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

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 $z_{\mathrm{eq}}(T)$, for which previous studies have found a divergent behavior; $z_{\mathrm{eq}}(T\to 0)\to \infty$. 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.

13 pages, 16 figures

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• Critical percolation in the slow cooling of the bi-dimensional ferromagnetic Ising model
• An experiment-oriented analysis of 2D spin-glass dynamics: a twelve time-decades scaling study
• Spin Glass in the Bond-Diluted J1-J2 Ising Model on the Square Lattice
• Dynamic scaling in the 2D Ising spin glass with Gaussian couplings
• Amorphous quantum magnets in a two-dimensional Rydberg atom array
• Quantum Vertex Model for Reversible Classical Computing
• SWAP algorithm for lattice spin models