Bimodal and Gaussian Ising spin glasses in dimension two

摘要

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 criticalexponent $η ≡ 0$, 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 witha scaling variable $\tau_b = T^2/(1 + T^2)$ suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility $χ(\tau_b,L)$ and second moment correlation length $ξ (\tau_b,L)$ in the thermodynamic limit regime are extrapolated to the $\tau_b = 0$ 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.