Irreversible Markov chains in spin models: Topological excitations

TL;DR

This paper investigates how the event-chain Monte Carlo algorithm performs in spin models with topological excitations, revealing slow vortex-antivortex correlation decay and varying dynamical exponents across temperatures.

Contribution

It provides a detailed analysis of the impact of topological excitations on the convergence and dynamics of the event-chain Monte Carlo algorithm in spin models.

Findings

Vortex-antivortex correlations decay slowly at critical temperatures.

Spin waves relax quickly under the event-chain algorithm.

Mixing times are significantly larger than equilibrium correlation times at low temperatures.

Abstract

We analyze the convergence of the irreversible event-chain Monte Carlo algorithm for continuous spin models in the presence of topological excitations. In the two-dimensional XY model, we show that the local nature of the Markov-chain dynamics leads to slow decay of vortex-antivortex correlations while spin waves decorrelate very quickly. Using a Frechet description of the maximum vortex-antivortex distance, we quantify the contributions of topological excitations to the equilibrium correlations, and show that they vary from a dynamical critical exponent z \sim 2 at the critical temperature to z \sim 0 in the limit of zero temperature. We confirm the event-chain algorithm's fast relaxation (corresponding to z = 0) of spin waves in the harmonic approximation to the XY model. Mixing times (describing the approach towards equilibrium from the least favorable initial state) however remain much larger than equilibrium correlation times at low temperatures. We also describe the respective influence of topological monopole-antimonopole excitations and of spin waves on the event-chain dynamics in the three-dimensional Heisenberg model.