Replica Monte Carlo Simulation (Revisited)

In 1986, Swendsen and Wang proposed a replica Monte Carlo algorithm for spin glasses [Phys. Rev. Lett. 57 (1986) 2607]. Two important ingredients are present, (1) the use of a collection of systems (replicas) at different of temperatures, but with the same random couplings, (2) defining and flipping clusters. Exchange of information between the systems is facilitated by fixing the τ spin (τ = σ 1 σ 2) and flipping the two neighboring systems simultaneously. In this talk, we discuss this algorithm and its relationship to replica exchange (also known as parallel tempering) and Houdayer’s cluster algorithm for spin glasses. We review some of the early results obtained using this algorithm. We also present new results for the correlation times of replica Monte Carlo dynamics in two and three dimensions and compare them with replica exchange. The spin glass problem has been studied extensively over the past 30 years. 1)–3) Monte Carlo simulation has been one of the main tools. However, spin glass models are among the most difficult to simulate, due to their extremely slow dynamics at low temperatures. Unlike the ferromagnetic models, 4) cluster algorithms for spin