Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

TL;DR

This paper analytically characterizes all stationary points of the one-dimensional lattice Landau gauge fixing functional, linking gauge theory and statistical physics, and explores implications for phase transitions and the Neuberger problem.

Contribution

It provides an explicit analytic solution for stationary points in odd-site lattices and connects these to gauge fixing and phase transition predictions.

Findings

Analytic solutions for stationary points with odd lattice sites.

Connection between stationary points and gauge fixing partition function.

Predictions on phase transition existence and critical energies.

Abstract

We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit.