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.

arXiv comment13 pages, 16 figures

References in corpus (32)

• Exchange Monte Carlo Method and Application to Spin Glass Simulations
• Quantum Annealing in the Transverse Ising Model
• A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
• Quantum annealing with more than one hundred qubits
• Dynamics of a Quantum Phase Transition
• Dynamics of quantum phase transition: exact solution in quantum Ising model
• Universal adiabatic dynamics across a quantum critical point
• Spontaneous creation of Kibble-Zurek solitons in a Bose-Einstein condensate
• The Kibble-Zurek Problem: Universality and the Scaling Limit
• Quantum versus Classical Annealing of Ising Spin Glasses
• Universal space-time scaling symmetry in the dynamics of bosons across a quantum phase transition
• Unreasonable Effectiveness of Learning Neural Networks: From Accessible States and Robust Ensembles to Basic Algorithmic Schemes
• Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines
• Subdominant Dense Clusters Allow for Simple Learning and High Computational Performance in Neural Networks with Discrete Synapses
• The critical exponents of the two-dimensional Ising spin glass revisited: Exact Ground State Calculations and Monte Carlo Simulations
• Seeking Quantum Speedup Through Spin Glasses: The Good, the Bad, and the Ugly
• Dynamic scaling at classical phase transitions approached through non-equilibrium quenching
• Strong universality and algebraic scaling in two-dimensional Ising spin glasses
• Quantum versus classical annealing: insights from scaling theory and results for spin glasses on 3-regular graphs
• Chaos and universality in two-dimensional Ising spin glasses
• Zero and low temperature behavior of the two-dimensional $\pm J$ Ising spin glass
• Energy size effects of two-dimensional Ising spin glasses
• Universal critical behavior of the 2d Ising spin glass
• Effective critical behavior of the two-dimensional Ising spin glass with bimodal interactions
• Dynamical scaling in Ising and vector spin glasses
• Correlation Length of the Two-Dimensional Ising Spin Glass with Gaussian Interactions
• Universal dynamic scaling in three-dimensional Ising spin glasses
• Finite-size scaling in two-dimensional Ising spin glass models
• Ground-state topology of the Edwards-Anderson +/-J spin glass model
• An exact algorithm for spin correlation functions of the two dimensional +/-J Ising spin glass in the ground state
• Comment on "Evidence for nontrivial ground-state structure of 3d +/- J spin glasses"
• Quantum Vertex Model for Reversible Classical Computing

Cited by in corpus (11)

• Randomness induced spin-liquid-like phase in the spin-$1/2$ $J_1-J_2$ triangular Heisenberg model
• Scaling Advantage in Approximate Optimization with Quantum Annealing
• Dynamic scaling of topological ordering in classical systems
• Coarsening and percolation in the kinetic $2d$ Ising model with spin exchange updates and the voter model
• 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