Algorithmic overlaps as thermodynamic variables: from local to cluster Monte Carlo dynamics in critical phenomena

TL;DR

This paper explores how overlaps of spin configurations in Monte Carlo algorithms reflect critical phenomena and phase transitions in models like Ising and Potts, linking geometric overlaps to thermodynamic behavior.

Contribution

It demonstrates that cluster overlaps serve as order parameters for Wolff and Swendsen-Wang algorithms, revealing critical behavior related to geometric cluster properties.

Findings

Wolff cluster overlap reflects critical behavior as an order parameter.

Swendsen-Wang configuration overlap varies with critical dynamics.

Metropolis algorithm's dynamics are governed by spin flip frequency, not overlaps.

Abstract

We investigate the spatial overlap of successive spin configurations in Markov chain Monte Carlo simulations using the local Metropolis algorithm and the Swendsen-Wang and Wolff cluster algorithms. We examine the dynamics of these algorithms for two models in different universality classes: the Ising model and the Potts model with three components. The overlap of two successive Wolff clusters reflects critical behavior and can be used as an order parameter for the algorithm's dynamics. In the case of the Swendsen-Wang algorithm, similar behavior is demonstrated by the variation in the overlap of two consecutive lattice configurations, which behaves like order parameter. Nothing similar is observed for the Metropolis algorithm, and the dynamics in the critical region are determined by the spin flip frequency, which is equivalent to the acceptance rate. Thus, the critical behavior of Wolff cluster overlap and the variation of configuration overlap in the Swendsen-Wang algorithm are naturally related to the critical behavior of geometric objects - Fortuin-Kastelein clusters. Interestingly, in all cases, the geometric quantity - configuration overlap or its variation - reflects the thermodynamics of the phase transition.