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arXiv:cond-mat/0111174

Ising Spin Glasses : interaction distribution dependence of the critical exponents

Pierre‐Olivier Mari, Ian A. Campbell

Published Nov 9, 2001• 5 citations
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Abstract

Continuous phase transitions are catalogued into universality classes, families of systems having identical values of all the exponents governing the critical behaviour of their different physical properties. Numerical simulations have been carried out on Ising Spin Glasses in dimension three, using a technique where corrections to finite size scaling can be controled. The data show that the critical exponents vary strongly as a function of the kurtosis of the interaction distribution, a parameter which from the standard point of view should not be pertinent. This observation implies that for spin glasses the renormalization group analysis should not be approached in a the same way as in the case of canonical second order transitions; a much richer structure of universality classes would appear to exist for spin glasses.

DOI
10.48550/arxiv.cond-mat/0111174

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Continuous phase transitions are catalogued into universality classes, families of systems having identical values of all the exponents governing the critical behaviour of their different physical properties. Numerical simulations have been carried out on Ising Spin Glasses in dimension three, using a technique where corrections to finite size scaling can be controled. The data show that the critical exponents vary strongly as a function of the kurtosis of the interaction distribution, a parameter which from the standard point of view should not be pertinent. This observation implies that for spin glasses the renormalization group analysis should not be approached in a the same way as in the case of canonical second order transitions; a much richer structure of universality classes would appear to exist for spin glasses.
Ising spin-glass models with bimodal, Gaussian, uniform, and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin-glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two.
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.
Extensive simulations are made on Ising Spin Glasses (ISG) with Gaussian, Laplacian and bimodal interaction distributions in dimension four. Standard finite size scaling analyses near and at criticality provide estimates of the critical inverse temperatures βcβ_cβc​, critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for βcβ_cβc​ and the exponent ννν from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable τ=1−β2/βc2τ= 1-β^2/β_c^2τ=1−β2/βc2​ and the appropriate scaling expressions. Data over the entire paramagnetic range of temperatures are analysed in order to obtain further estimates of the critical exponents together with correction to scaling terms. The Privman-Fisher ansatz then leads to compact scaling expressions for the whole paramagnetic regime and for all sample sizes L. Comparisons between the 4d ISG models show that the critical dimensionless parameters characteristic of a universality class, and the susceptibility and correlation length critical exponents γγγ and ννν, depend on the form of the interaction distribution. From these observations it can be deduced that critical exponents are not universal in ISGs, at least in dimension four.
We determine accurate values of ordering temperatures and critical exponents for Ising-spin-glass transitions in four dimensions, using a combination of finite-size scaling and nonequilibrium scaling techniques. We find that the exponents $\ensuremath{\eta}$ and zzz vary with the form of the interaction distribution, indicating nonuniversality at Ising-spin-glass transitions. These results confirm conclusions drawn from numerical data for three dimensions.
Extensive simulations are made of the spin glass susceptibility and correlation length in five dimension Ising Spin Glasses (ISGs) with Gaussian and bimodal interaction distributions. Once the transition temperature is accurately established using a standard criterion, critical exponents and correction terms can be readily estimated by extrapolating measurements made in the thermodynamic limit regime. The data show that the critical exponents of the susceptibility γγγ and of the correlation length ννν depend on the form of the interaction distribution. This observation implies that quite generally critical exponents are not universal in ISGs.