be found in a similar fashion. The dependence on z is changed to one on K by writing equation 2.15 as

2K = (: dt exp (- 3zt) Io(t)3

for A = 0. Since 3112

21/2, 2K = 2K, - ~ ( Z - 1)”’ + O(Z - 1)

where K , = K,(O) (see Appendix), we have

871’ z - 1 % +Kc - K)2

to leading order. The final result is the expansion

(2.22)

(2.23)

A similar expansion can be derived for the susceptibility which differs from (2.23) in that odd powers of 1- also appear

The coefficients of each power of 3, in either series diverge as the unperturbed critical point is approached, so clearly these expansions break down when K is sufficiently close to K , that ( K , - K)’ z A. However, for small values of 3, one would need to be well into the critical region before noticing that the behaviour is not really that of the three dimensional model. Away from the critical point the terms which have been neglected- for each power of II a whole series of terms in K , - K less singular than the term retained together with a regular function of K-become important.

940 D C Rapaport

3. Predictions of scaling theory

How can perturbation expansions of the kind we have just derived be used to learn something about the properties of the perturbed system? One way, of course, is to sum the terms of the series, but in most cases this step is rendered impracticable by the sheer effort required in generating the terms. The exception is the spherical model which we shall discuss later. An alternative possibility is to resort to scaling type arguments which attempt to predict the dominant behaviour of the individual terms in the critical region and then to sum the resultant simplified series. The consequences of this kind of approach have already been discussed by a number of authors (Abe 1970, Suzuki 1971, Coniglio 1971) and in this section we review this approach to the problem; the predictions will later be tested against the spherical model.

Consider a classical d dimensional spin system with hamiltonian .Xo. If a perturba- tion %Pl is imposed, the free energy per spin is given by

- PF(,?) = lim N - l In Tr [exp( - P ( S o + lwXl)}] N + m

and it can be expressed as a series expansion in the parameter 3, whose coefficients involve the cumulants of Sl

A particular instance of such an expansion occurs for the Ising model above the critical temperature when So is the hamiltonian in zero field, and A S l the interaction between the spins and a uniform magnetic field. In this case equation 3.1 becomes

Oc h2n -PF(h) = -PF(O) +

~ (S2n)c ,,= 1 (2n)!

where h = ,BpoH and (S2n)c denotes a sum over lattice sites of cumulants of 2n spins evaluated at h = 0. It was this problem which Patashinskii and Pokrovskii (1966) discussed (see also Kadanoff 1966, 1970), and their arguments suggested that in the critical region (with a redefinition of exponents to conform to current usage) the terms of the expansion diverged:

where t = ( T - Tc)/Tc(Tc = Tc(0)), v and A are the coherence length and free energy gap exponents respectively. An assumption implicit in scaling theory is that 1 < dv 6 3, and that if dv = 2 or 3, tdv appears multiplied by a factor In t (see Stell (1969) for a discus- sion related to this point). We are then able to concentrate on the singular part of the free energy and to write

-PF(O) z aOtdV + terms analytic at t = 0 (3.3) neglecting higher order singular terms. Consequently

00

-PF(h) f i tdv 1 a2n(ht-A)2n n=O

(3.4)

where the uZn are constants. In this series only the most strongly singular terms are retained for each power of h; what has been neglected is a doubly infinite series in h and t,

Perturbation theory, scaling and the spherical model 94 1

all of whose terms are less singular in t for any given power of h than the term present in (3.4). In the critical region it is, of course, these leading order terms which dominate. Assuming that F is an analytic function of temperature in nonzero field, then (3.4) can be regarded as an expansion of the function

-PF(h) z td"f( th- (3.5) in which f ( x ) is analytic for x # 0. There are of course, a number of other ways of arriving at this result (eg Griffiths 1967).

We have just demonstrated that a perturbation can eliminate the phase transition entirely. Let us now assume that even with the perturbation applied the transition is of second order, although the form of F and possibly also the values of the critical exponents will change. Once again a scaling type argument is used to provide the form of the ex- pansion coefficients in the critical region. We write the free energy as a double series in the variables 3, and h; for T > T, the series is

Since both S and XI depend on the spin variables, ( S 2 n 2 f y ) c will involve a sum over the lattice sites of cumulants evaluated at h = A = 0, the precise details of which will depend on the exact nature of the perturbation. If H I contains k spin interactions, each term of the sum denoted by (S2n#y) , will be a cumulant of the form

(Sr,Sr, * * * Sr2nSrZn+ 1 . . ' srz,+->c where sr is the spin located at site r.

