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Journal of Physics A Mathematical and General

On universality between two-dimensional Ising-like systems

Joan Adler

Published Aug 1, 1987• 3 citations
Publisher

Abstract

Questions concerning the universality between different two-dimensional Ising-like systems have recently been raised, with evidence for the existence of several different subclasses of Ising-like systems being presented. Some new facts concerning similarity between border model and low-temperature spin-1 Ising model series analyses are noted. For the spin-1 case an alternative determination of the critical temperature determines unequivocally that this model is in the Ising universality class and it is suggested that a similar independent result could lead to a similar conclusion in the border model case. The possibility of an analytic correction to scaling in the spin-1 model is proposed.

DOI
10.1088/0305-4470/20/11/043

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Questions concerning the universality between different two-dimensional Ising-like systems have recently been raised, with evidence for the existence of several different subclasses of Ising-like systems being presented. Some new facts concerning similarity between border model and low-temperature spin-1 Ising model series analyses are noted. For the spin-1 case an alternative determination of the critical temperature determines unequivocally that this model is in the Ising universality class and it is suggested that a similar independent result could lead to a similar conclusion in the border model case. The possibility of an analytic correction to scaling in the spin-1 model is proposed.
Although the two-dimensional spin- 1 / 2 Ising model was solved in zero field in 1944, no exact results are yet available for the spin-1 model. This model should have the same dominant critical exponents as the spin- 1 / 2 model, but appears to exhibit non-analytic corrections to scaling as expected for a phi 4 model and unlike the spin- 1 / 2 model. In this paper new 45-term low temperature series for the magnetisation, specific heat and susceptibility of the spin-1 model on the square lattice are presented and analysed. A new 24-term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model (which is also in the phi 4 universality class) are also considered. For both models a non-analytic confluent correction with exponent 1.0< Delta 1 <1.3 is found. The validity of this result is enhanced by a comparison with the S= 1 / 2 case.
The aim of this paper is to determine the behavior of the specific heat of the 4-dimensional Ising model in a region around the critical temperature, and via that determine if the Ising model and the ϕ4-model belong to the same universality class in dimension 4. In order to do this we have carried out what is currently the largest scale simulations of the 4-dimensional Ising model, extending the lattices size up to L=256 and the number of samples per size by several orders of magnitude compared to earlier works, keeping track of data for both the canonical and microcanonical ensembles. Our conclusion is that the Ising model has a bounded specific heat, while the ϕ4-model is known to have a logarithmic divergence at the critical point. Hence the two models belong to distinct universality classes in dimension 4.
Order parameter fluctuations for the two-dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T(*) (L) and a locus of magnetic fields B(*) (L) are identified, for which the probability density function is similar to that for the two-dimensional XY model in the spin wave approximation. The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality-class-dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.
We compare simulation data from different sources on two canonical three-dimensional Ising spin glasses (ISGs): the binomial $\ifmmode\pm\else\textpm\fi{}J$ near-neighbor interaction ISG and the Gaussian interaction ISG. We allow for the possibility of corrections to finite size scaling and estimate the correction exponent w. Consistent estimates for the critical temperatures Tg{T}_{g}Tg​ and for the critical exponents for each system are obtained. The data strongly indicate that critical exponents in the two systems are significantly different from each other. These results thus confirm a breakdown of standard universality rules in Ising spin glasses.