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arXiv:1709.06153 · Physical review. B./Physical review. B

Bimodal Ising spin glass in two dimensions: The anomalous dimension η

P. H. Lundow, Ian A. Campbell

Published Jan 22, 2018• 5 citations
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Abstract

Direct measurements of the spin glass correlation function $G(R)$ 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 $T>{T}^{*}(L)$, well above the crossover ${T}^{*}(L)$ 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 ${U}_{4}(T,L)$ 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 ${T}^{*}(L)$. The 2D bimodal and Gaussian interaction distribution Ising spin glasses are not in the same universality class.

DOI
10.1103/physrevb.97.024203

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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∼1T\ensuremath{\sim}1T\ensuremath∼1. In the Gaussian case the data are consistent with the known anomalous dimension value \ensuremathη\ensuremath≡0\ensuremath{\eta}\ensuremath{\equiv}0\ensuremathη\ensuremath≡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 \ensuremathη\ensuremath{\eta}\ensuremathη is clearly nonzero and the data are consistent with the estimate \ensuremathη\ensuremath∼0.28(4)\ensuremath{\eta}\ensuremath{\sim}0.28(4)\ensuremathη\ensuremath∼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 \ensuremathξ(T,L)/L\ensuremath{\xi}(T,L)/L\ensuremathξ(T,L)/L for the two models confirms the conclusion that the two-dimensional (2D) bimodal model has a nonzero effective \ensuremathη\ensuremath{\eta}\ensuremathη 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.
Monte Carlo data of the two-dimensional Ising spin glass with bimodal interactions are presented with the aim of understanding the low-temperature physics of the model. An analysis of the specific heat, spin-glass susceptibility, finite-size correlation length, and the Binder ratio is performed to try to verify a recent proposal in which for large system sizes and finite but low temperatures the effective critical exponents are identical to the critical exponents of the two-dimensional Ising spin glass with Gaussian interactions. Our results show that with present system sizes the recently proposed scenario, in which the two-dimensional Ising spin glass with bimodally distributed interactions is in the same universality class as the model with Gaussian-distributed disorder at low but finite temperatures, cannot be reliably proven.
An analysis is given of numerical simulation data to size L=128 on the archetype square lattice Ising spin glasses (ISGs) with bimodal (±J) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent η≡0, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover temperature T(*)(L) there is a regime of effectively continuous energy levels; below T(*)(L) there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. T(*)(L) tends to zero at very large L, leaving only the effectively continuous regime in the thermodynamic limit. The simulation data on both models are analyzed with the conventional scaling variable t=T and with a scaling variable τ(b)=T(2)/(1+T(2)) suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility χ(τ(b),L) and second moment correlation length ξ(τ(b),L) in the thermodynamic limit regime are extrapolated to the τ(b)=0 critical limit. The Gaussian critical exponent estimates from the simulations, η=0 and ν=3.55(5), are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are η=0.20(2) and ν=4.8(3), distinctly different from the Gaussian critical exponents.
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.
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.
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.
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.