Near the critical point we can picture the system (for h = A = 0) as a collection of microdomains ('droplets') of ordered spins. By generalizing Kadanoffs (1970) argument we can write

where r = max I ri - rjl is the greatest separation between the spins appearing in (. . .),, 5 is the coherence length, and the function f n m is of order one if r 6 5, that is if all the spins are much closer together than the coherence length, but very small if any of the spins are separated by a distance much greater than 5. The indices xu and xu will be defined below. Since 5 diverges as T -+ T, (5 - t -") we can replace the sums over sites by integrations over the ri with the result that

( S 2 n ~ y ) ~ - 5 2 n W x , ) + m ( d - x u ) - d

The index x, is related to the gap exponent of the unperturbed system, A = (d - xu) v, (Kadanoff 1970). If we introduce a new 'gap' exponent Au

A" = (d - X ~ ) V

associated with the 2 derivatives of the free energy we obtain ( s 2 n # y ) , t d v - 2 n A - m A a (3.7)

which is a fairly obvious generalization of (3.2). We make the assumption that the perturbed system undergoes a transition only in

zero field. Furthermore, because the critical temperature is altered by the perturbation, F(h, %) will be analytic at t = 0 irrespective of h. This suggests formally summing the

942 D C Rapaport

terms of equation 3.6 to give

@(t, h, A) 2 tdv4(t3L-'!AU, th - l iA) (3.8) where @ is just -pF together with an analytic function oft, h and J. which cannot affect the critical exponents. As was the case with (3.5) only the singular terms dominant near t = 0 are taken into account, and (3.8) will not apply away from the immediate vicinity of t = 0. If, as in the three dimensional spherical model, the unperturbed free energy does not display any singular behaviour as the critical temperature is approached from above (d = 3 and there is no In t term), the argument leading to (3.8) proceeds without using (3.3).

Let d de the dimensionality of the perturbed system and Tc(A) the critical temperature; then if we define

the free energy for f 2 0 should be expressible in a form similar to (3.5):

@(Z, h, A) e td'Y(l, Zh-'Ii) (3.9) in which it is assumed that so long as J. # 0 the explicit dependence of Y on 3, does not change the exponent values.? Hence ? and x, the analogues of v and A for the perturbed system, are independent of A for A # 0. The expression (3.9) is equivalent to the statement that a change in 2 (providing A # 0) does not affect the universality class of the system. Tc(J.) may reasonably be expected to be continuous in /2, which means that for A sufficiently small there is an overlap of the regions in which (3.8) and (3.9) apply. Thus, for h = 0 and J.41

from which it follows that

t d v 4 ( t J . - 1'*L, a) = fJiY(J., CO)

T&) z Tc(0) (1 + zlA1iAu) (3.10)

A change of variable in (3.8) (Coniglio 1971) leads to an expression for @ in which with z1 satisfying 4(zl, CO) = 0.

the dependence on t is replaced by one on 2

2 (3.1 1) @ ( I , h, 2) ~ ( d v - h) , 'A.L'Zdi$l(fJ . - l / A u i ( A - & / A u i Eh- l l i )

For J. # 0 this is clearly a function whose behaviour in the shifted critical region is covered by the class of functions defined in (3.9). The critical properties of the perturbed system for small 2 follow immediately from equation 3.11: if the normal exponent equalities (eg Fisher 1967) are satisfied by both the perturbed and unperturbed systems, the zero field specific heat and susceptibility are

c Iv j l ( 6 -a ) /Au I -C (3.12)

A ( j - ; ) / A u ? - ? (3.13)

and the magnetization on the critical isotherm ( E = 0) is

M 'i-' - 8 'J A , ' A u h h ' (3.14) t If the index xu = d this assumption is untenable, as the exponents can be shown to depend on, (Kadanoff and Wegner 1971); we exclude this case from our discussion. The po\sibility that xu > d is also excluded, because €or sufficiently large w z (3.7) no longer diverges at t = 0, and the critical properties are, therefore, unaffected by the perturbation.

