Scaling behavior at zero-temperature critical points

A scaling form for the logarithm of the partition function suitable for a zero-temperature critical point is obtained and found to hold for the spherical model in less than two dimensions and the classical n-component Heisenberg linear chain. Nevertheless, several cases are found where the critical-exponent relations involving the specific heat fail. These anomalous cases do not imply a breakdown of the scaling implicit in the basic formulation of renormalization-group theory.


I. INTRODUCTION
The renormalization-group approach 1 to the problem of critical phenomena depends explicitly on an assumption which requires a certain scaling form for the logarithm of the partition function. In this paper we investigate the spherical model and the n-component classical Heisehberg model for those spatial dimensions for which the critical temperature goes to zero. We find in the case of the linear three-component classical Heisenberg model and also the linear spin-i Ising model that anomalies do occur. These anomalies imply the failure of the expected scaling relations among the critical indices which involve O!, the specific-heat index. However, this failure does not invalidate the scaling form assumed by the renormalization-group approach, as one might perhaps initially have feared, but reflects certain special situations.
Hyperscaling for a zero-temperature critical point implies that lim (ln( C H~tl)) = Q ' using the usual definition of critical exponents. Here ~ is the range of correlation, C 8 is the speciic heat at constant magnetic field, d is the spatial dimension, and T is the absolute temperature, Specifically, the anomalies that occur are that instead of zero for the right-hand side of Eq. (1), we obtain 2 for the one-dimensional Ising model and 1 for the classical Heisenberg chain. If the "singular part" of the specific heat is used instead of the dominant part, the result,oo, is obtained instead for the linear classical Heisenberg model! II. REFORMULATED SCALING AND THE RENORMALIZATION GROUP Wilson 2 explicitly assumes that there exists a renormalization-group transformation with a fixed point and a particular, simple form near the fixed point. He uses this assumption to show that r(q1, Ch, · · ·, ' ii.n; T) = ~ <n.ll11.ndafu1F(~q 1 , ~~' , , , , ~C'in) ,

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where tA is a normalization factor, F is a function assumed not to vanish (but it can in anomalous cases), and the r's are the Fourier transforms of the truncated spin-spin correlation functions. By truncated correlation functions, we mean the Ursell functions, 3 or, in the language of statistics, 4 the cumulants, instead of the moments ( correlation functions). The rest of Wilson's renormalization-group derivations appear to depend in an essential way on this result. As far as the thermodynamics of the system is concerned, we need to know the logarithm of the partition function where (3=1/kT, :JC is the Hamiltonian, His the magnetic field, mt is the magnetization variable, and m is the magnetic moment per spin. The form of (lnZ)/N, where N is the number of spins, can be deduced from (2) by setting Ci; =0. We then conclude (4) where d,, is called the anomalous dimension of the spin field. The function Y is the formal sum of a power series in H 2 whose coefficients are derived from (2). This form is supposed to hold at least forT~T 0 , andH«~1 1 a- 11 • In order. to extend the form (4) to the case of zero critical temperature, it is helpful to remember the various definitions for the zero-'field susciiptibility, specific heat, spin-spin correlation functions, correlation length, and the magnetization along the critical isotherm In concert with these definitions (5) and the scaling hypothesis (6), we may write (4) to display explicitly the dependence on inverse temperature (3. Thus we have (7) Form (7) insures the definition of x and the scaling hypothesis (6) with and is a form of weak scaling. 5 When, in addition, we impose the (renormalization-group) assumption z =d, we have hyperscaling. The usual exponent relations 6 then become by form (7) and the definitions (5) and (6). The "singular-part" [relation (9) (iv) is equivalent to the limit formulation (1)] of the specific heat has index a 8 , where it differs from a. The modifications from the usual relations [only (9) (ii) remains the same] are due to the confluence of the singularity at T=O with that at T=T 0 when T 0 =0.

III. SPHERICAL MODEL
It is well known that the spherical model is the noo limit of the classical n-component Heisenberg model 7 and that it can be treated in such a way that the space dimension d plays the role of a continuous parameter. 8 The critical temperature falls to zero as d decreases to 2. Ford> 2, the scaling form (4) is known to hold; however, many of the critical indices diverge 9 as d-2. By use of the general formulas of Joyce, 9 we have verified by direct calculation for 0 < d < 2 the validity of form (7) with z =d. The critical indices for this model are A=2/(2-d), o=oo, which satisfy the relations (9) and (11) The magnetization for T = 0 is a constant in H which gives [by {5)] the values of T/ and o.

