Scaling Relations for Logarithmic Corrections

Abstract

Multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical-mechanical systems. Here, a Lee-Yang zero approach is used to systematically analyse the exponents of such logarithms and to propose scaling relations between them. These proposed relations are then confronted with a variety of results from the literature. 05.50.+q, 05.70.Jk, 75.10.Hk Conventional leading scaling behavior at a secondorder phase transition is described by power laws in the reduced temperature t and field h. With h = 0, the correlation length, specific heat, and susceptibility behave as ξ ∞ (t) ∼ |t| −ν , C ∞ (t) ∼ |t| −α , and χ ∞ (t) ∼ |t| −γ , while the magnetization in the broken phase has m ∞ (t) ∼ |t| β . Here the subscript indicates the extent of the system. At t = 0 the magnetization scales as m ∞ (h) ∼ h 1/δ while the anomalous dimension η characterizes the correlation function at criticality. In the 1960's, it was shown that these six critical exponents are related via four scaling relations (see e.g. Ref.[1] and references therein), which are now firmly established and fundamentally important in the theory of critical phenomena. With d representing the dimensionality of the system, the scaling relations areIn the conventional scaling scenario, (2) and (3) can, in fact, be deduced from the Widom scaling hypothesis that the Helmholtz free energy is a homogeneous function [2]. Widom scaling and the remaining two laws can, in turn, be derived from the Kadanoff block-spin construction [3] and ultimately from Wilson's renormalization group (RG) [4]. The relation (1) can also be derived from the hyperscaling hypothesis, namely, that the free energy behaves near criticality as the inverse correlation volume:. Twice differentiating this relation recovers (1).The scaling relations, (2) and (3), were both rederived using an alternative route by Abe [5] and Suzuki [6] exploiting the fact that the even and odd scaling fields can be linked by Lee-Yang zeros [7]. The locus of these zeros in the magnetic-field plane is controlled by the temperature. In the t > 0 (disordered) phase this locus terminates at the Yang-Lee edge [7], the distance of which from the critical point is denoted by r YL (t). At a conventional second-order phase transition r YL (t) ∼ t ∆ for t > 0, and the gap exponent ∆ is related to the other exponents through [5,6] Logarithmic corrections are characteristic of a number of marginal scenarios (see, e.g., Ref.[8] and references therein). Hyperscaling fails at and above the upper critical dimension d c and, while (1) holds there, it too fails above d c , where mean-field behavior (which is independent of d) prevails. At d c itself, multiplicative logarithmic corrections to scaling are manifest. Such corrections are found in marginal d < d c situations too [8,9]. The q-state Potts model in d = 2 dimensions possesses a firstorder transition for q > 4 and a second-order one when q < 4. The q = 4 case is also characterized by a transition of second order, alb...

Scaling analysis of the site-diluted Ising model in two dimensions

“…This comes about through the delicate manner in which the exponents of the logarithms, which are nonzero in thermal scaling, balance each other out. In this way, it is established that the Lee-Yang zeros of disordered systems can be precisely determined numerically, a density-of-zeros analysis is applicable to such a system, also at the level of logarithms, new scaling relations for the logarithmic corrections [5,6] in this model are confirmed and a negative exponent for the contentious specific heat or its multiplicative logarithmic correction is made unlikely.…”

“…The theory presented in [5,6] relates the exponents of the logarithmic corrections in an analogous manner. With the strong universality hypothesis, the leading critical exponents for the dilute Ising models are identical to their pure counterparts:…”

“…Recently a self-consistent scaling theory for logarithmic-correction exponents has been presented [5,6]. Denoting the reduced temperature by t and the reduced external field by h, this theory deals with the circumstances where, in the absence of field, the specific heat, magnetization, susceptibility and correlation length scale respectively as…”

“…Finally, a self-consistent scaling theory which links the exponents characterising the logarithmic corrections to scaling was recently presented in Refs. [5,6].…”

“…Here, these numerical [3,4] and analytical [5,6] advances are combined to unambiguously determine the leading scaling behaviour and potential multiplicative logarithmic corrections in the specific heat through the density of Lee-Yang zeros and the scaling relations. This and all multiplicative logarithmic correction-to-scaling exponents for FSS of odd functions are zero.…”