Perturbation theory, scaling and the spherical model 943

where, once again, 7 and 6 are the exponents for the perturbed system corresponding to y and 6. If the critical behaviour involves a dependence on In t or In h the way in which the critical amplitudes vary with il must be looked at more carefully. We shall retum to this problem later in our discussion of the spherical model and also consider what happens if the exponent equalities are not all satisfied.

Scaling theory, therefore, is able to predict the variation of the critical amplitude of a quantity given that the perturbation changes the value of the exponent. No relations between the exponents of the unperturbed and perturbed systems are indicated, however, but this is a similar kind of problem to the prediction of the exponents from the hamiltonian itself-a problem which has yet to yield to solution. Finally, there is the problem of determining the value of the exponent A,. It has been argued (Abe 1970) that in the case of a perturbation which couples an infinite set of Ising layers A, = 3 (the susceptibility exponent, y, of the unperturbed system)-+ conclusion supported by the results of a recent numerical study (Rapaport 1971). A similar argument suggests that for the spherical model discussed in the previous section Au = 2; expression (2.23) confirms this. Other values of A, are appropriate for perturbations which change symmetry, etc (Suzuki 1971).

The case where neither dimensionality nor the exponents are altered by the perturba- tion deserves some attention. Equation 3.1 1 becomes

@(t, h, A ) x Zd”g51(til- th-”A)

and as T approaches T,(il)

@ ( I , h, A ) = tdv41(0, th-’IA)

for A > 0, whereas for A = 0

@(t, h, A ) z td”41(co, th-’iA)

In the event that

4 , ( a , x) = 41(@ x) (3.15)

we have

@ ( I , h, A ) = @(t - T ~ A ” ~ ~ , h, 0) (3.16)

and since @ has an expansion in integer powers of il it follows that l/A, must be a positive integer (Coniglio 1971). So the limits T -+ Tc(il) and il -+ 0 commute, and there is nothing special about the point il = 0 in so far as the critical behaviour is concerned: in other words, the smoothness postulate (Griffiths 1971) applies. Smoothness requires that for a system whose free energy is of the form (1.2) a, b and T, are analytic functions of 1 (over a suitable range), and that the singular behaviour is determined by theA independent two variable function f ( x , y). Under these conditions equations 1.2 and 3.16 are equivalent for 3, < 1, providing, as before, that l/A, is a positive integer. In $1 we men- tioned two ex~.mples of systems which apparently obey the smoothness postulate. One, the square Ising model with a perturbation which couples next-nearest neighbour spins (Herman and Dorfman 1968), is found to have l/A, = 1; the other, the Ising antiferro- magnet in a uniform field (denoted by 1 H itself becomes the staggered magnetic field) (Rapaport and Domb 1971), has l/A, = 2 because of the symmetry under a change of field direction. A further example of smoothness is given in 56. There is no reason why equation 3.15 must be true in general, and in @ we also describe a case where, although the exponents are not affected by the perturbation smoothness is found not to apply.

944 D G Rapaport

4. Summation of the perturbation series

The perturbation series derived in $2 tell us little about the properties of the perturbed system because each term of the series diverges at the unperturbed critical temperature T,. However, it is reasonable to expect that the critical point is shifted by the perturbation -an assumption supported by the scaling argument; thus the divergence of the individual terms at T, is attributable to expansion of the free energy about the wrong point and does not reflect on the behaviour of the perturbed system. What is required is a method of analytically continuing the function represented by the expansion into a region centred on the perturbed critical point. Such a method is available for the spherical model and will be described in this section. But rather than concentrate on the free energy we shall expand the equation for the saddle point variable (2.3) in terms of A, the aim being to express z as a function of T (or H if the critical isotherm is being studied) and I . Once z is known the thermodynamic properties follow from equations 2.5, 2.6 and 2.7.

We start by introducing the function?

