Infinite volume extrapolation in the one-dimensional bond diluted Levy spin-glass model near its lower critical dimension

TL;DR

This paper uses numerical simulations to analyze the critical behavior of one-dimensional bond diluted Levy spin glasses near their lower critical dimension, estimating correlation lengths and susceptibilities.

Contribution

It provides the first detailed finite size scaling analysis of these models outside mean field theory, especially near the critical threshold at b6=2.

Findings

Evidence of non-zero critical temperature for b6=5/3 and b6=9/5

Divergences in correlation length and susceptibility at critical points

Behavior of critical exponents near the threshold b6=2

Abstract

We revisited, by means of numerical simulations, the one dimensional bond diluted Levy Ising spin glasses outside the limit of validity of mean field theories. In these models the probability that two spins at distance $r$ interact (via a disordered interactions, $J_{ij}=\pm 1$) decays as $r^{-\rho}$. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for $\rho=5/3$ and $\rho=9/5$. We have obtained strong evidence for divergences of the previous observables at a non zero critical temperature. We discuss the behavior of the critical exponents, especially when approaching the value $\rho=2$, corresponding to a critical threshold beyond which the model has no phase transition. Finally, we numerically study the model right at the threshold value $\rho=2$.