The bimodal and Gaussian Ising Spin Glasses in dimension two revisited

摘要

A new analysis is given of numerical simulation data on the archetype squarelattice Ising Spin Glasses (ISG) with a bimodal ($\pm J$) and Gaussianinteraction distributions. It is well established that the ordering temperatureof both models is zero. The Gaussian has a non-degenerate ground state soexponent $\eta \equiv 0$ and it has a continuous distribution of energy levels.For the bimodal model, above a size dependent cross-over 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 plusan 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. Weshow that in this regime the critical exponent $\eta$ is not zero, so theeffectively continuous regime $2$D bimodal ISG is not in the same universalityclass as the $2$D Gaussian ISG. The simulation data on both models are analyzedusing a scaling variable $\tau = T^2/(1+T^2)$ suitable for zero temperaturetransition ISGs, together with appropriate scaling expressions. Accuratesimulation estimates can be obtained for the temperature dependence of thethermodynamic limit reduced susceptibility $\chi(\tau)$ and second momentcorrelation length $\xi(\tau)$ over the entire range of temperature from zeroto infinity. The Gaussian critical exponent from the simulations $\nu = 3.5(1)$is in full agreement with the well established value from the literature. Thebimodal exponent from the thermodynamic limit regime analysis is $\nu =4.2(1)$, once again different from the Gaussian value.