3

~ ( A , z ) = ~ - ~ j o { d ~ ~ { ( 3 + I)z - i = l C O S W ~ - ICOSW, I-' (4.1)

so that for H = 0, equation 2.3 is equivalent to

2K = 4 ( I , z ) ( 4 4 The integrand of equation 4.1 can be expanded as a series in 1. cos w4 and integrated over w4, with the result

(4.3)

where X

0 Equation 4.4 is a consequence of 2.10. The range of integration in equation 4.4 is divided into two parts at some point to, and for t > to the Bessel function product is replaced by its asymptotic form

(4.5) ~ ~ ( t ) ~ - (2~ct)-~" exp (3t) 1 c,t-' CO = 1 1 3 0

In terms of the exponential integral E,(x)

Y(y) = jd. dt exp (-yt) Io(t)3 + (271)-3'2 1 ~ , t , - ' - ~ E ~ + * [ ( y - 3) to] 1>0

Using the expansion

O0 (-x)" n = O n ! ( n - p + 1) E&) = - 1 + X p - ' r ( i - p) p # 1,2,3, . . .

t Y(l., z) is proportional to the random walk generating function (eg Montroll and Weiss 1965) ?or the four dimensional lattice with an anisotropic single step probability whose value in the a4 direction is i t imes that in the other three. The results of this section can be used in a study of the effects of a change in dimension on the properties of random walks.

Perturbation theory, scaling and the spherical model 945

and expanding exp ( - yt) about y = 3 we find that m

where

The coefficients a, are independent of the value of to provided that it is sufficiently large, and the upper limit of the sum over n suitably chosen. (ao is just Watson's integral for the simple cubic lattice (Watson 1939) and is the value of 9(0, l).) After substituting equation 4.6 into 4.3 and a certain amount of manipulation we find that 9 ( A , z) can be expressed as the sum of a pair of double series in A and a new variable

p = (3 + A)(z - 1) (4.8) one of which is regular in p , the other not. Thus

9 ( A , z) = P ( A , z ) + Y ( A , z ) (4.9) The regular part is

(4.10)

and, assuming p > A, the singular part is

Consider what happens i f z is allowed to approach unity for fixed A > 0. Treating equation 4.11 as a power series in A alone we see that each coefficient diverges in this limit, and the dominant contribution for z - 1 < A comes from the most negative power of 0 for each Dower of A

(4.12) Using equation 4.2, and because 9(0 , 1) = 2Kc, we can write equation 4.12 as

and by reversion we obtain an expansion for z - 1

(4.13)

The terms we have ignored appear as corrections of the form A"(Kc - K)-Z"+m+2 with m > 0, but even though some of these will diverge as K -+ K, the divergence is less rapid than that of the term in (4.13) with the same power of A. The first few terms of (4.13) are

1 A2 + A 8 z 2 ( K , - K)' 512z4(K, - K)4 ' ' '

946 D C Rupuport

and this result can be used.in equations 2.5 and 2.6 to produce expansions for the specific heat and susceptibility, each of whose higher order terms diverges on approaching the critical point of the unperturbed system. This is exactly the problem we encountered earlier and we now show that it can be overcome.

Equation 4.11 for the singular part of $(,I, z ) can be expressed as a sum over hyper- geometric functions provided that 2%/p < 1,

In order to use this result in the critical region when A is nonzero and consequently p < 2 we must look for an alternative form which has an expansion about p/A = 0. A linear transformation of the hypergeometric function which changes its argument from x to x / ( x - 1) yields

(4.14) The critical behaviour can be seen to depend on the properties of the 2 F , hyper-

geometric function when its argument approaches unity from below. A linear trans- formation is available whose result is a series which is useful in precisely this region. Care must be taken because the three parameters of the zFl add to an integer, but the required result is well known (ErdClyi 1953) and after simplification we find that the ,F1 of equation 4.14 is replaced by

1 r ( n + 4) ( I - n)! 7 1 - 1 1

n = O r(1 - n + $ ) n !

