Finding Low-Temperature States with Parallel Tempering, Simulated Annealing and Simple Monte Carlo

TL;DR

This paper compares Monte Carlo methods like simulated annealing and parallel tempering for finding ground states in disordered systems, showing parallel tempering's superiority for certain models but highlighting challenges in achieving true Boltzmann sampling.

Contribution

It provides a direct comparison of these methods on 3D disordered magnetic systems and discusses their effectiveness and limitations in finding ground states and sampling degeneracies.

Findings

Parallel tempering outperforms simple Monte Carlo and simulated annealing for ground state search in DAFF.

Finding all degenerate ground states with correct probabilities is very difficult for large systems.

Equilibration becomes harder as the external field strength increases.

Abstract

Monte Carlo simulation techniques, like simulated annealing and parallel tempering, are often used to evaluate low-temperature properties and find ground states of disordered systems. Here we compare these methods using direct calculations of ground states for three-dimensional Ising diluted antiferromagnets in a field (DAFF) and three-dimensional Ising spin glasses (ISG). For the DAFF, we find that, with respect to obtaining ground states, parallel tempering is superior to simple Monte-Carlo and to simulated annealing. However, equilibration becomes more difficult with increasing magnitude of the externally applied field. For the ISG with bimodal couplings, which exhibits a high degeneracy, we conclude that finding true ground states is easy for small systems, as is already known. But finding each of the degenerate ground states with the same probability (or frequency), as required by Boltzmann statistics, is considerably harder and becomes almost impossible for larger systems.