Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points

TL;DR

This study explores the energy landscape of the 2D XY model by numerically identifying stationary points of various indices, analyzing their distribution and scaling behavior to test existing conjectures about such landscapes.

Contribution

It provides a comprehensive numerical analysis of stationary points in the 2D XY model, including their energies and indices, and examines how these properties scale with system size.

Findings

Number of stationary points grows with system size

Energy and index distributions follow specific scaling laws

Results support or challenge existing theoretical conjectures

Abstract

The application of numerical techniques to the study of energy landscapes of large systems relies on sufficient sampling of the stationary points. Since the number of stationary points is believed to grow exponentially with system size, we can only sample a small fraction. We investigate the interplay between this restricted sample size and the physical features of the potential energy landscape for the two-dimensional $XY$ model in the absence of disorder with up to $N=100$ spins. Using an eigenvector-following technique, we numerically compute stationary points with a given Hessian index $I$ for all possible values of $I$. We investigate the number of stationary points, their energy and index distributions, and other related quantities, with particular focus on the scaling with $N$. The results are used to test a number of conjectures and approximate analytic results for the general properties of energy landscapes.