(4.15) where h,, consists of a sum of four digamma functions,

h,, = Y ( n + 3) + Y ( n + 1 + 3) - Y ( n + 1) - Y ( n + 1 + 2)

The logarithm in (4.15) can be expanded as

Then, with the exception of the term involving ln(p/A), the remainder of (4.15) is an analytic function of p for A > 0 and p sufficiently small. The end result is that for p 4 ,I, $(A, z ) can be expressed as the sum of an analytic and a singular function of p, the latter having the form

For z = 1 we obtain

(4.16)

(4.17)

Perturbation theory, scaling and the spherical model 947

We have discussed this expansion in some detail in order to emphasize the great simplification which takes place near the critical point. First the new transition tempera- ture Tc(%): the left side of equation 4.17 is 2Kc(A), so that equation 4.17 is the expansion for the shift in critical temperature; for 1 @ 1 we have

(4.18)

The T dependence of z near T#,) is obtained with the aid of (4.16) and (4.17); to leading order

(4.19)

The terms of the analytic part of $(A, z) are of higher order and only contribute to the corrections. As a function of K, (4.19) is

an equation which may be solved approximately for z to give 8 2 l j 2 -371 A [K,(1) - K]

z - 1 % ln[{K,(A) - K}/1'/']

(4.20)

Corrections to (4.20) involve powers and logarithms of logarithms, etc, all of which can be neglected close to Tc(A), as can the dependence on In 1.

The specific heat and susceptibility at H = 0 follow immediately:

87c2k,K,Z11i2 In Z

C NN $k , +

where Z = T/T,(A) - 1. z can also be expressed as a function of H and A on the critical isotherm of the perturbed system, from which we obtain the magnetization

M % -m,,A- 1/6(H In H)1/3

where m, is a positive constant. We are now in a position to test the predictions made by scaling theory. One difficulty

which arises is that the scaling treatment normally assumes that the specific heat diverges at the critical point. But, at least for the three dimensional spherical model, the argument leading to (3.8), and hence to (3.1 l), does not depend on this feature. The resulting altera- tion of the scaling predictions is that (3.12) is replaced by

c = ~ ( 0 ) + CIA 'n , -d i j /A.uZd;-2

with C(0) the limiting value of C as i ---f 0 + : c1 is a constant. In four dimensions the critical behaviour involves a logarithmic dependence on Z and H (see Appendix for the spherical model exponents); if we ignore this for the purpose of determining the 1. dependence of the amplitudes we find that the scaling theory predictions agree with the spherical model results. Finally, because AV = y = 2, scaling predicts that the leading order shift in T, is proportional to Al l2 , which ,is just the result given in equation 4.18.

948 D C Rapaport

So, even though the critical temperature varies continuously with 2, the function TC(3,) is singular at A = 0.

5. The critical properties by a more general method

It is to be expected that the properties of the perturbed system can be derived if sufficient information about the structure of the perturbation series is available. This is confirmed by the results of $4. Looking again at these results we see that for i < 1 the critical behaviour follows from considering only the first term in the asymptotic expansion (4.5) and all subsequent equations involving the coefficients c,. As A is increased the higher order terms can no longer be neglected and, though the critical exponents remain fixed, both and the amplitudes change with A. The solution for arbitrary 3, appears in the Appendix; the result of the perturbation approach is seen to agree with the exact result when 3. 3 1.

There is a second more general method for extracting the critical properties which does not rely on the transformation properties of the hypergeometric function, and which can, therefore, be used to treat other problems. We start by showing that this method can reproduce the above results and then use it in a discussion of the five dimensional spherical model.

We first rewrite equation 4.14 as

where

CO T(k - 41 - a) T(k - $1 + a) ( ;)-'* 1 + - T(k + 1)' G , = c

k = O (5.2)

The series (5.2) is convergent for all 1 > 0, even at p = 0; however a repeated term by term differentiation with respect to p eventually yields a series which diverges at p = 0. This implies that for A > 0 G, is itself singular at p = 0. Assume that for fixed A > 0 the form of G, near p = 0 (ie near z = 1) is either

41w + ( z - 42(4 e + 0, i ,2 , . . . (5.3)

41k) + ( z - l )Oln(z - 1)42(4 (5.4) or

where both 41 and 42 are analytic at z = 1. A theorem due to Darboux (eg Ninham 1963) relates the coefficients of the Taylor expansion of 42 about z = 1 to the asymptotic form of the coefficients of equation 5.2. The converse process of replacing the large k terms of equation 5.2 by their asymptotic expressions can be performed, yielding a series for q52 in addition to determining which of equations 5.3 and 5.4 applies and the value of the index 0. A systematic procedure for determining the behaviour of a function near a singularity of this type has been proposed by Joyce (1972); in our problem we are interested only in the leading order singular term and this may be found directly.