IV. n-COMPONENT CLASSICAL HEISENBERG MODEL
In the calculation of the properties of the n-component classical Heisenberg model of Stanley,7 we use the transfer-matrix method to compute the solution for the linear chain. The transfer matrix for this problem is T = exp(isJSln> • s::n (12) and its eigenvalues are 1 0-12 x:n)~J) = (2'11")n/2(n{3J)1-<n/2)J(n/2).1+J(n{3J) ' (13) where by the asymptotic expansion of the Bessel functions for large arguments. 14 It is to be noted 13 that when n is 1or3, the series (15) does not have the expected leading order terms, and, in fact, the correction terms to those explicitly given in (15) are of the form e<-2.Bn. If we follow the analysis of McLean and Blume 15 for the four-spin correlation function, which generalizes directly from their case n = 3 to general n ;z: 1 in terms of the eigenvalues (13), we can compute, using that a 4 lnZ/N ( K K' .\ aH4 cc(3s (l-Yn)2{l-zn) + (1-yn)a;' (17) where K andK' are constants. Thus by substitution of (15) and (16) for all n except 1 and 3. (The definition of the magnetization in terms of the limit of the spin-spin correlations has been used.) Comparison with (10) shows that these results agree identically with those SCALING BEHAVIOR AT ZERO-TEMPERATURE CRITICAL POINTS 3743 for the spherical model with d = 1.
In the case n = 3, the form C H/N""'-k(n -l)(~ + c 1T + c 2 T 2 + • .. ) (19) for specific heat fails, as all the ci coefficients vanish. The correct formula 13 • 16 is· (20) which yields the anomalous value a 8 =oo, In the case n= 1, Eq. (15) fails as every coefficient vanishes, and we get Y 1 = tanh(j3J) ""' 1 -2e· 28 J. (21) This exponential approach causes, y, v, and a to be infinite. [The coefficient K in Eq. (17) vanishes for this case as well. ] Even so, all the relations (9) continue to hold, with the exception of (9) '{iv), if' we interpret them as limits in the sense of Eq. (1}. First we have T/ = 1, since y 1 goes to unity as T-0. Then relation (9)  with error terms of e· 213 J in comparison with the terms retained, allows us to observe that "i3"'"' is replaced by j3e 2 13J; so relation (9) (ii) correctly yields 5 = «>, Relation (9) (iii) becomes . ln(Nl3 2 e 4 13J,/ ~x{3) we see that relation (9) (iv) fails, i.e., Eq. (1). The reason is clearly that the derivative with respect to j3 of terms like e· 2 13J does not increase the divergence, as it does for a power of j3. Thompson 18 has considered in detail the critical properties of the one-dimensional, spin-oo Ising model (see also Joyce 19 ). nition A= 2 -a -y, then relations (9) hold for this model! This result differs, of course, from the spin-~Isingcasewhere (9) (iv) failso Wewouldlike to point out in connection with the n-vector models that the assertation of Balian and Toulouse 12 that a = 1 for n < 1 in these models is somewhat artificial, as the coefficient of j3 1 in CH exactly vanishes. This analytic continuation of the one-dimensional Stanley sequence is interesting in that Tc> 0 for n < 1.
[Note that if Tc continued to stick at zero, Eq. (19) would indicate a negative zero-point specific heat.] The scaling form for the logarithm of the partition function continues to hold, but a = 0 and a failure of the usual (Tc> O) scaling law dv = 2 -a for these models is the correct conclusion. It is to be noted that for n = O, the second-and third-largest eigenvalues are exactly degenerate for all temperatures, since for integral order Im= 1.m• and so the critical behavior may not be given by the above analysis.

V. SUMMARY
In conclusion, we point out that for all the models considered here, the scaling form (7) holds with z = d and relations (9) (i)-(9) (iii) are valid. The adependent relation (9) (iv) fails in a variety of cases for a number of reasons unrelated to the validity of the scaling form. For this class of models (n vector, d dimensional, d < 4) it appears, as is well known, that the only substantial evidence for the failure of hyperscaling as distinguished from the failure of a particular exponent relation, is the numerical evidence for the three-dimensional Ising model, where the best exponent values are 20 -22 A=l which shows a small but persistent difference from the value 3, i.e., an anomalous dimension of the vacuum! The determination of an index A is, of course, a verification of the scaling form (7).