The bimodal and Gaussian Ising Spin Glasses in dimension two revisited

Abstract

A new analysis is given of numerical simulation data on the archetype square lattice Ising Spin Glasses (ISG) with a bimodal ($\pm J$) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian has a non-degenerate ground state so exponent $\eta \equiv 0$ 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 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. We show that in this regime the critical exponent $\eta$ is not zero, so the effectively continuous regime $2$D bimodal ISG is not in the same universality class as the $2$D Gaussian ISG. The simulation data on both models are analyzed using a scaling variable $\tau = T^2/(1+T^2)$ suitable for zero temperature transition ISGs, together with appropriate scaling expressions. Accurate simulation estimates can be obtained for the temperature dependence of the thermodynamic limit reduced susceptibility $\chi(\tau)$ and second moment correlation length $\xi(\tau)$ over the entire range of temperature from zero to infinity. The Gaussian critical exponent from the simulations $\nu = 3.5(1)$ is in full agreement with the well established value from the literature. The bimodal exponent from the thermodynamic limit regime analysis is $\nu = 4.2(1)$, once again different from the Gaussian value.