Replacing the gamma functions of (5.2) by their leading order asymptotic forms valid for k 9 1 we obtain

e = 0,1,2, . . .

- 2k

G I = Gy + $ k-"(l +!) + ... k = l

Perturbation theory, scaling and the spherical model 949

where BJq) is the Bose function (Dingle 1957), GP is analytic in p for A > 0 and the terms neglected involve Bose functions of higher order. For positive integral values of s, Bs(q) consists of a series of positive powers of q and a singular term - q s In ( -q ) / s ! . So to leading order equation 5.1 is an analytic function of p together with a term

This term itself consists of an analytic function of p (or z ) together with a function which, for z 2 1 is

3 - / 2 - 1 / 2 ( z - 1) In 4Tc2

to leading order. This result is consistent with the assumed form of the singular behaviour (5.4) and agrees with the results of the previous section. It is perhaps worth pointing out that there appears to be a formal similarity between the steps of this procedure and the scaling treatment of $3.

The five dimensional model can be treated as a perturbation about the three dimen- sional model in a similar fashion. If ai(l < i d 5) are the basis vectors of the five dimen- sional hypercubic lattice and if J(ai) = J ( l < i < 3), J(a,) = J(a,) = AJ, then the form of equation 4.1 appropriate to this problem is

3

Z) = 71- j: d5w { ( 3 + 2 4 z - COS - ,?(cos w4 + cos w5) i = 1

y = (3 + 21)z

with Y ( y ) as in equation 4.4. Following the procedure for the four dimensional case we have for A > 0,

(5.5)

where

k = O T(k + 1)3 (1 + & ) - 2 k (5.6) r(k - $1 - $) r(k - 31 + $) r(k + 4) Tc.

G,=

and j = (3 + 2%)(z - 1). The sum denoted by G, is essentially the 3 F 2 hypergeometric function. Replacing the coefficients of equation 5.6 by their leading order asymptotic terms we find that

Equation 5.5 therefore consists of an analytic function of p and a term

H9

950 D C Rapaport

When s is not a positive integer gS(q) has a singular part r( - s) ( - q y ; hence the leading order singular term in g5(2> z ) is

3 112

2 7 1 I % - + - 113'2 (5.7)

This result is the same as that obtained by expanding the five dimensional saddle point equation in terms of 2 (see Appendix). However, since the expansion of gs(q) involves a power series in y in addition to the singular term, there will be a term in the expansion of POL, z ) linear in z - 1 which dominates over (5.7) near the critical point.

As it stands the method is capable of yielding the first singular term in the expansions of the four and five dimensional saddle point equations. In four dimensions the critical behaviour is determined by this singular term, but in five the leading order term of the expansion which controls the critical behaviour is analytic, and not predicted by this method; the term (5.7) appears only as a correction. It may well be possible to extend the method to yield the analytic term as well.

6. Perturbations which do not affect the exponents

Our discussion of the spherical model has so far concentrated on perturbations which produce a change of exponents and hence permit comparison with the scaling theoretical predictions of how the critical amplitudes depend on the perturbation strength. In this section we consider two examples of perturbations which leave the exponents unaltered.

The first involves the spherical model on the face centred cubic (FCC) lattice in which the nearest neighbour interactions directed along the bonds in two directions is J and along the third 3 J . The thermodynamics of this model can be expressed in terms of the integral

A closed form expression for this integral in terms of complete elliptic integrals has been obtained by Joyce (1971) who also points out that for ;1 = 0 the problem reduces to the spherical model on the body centred cubic (BCC) lattice with isotropic nearest neighbour interactions. An alternative way of viewing the anisotropic FCC model is as a BCC model with nearest neighbour interaction J together with interactions of strength AJ between certain pairs of next-nearest neighbours. This latter interaction can be regarded as the perturbation. Without going into detail, it is a straightforward matter to expand the elliptic integrals and to show that both the critical temperature and the thermodynamic amplitudes are analytic functions of A, even at 2 = 0. Thus we have found yet another system which satisfies the smoothness postulate.

