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

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.

8 pages, 10 figures. Published in Journal of Chemical Physics

References in corpus (10)

• Some Further Results for the Stationary Points and Dynamics of Supercooled Liquids
• Phase transitions and configuration space topology
• Phase transitions and topology changes in configuration space
• Enumerating Gribov copies on the lattice
• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian
• Phase transitions induced by saddle points of vanishing curvature
• Exploring the energy landscape of XY models
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Enumerating Copies in the First Gribov Region on the Lattice in up to four Dimensions
• Stationary point approach to the phase transition of the classical XY chain with power-law interactions

Cited by in corpus (2)

• Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
• Gauge-fixing on the Lattice via Orbifolding