Ergodicity properties of energy conserving single spin flip dynamics in the XY model

TL;DR

This paper investigates the ergodicity of energy-conserving single spin flip dynamics in the XY model, showing that phase space trajectories densely fill connected energy surface parts and identifying conditions for non-ergodic behavior.

Contribution

It provides a detailed analysis of the ergodicity properties of the XY model's dynamics, including the impact of energy surface topology and external fields on phase space connectivity.

Findings

Phase space trajectories densely fill connected energy surface regions.

Transitions between disconnected energy surface parts are not possible.

Regions of non-ergodic behavior are identified in the phase space.

Abstract

A single spin flip stochastic energy conserving dynamics for the XY model is considered. We study the ergodicity properties of the dynamics. It is shown that phase space trajectories densely fill the geometrically connected parts of the energy surface. We also show that while the dynamics is discrete and the phase point jumps around, it cannot make transitions between closed disconnected parts of the energy surface. Thus the number of distinct sectors depends on the number of geometrically disconnected parts of the energy surface. Information on the connectivity of the surfaces is obtained by studying the critical points of the energy function. We study in detail the case of two spins and find that the number of sectors can be either one or two, depending on the external fields and the energy. For a periodic lattice in $d$ dimensions, we find regions in phase space where the dynamics is non-ergodic and obtain a lower bound on the number of disconnected sectors. We provide some numerical evidence which suggests that such regions might be of small measure so that the dynamics is effectively ergodic.