The second example is a four dimensional hypercubic spherical model which is of infinite extent in three dimensions and finite in the fourth. The interaction in the first three directions is J and W in the fourth. The interaction I J can be treated as a perturba- tion which couples a finite set of three dimensional systems. Though strictly four di- mensional, this model can be regarded as three dimensional, with a fairly complicated

Perturbation theory, scaling and the spherical model 95 I

set of short range interactions. The perturbation, therefore, does not change the dimen- sionality, hence the exponent values should remain unaltered.

The analysis of this model follows the Berlin and Kac (1952) calculation, and it will suffice to mention the points at which the analyses differ. If the edges of the lattice contain ni sites (1 d i d 4), periodic boundary conditions are assumed, and the interactions are J i = J ( l 6 i 6 3), J4 = 25, we find that the saddle point equation depends on the function

N

&nL)(z) = N - ' 1 In ( ( 3 + A)z - p j } j = 2

where N = iiln2n3n4 is the number of sites in the lattice. The quantities p j ( l < j < N ) are proportional to the eigenvalues of the matrix which represents the spin-exchange interactions in the hamiltonian and have the form

2710' - 1) 2710'- 1) 2710' - 1) 2710' - 1) p . = cos + cos + cos + 3, cos

1 n2n3 n4 n2n3n4 n3n4 n4 J

If we allow all the ni to become infinite we recover equation 2.2. If on the other hand we keep n4 fixed at some finite value we obtain

lim &n,j(z) fn,(z) = nlnzns-'m

It follows that in zero field

and if we proceed in the same manner which led to equation 4.6 we find that 6.1 is the sum of a regular function of z and A and an expression

(271)-3'2 1 c,r(-+ - 1 ) n i ' y [ ( 3 + A)(z - 1) + A 1 3 0 j , = 0

For the particular case n4 = 2 we obtain to leading order

1 2K = 2K, - __ {(3 + A)+ (z - 1)+ + (24+ . . . }

2 3 1 2 ~

so that

Compare this result with equation 4.18 which applies in the limit n4 -+ c/; : in both cases the leading order shift is proportional to A+, but the coefficients are different. The specific heat and susceptibility in the critical region are found by sol\ ing

1 3 25un 2312 K,(R) - K z -{3i(z - l)+ + -l"-+(z - 1))

952 D C Rapaport

for z : to leading order we find that

and this can be used in equations 2.5 and 2.6. The H dependence of z on the critical iso- therm can also be obtained; the magnetization then follows from equation 2.7. The critical behaviour is found to be qualitatively the same as that of the ordinary three dimensional model, but the amplitude values are different, even for 1. < 1. The same holds true for any finite value of n4.

In both the examples of this section the perturbation fails to change the exponents. but there the similarity ends. Whereas in the first example the critical temperature and the amplitudes vary analytically with A, the amplitude values in the second example change discontinuously when the perturbation is applied, no matter how weak it is. The critical temperature in the second case is continuous. but no longer analytic at 2 = 0. In both cases we could have employed the methods of $2 to derive perturbation expansions qualitatively similar to (2.23). but on account of the very different kinds of behaviour which can arise from different perturbations it is clear that the result of resumming such expansions is, not surprisingly, very sensitive to the nature of the coefficients.

7. Concluding remarks

Our primary aim has been to show that it is possible to determine the effects of a per- turbation on the critical properties of a system, given a certain amount of information about the structure of the terms of the perturbation expansion. Though it is possible, as in $4, to consider the expansion terms in considerable detail, it turns out that much of this is irrelevant if one is only interested in the immediate neighbourhood of the critical point: should corrections to the critical behaviour be needed as well: then, of course? a more detailed knowledge of the term structure must be sought. Thus, in 95, we are able to deduce the critical properties simply be replacing the terms of the perturbation expansion by their leading order asymptotic forms.

