Exploring the energy landscape of XY models

TL;DR

This paper analytically studies the energy landscape of 2D and 3D XY models, revealing complex stationary points that impact understanding of phase transitions.

Contribution

It constructs and analyzes classes of stationary points in XY models, highlighting their complexity and implications for phase transition analysis.

Findings

Existence of exponentially many singular stationary points

Stationary points form continuous one-parameter families

Implications for theoretical analysis of phase transitions

Abstract

We investigate the energy landscape of two- and three-dimensional XY models with nearest-neighbor interactions by analytically constructing several classes of stationary points of the Hamiltonian. These classes are analyzed, in particular with respect to possible signatures of the thermodynamic phase transitions of the models. We find that, even after explicitly breaking the global O(2) symmetry of the XY spins, an exponentially large class of stationary points are singular and occur in continuous one-parameter families. This property may complicate the use of theoretical tools developed for the investigation of phase transitions based on stationary points of the energy landscape, and we discuss strategies to avoid these difficulties.