Dual time scales in simulated annealing of a two-dimensional Ising spin glass

TL;DR

This paper investigates the out-of-equilibrium dynamics of simulated annealing in a 2D Ising spin glass, revealing dual time scales in relaxation processes and highlighting differences from equilibrium dynamics.

Contribution

It introduces a generalized Kibble-Zurek scaling approach to identify two distinct dynamic exponents for energy and order parameter relaxation in spin glasses.

Findings

Energy relaxes slower than the order parameter, indicating two dynamic exponents.

The dynamic exponents are approximately 8.28 for the order parameter and 10.31 for energy.

The dual time scale behavior is linked to entropy-driven droplet ordering mechanisms.

Abstract

We apply a generalized Kibble-Zurek out-of-equilibrium scaling ansatz to simulated annealing when approaching the spin-glass transition at temperature $T=0$ of the two-dimensional Ising model with random $J= \pm 1$ couplings. Analyzing the spin-glass order parameter and the excess energy as functions of the system size and the annealing velocity in Monte Carlo simulations with Metropolis dynamics, we find scaling where the energy relaxes slower than the spin-glass order parameter, i.e., there are two different dynamic exponents. The values of the exponents relating the relaxation time scales to the system length, $\tau \sim L^z$, are $z=8.28 \pm 0.03$ for the relaxation of the order parameter and $z=10.31 \pm 0.04$ for the energy relaxation. We argue that the behavior with dual time scales arises as a consequence of the entropy-driven ordering mechanism within droplet theory. We point out that the dynamic exponents found here for $T \to 0$ simulated annealing are different from the temperature-dependent equilibrium dynamic exponent $z_{\rm eq}(T)$, for which previous studies have found a divergent behavior; $z_{\rm eq}(T\to 0) \to \infty$. Thus, our study shows that, within Metropolis dynamics, it is easier to relax the system to one of its degenerate ground states than to migrate at low temperatures between regions of the configuration space surrounding different ground states. In a more general context of optimization, our study provides an example of robust dense-region solutions for which the excess energy (the conventional cost function) may not be the best measure of success.