It is in a sense unfortunate that the spherical model, which has served as a basis for testing the series summation techniques, shares only some of the properties of the more interesting and ‘realistic’ systems such as the Ising model. For example, the spherical model’s low temperature behaviour has prevented us from studying the effects of a change of dimension on the long range order; the logarithmic terms and the finite specific heat maximum have also tended to complicate matters. It will be interesting to see whether methods of the kind described here can be put to use in a study of systems for which the solution is not already known.

We have also discussed the ideas of smoothness and universality and shown how they can be used to classify some of the possible effects that a perturbation can have on the critical behaviour. The spherical model in its variety of forms supports universality and, in certain cases, provides an application for the smoothness postulate as well. As we mentioned in $1, there is evidence to suggest (but not necessarily confirm) that uni- versality is a feature common to a wide class of systems. The somewhat restricted property of smoothness has also found support. Like scaling, these ideas stand or fall as more information becomes available from series studies and exact solutions; meanwhile they serve as a convenient framework for classifying and (hopefully) interpreting available results, and as a source of ideas for future research.

Perturbation theory, scaling and the spherical model 953

Acknowledgments

The author wises to thank Professor C Domb for his suggestions which led to the work described in this article and for his continued interest. Dr G S Joyce and Dr A Coniglio are thanked for useful discussion. The financial support of a Commonwealth Scholar- ship is gratefully acknowledged.

Appendix. Saddle point equation for arbitrary 3.

Consider the nearest neighbour spherical model on a d (= 3 , 4 or 5) dimensional hyper- cubic lattice. The interaction strengths are chosen to be J(ai) = J for 1 6 i < 3 and i J otherwise, where the ai are the lattice basis vectors. Then for H = 0 we obtain the saddle point equation by generalizing equation 2.2 to give

2K = {: dt exp [ - { 3 + (d - 3 ) i ) z t I Zo(t)3Zo(2t)d-3

Following Maradudin et a1 (1960) we divide the integration range into two parts at the point to, with to chosen so that ,?to % 1. We find that equation A.l is the sum of a regular function of 3. and z together with the expression

(2n)-d'2 1 ) . - I - + ( d - 3 ) c I ( i ) tA-3d-1 El+3d (p to) 1 3 0

where p = ( 3 + (d - 3 ) i ) ( z - 1) and cl().) is a polynomial of degree 1 in R arising out of the asymptotic expansion of the Bessel function product. Since the singular part of E,,(x) is - ( -x){~-' In x/T(p) if p = 1,2 ,3 , . . . and XI'- ' r (1 - p) otherwise, the singular z dependence of equation A.l near z = 1 is

d = 3

Expansions for the saddle point equation near the critical point when 2 = 1 (the isotropic case) follow

3; ,Kc - 23'271 ( z - 1)+ + O(z - 1) d = 3

d = 4

I 5 % ~ IK, - b,(z - 1) + 323:2n?(Z - 1p2 + . . . d = 5

In five dimensions the singular term appears only as a correction and b,, the coefficient of the leading order term, must be evaluated numerically. Expansions for z - 1 in terms of t (= T/T , - 1) may be obtained from the above expressions. Similarly, along the

954 D C Rapaport

critical isotherm ( t = 0), z - 1 can be expressed as a function of H. The critical exponents are readily deduced. Their values have been tabulated by Gunton and Buckingham (1967) (they are actually the value for the ideal Bose gas, but the two problems are mathe- matically isomorphic in the critical region).

For completeness we list the critical forms (for i = 1) of the quantities which are of interest here, and also give the exponent values required in $4. Specific heat :

3 - 16n2K;t d = 3

1 8n2K2 -+ - k ; ' C * 2 l n t I d = 4

Susceptibility:

( y = 2 ) d = 3 1 16n2K2t2

[L 10K,t

( p = 1 ) d = 5

Magnetization :

m,H1I5 ( 6 = 5 ) d = 3

M = - m , ( ~ 1n ~ p 3 ( 6 = 3 ) d = 4 1 m 5 ~ 1 / 3 ( 6 = 3 ) d = 5

where the mi are positive constants.

Coherence length:

1 t-' (v = 1) d = 3

I t - + (v = 3) d = 5

In giving the exponent values we have ignored the logarithmic dependence which appears when d = 4. For d = 3 the gap exponent A = 5.

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