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=±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, τ∼L^{z}, are z=8.28±0.03 for the relaxation of the order parameter and z=10.31±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→0 simulated annealing are different from the temperature-dependent equilibrium dynamic exponent z_{eq}(T), for which previous studies have found a divergent behavior: z_{eq}(T→0)→∞. 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.

We carry out simulated annealing and employ a generalized Kibble-Zurek scaling hypothesis to study the two-dimensional Ising spin glass with normal-distributed couplings. The system has an equilibrium glass transition at temperature T = 0. From a scaling analysis when T → 0 at different annealing velocities v, we find power-law scaling in the system size for the velocity required in order to relax toward the ground state; v ∼ L −(z+1/ν) , the Kibble-Zurek ansatz where z is the dynamic critical exponent and ν the previously known correlation-length exponent, ν ≈ 3.6. We find z ≈ 13.6 for both the Edwards-Anderson spin-glass order parameter and the excess energy. This is different from a previous study of the system with bimodal couplings [S. J. Rubin, N. Xu, and A. W. Sandvik, Phys. Rev. E 95, 052133 (2017)] where the dynamics is faster (z is smaller) and the above two quantities relax with different dynamic exponents (with that of the energy being larger). We argue that the different behaviors arise as a consequence of the different low-energy landscapes-for normal-distributed couplings the ground state is unique (up to a spin reflection) while the system with bimodal couplings is massively degenerate. Our results reinforce the conclusion of anomalous entropy-driven relaxation behavior in the bimodal Ising glass. In the case of a continuous coupling distribution, our results presented here also indicate that, although Kibble-Zurek scaling holds, the perturbative behavior normally applying in the slow limit breaks down, likely due to quasi-degenerate states, and the scaling function takes a different form.