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arXiv:1411.2155 · Physical Review E

Evidence for nonuniversal scaling in dimension-four Ising spin glasses

P. H. Lundow, Ian A. Campbell

Published Apr 17, 2015• 7 citations
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Abstract

The critical behavior of the Binder cumulant for Ising spin glasses in dimension four is studied through simulation measurements. Data for the bimodal interaction model are compared with those for the Laplacian interaction model. Special attention is paid to scaling corrections. The limiting infinite size value at criticality for this dimensionless variable is a parameter characteristic of a universality class. This critical limit is estimated to be equal to 0.523(3) in the bimodal model and to 0.473(3) in the Laplacian model.

DOI
10.1103/physreve.91.042121

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The critical behavior of the Binder cumulant for Ising spin glasses in dimension four is studied through simulation measurements. Data for the bimodal interaction model are compared with those for the Laplacian interaction model. Special attention is paid to scaling corrections. The limiting infinite size value at criticality for this dimensionless variable is a parameter characteristic of a universality class. This critical limit is estimated to be equal to 0.523(3) in the bimodal model and to 0.473(3) in the Laplacian model.
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.
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.
Equilibrium numerical data on the three dimensional bimodal interaction Ising spin glass up to size L=48 show that corrections to scaling, which are known to be strong, behave in a non-monotonic manner with size. Extrapolation to the infinite size thermodynamic limit is difficult; however the large L data indicate that the ordering temperature Tc lies significantly higher than the values which have been estimated from previous numerical work limited to smaller sizes. In view of the present results it is at the least premature to conclude that the three dimensional bimodal and Gaussian Ising spin glasses lie in the same universality class.
Extensive simulations are made on the bimodal Ising Spin Glass (ISG) in dimension four. The transition temperature is established using a combination of standard finite size scaling and of thermodynamic derivative peak data. Measurements in the thermodynamic limit regime are analysed so as to estimate critical exponents and confluent correction terms. Comparisons with results on other 4d ISGs show that the susceptibility and correlation length critical exponents γγγ and ννν depend on the form of the interaction distribution. From this observation it can be deduced that critical exponents are not universal in ISGs.
Direct measurements of the spin glass correlation function G(R)G(R)G(R) for Gaussian and bimodal Ising spin glasses in dimension two have been carried out in the temperature region $T\ensuremath{\sim}1$. In the Gaussian case the data are consistent with the known anomalous dimension value $\ensuremath{\eta}\ensuremath{\equiv}0$. For the bimodal spin glass in this temperature region T>T∗(L)T>{T}^{*}(L)T>T∗(L), well above the crossover T∗(L){T}^{*}(L)T∗(L) to the ground-state-dominated regime, the effective exponent $\ensuremath{\eta}$ is clearly nonzero and the data are consistent with the estimate $\ensuremath{\eta}\ensuremath{\sim}0.28(4)$ given by McMillan in 1983 from similar measurements. Measurements of the temperature dependence of the Binder cumulant U4(T,L){U}_{4}(T,L)U4​(T,L) and the normalized correlation length $\ensuremath{\xi}(T,L)/L$ for the two models confirms the conclusion that the two-dimensional (2D) bimodal model has a nonzero effective $\ensuremath{\eta}$ both below and above T∗(L){T}^{*}(L)T∗(L). The 2D bimodal and Gaussian interaction distribution Ising spin glasses are not in the same universality class.
Following numerous earlier studies, extensive simulations and analyses were made on the continuous interaction distribution Gaussian model and the discrete bimodal interaction distribution Ising spin glass (ISG) models in two dimensions [Lundow and Campbell, Phys. Rev. E 93, 022119 (2016)1539-375510.1103/PhysRevE.93.022119]. Here we further analyze the bimodal and Gaussian data together with data on two other continuous interaction distribution two-dimensional ISG models, the uniform and the Laplacian models, and three other discrete interaction distribution models, a diluted bimodal model, an "antidiluted" model, and a more exotic symmetric Poisson model. Comparisons between the three continuous distribution models show that not only do they share the same exponent η≡0 but that to within the present numerical precision they share the same critical exponent ν also, and so lie in a single universality class. On the other hand the critical exponents of the four discrete distribution models are not the same as those of the continuous distributions, and the present data strongly indicate that they differ from one discrete distribution model to another. This is evidence that discrete distribution ISG models in two dimensions have nonzero values of the critical exponent η and do not lie in a single universality class.