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arXiv:cond-mat/0510668

Nontrivial critical behavior of the free energy in the two-dimensional Ising spin glass with bimodal interactions

Helmut G. Katzgraber, L. W. Lee, Ian A. Campbell

Published Oct 25, 2005• 1 citation
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Abstract

A detailed analysis of Monte Carlo data on the two-dimensional Ising spin glass with bimodal interactions shows that the free energy of the model has a nontrivial scaling. In particular, we show by studying the correlation length that much larger system sizes and lower temperatures are required to see the true critical behavior of the model in the thermodynamic limit. Our results agree with data by Lukic et al. in that the degenerate ground state is separated from the first excited state by an energy gap of 2J.

DOI
10.48550/arxiv.cond-mat/0510668

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A detailed analysis of Monte Carlo data on the two-dimensional Ising spin glass with bimodal interactions shows that the free energy of the model has a nontrivial scaling. In particular, we show by studying the correlation length that much larger system sizes and lower temperatures are required to see the true critical behavior of the model in the thermodynamic limit. Our results agree with data by Lukic et al. in that the degenerate ground state is separated from the first excited state by an energy gap of 2J.
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.
We compute the exact partition function of 2d Ising spin glasses with binary couplings. In these systems, the ground state is highly degenerate and is separated from the first excited state by a gap of size 4J. Nevertheless, we find that the low temperature specific heat density scales as exp(-2J/T), corresponding to an "effective" gap of size 2J; in addition, an associated crossover length scale grows as exp(J/T). We justify these scalings via the degeneracy of the low lying excitations and by the way low energy domain walls proliferate in this model.
We study the correlation length of the two-dimensional Edwards-Anderson Ising spin glass with bimodal interactions using a combination of parallel tempering Monte Carlo updates and a rejection-free cluster algorithm in order to speed up equilibration. Our results show that the correlation length grows $\ensuremath{\sim}\mathrm{exp}(2J∕T)$ suggesting through hyperscaling that the degenerate ground state is separated from the first excited state by an energy gap $\ensuremath{\sim}4J$, as would naively be expected.
An exact algorithm is used to compute the degeneracies of the excited states of the bimodal Ising spin glass in two dimensions. It is found that the specific heat at arbitrary low temperature is not a self-averaging quantity and has a distribution that is neither normal nor lognormal. Nevertheless, it is possible to estimate the most likely value and this is found to scale as L 3 T -2 exp (−4 J / kT ), for a L × L lattice. Our analysis also explains, for the first time, why a correlation length ξ∼exp (2 J / kT ) is consistent with an energy gap of 2 J . Our method allows us to obtain results for up to 10 5 disorder realizations with L ⩽96. Distributions of second and third excitations are also shown.
We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size LLL: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter ppp, p=0.5p=0.5p=0.5 and p=0.7p=0.7p=0.7 (up to L=28L=28L=28 and L=20L=20L=20, respectively), and the bond-diluted bimodal model for bond-occupation probability pb=0.45{p}_{b}=0.45pb​=0.45 (up to L=16L=16L=16). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at pb=0.7{p}_{b}=0.7pb​=0.7 and pb=0.35{p}_{b}=0.35pb​=0.35 (up to L=10L=10L=10) and of the Ising spin glass model with Gaussian bond distribution (up to L=8L=8L=8) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.