Bimodal and Gaussian Ising spin glasses in dimension two

Abstract

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.

Ising spin glasses in dimension five

“…These dimension d = 5 ISG data thus confirm the empirical conclusion reached from dimension d = 4 and dimension d = 2 studies [4][5][6] that ISG models in a fixed dimension but with different interaction distributions do not lie in the same universality class.…”

Ising spin glasses in two dimensions: Universality and nonuniversality

“…From both approaches we deduce estimates for the bimodal ISG exponents in the T > T * (L) regime which are fully compatible with our previous conclusions Ref. [1] including η ≈ 0.20.…”

“…There are two limiting regimes, with a size dependent crossover temperature T * (L) [10], a T < T * (L) ground state plus gap dominated regime and an effectively continuous energy level regime T > T * (L). There have been consistent estimates over decades from correlation function measurements [11,12], Monte Carlo renormalization-group measurements [13], transfer matrix calculations [14], numerical simulations [1,[15][16][17],…”

“…The canonical dimension d = 2 Edwards-Anderson (EA) model Ising spin glasses (ISGs) on square lattices with either Gaussian or bimodal (±J) nearest neighbor interaction distributions have been the subject of numerous studies over many years. Below we will refer in particular to our own measurements on these two models [1]. There are analytic arguments that these two archetype models (and by extension all 2D ISG models with other distributions) have zero-temperature transitions [2,3].…”

Hyperscaling Violation in Ising Spin Glasses

“…The reduced second-moment correlation length is defined as in Ising models and as in ISG models [ 20 ]. Note that the critical-limit ThL exponent is unaltered by this normalization (models with zero-temperature critical points are a special case [ 21 ]). From exact and general HTSE for either model this reduced correlation length tends to 1 at infinite temperature [ 2 , 19 ].…”