Projected single-spin flip dynamics in the Ising Model

TL;DR

This paper analyzes projected single-spin flip dynamics in the Ising Model, evaluating relaxation times and tunneling times across various ensembles, and introduces an efficient Monte Carlo method for comparing different local spin-flip dynamics.

Contribution

It introduces a method to efficiently evaluate and compare the performance of arbitrary single-spin flip dynamics using projected transition matrices and Monte Carlo estimates.

Findings

Projection in magnetization space accurately approximates full dynamics.

Largest relaxation times scale with system size as shown by eigenvalue analysis.

Finite-size scaling exponents for tunneling times are computed and compared.

Abstract

We study transition matrices for projected dynamics in the energy-magnetization space, magnetization space and energy space. Several single spin flip dynamics are considered such as the Glauber and Metropolis canonical ensemble dynamics and the Metropolis dynamics for three multicanonical ensembles: the flat energy-magnetization histogram, the flat energy histogram and the flat magnetization histogram. From the numerical diagonalization of the matrices for the projected dynamics we obtain the sub-dominant eigenvalue and the largest relaxation times for systems of varying size. Although, the projected dynamics is an approximation to the full state space dynamics comparison with some available results, obtained by other authors, shows that projection in the magnetization space is a reasonably accurate method to study the scaling of relaxation times with system size. The transition matrices for arbitrary single-spin flip dynamics are obtained from a single Monte-Carlo estimate of the infinite temperature transition-matrix, for each system size, which makes the method an efficient tool to evaluate the relative performance of any arbitrary local spin-flip dynamics. We also present new results for appropriately defined average tunnelling times of magnetization and compute their finite-size scaling exponents that we compare with results of energy tunnelling exponents available for the flat energy histogram multicanonical ensemble.