Numerical study of the dynamics of some long range spin glass models

TL;DR

This study uses Monte Carlo simulations to analyze the equilibrium dynamics of long-range spin glass models, revealing insights into relaxation times and scaling behaviors across different regimes.

Contribution

It provides a detailed numerical analysis of the dynamical scaling laws in long-range spin glasses, bridging mean field and finite-dimensional behaviors.

Findings

Activated scaling fits are unstable with subleading corrections.

Critical scaling offers more stable fits in non mean field regimes.

Large system simulations highlight challenges in determining dynamical exponents.

Abstract

We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter $\sigma$, this model interpolates between the mean field Sherrington-Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time $\tau$ as a function of the number of spins $N$ (Namely $\ln(\tau)\propto N^{\psi}$) give values of $\psi$ that are not stable against inclusion of subleading corrections. Critical scaling ($\tau\propto N^{\rho}$) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington-Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.