Topological terms in abelian lattice field theories

TL;DR

This paper develops a lattice discretization of the topological charge for abelian theories, enabling better numerical studies of topological effects and phenomena like the Witten effect.

Contribution

It introduces a generalized Villain action including the topological term, expressed via integer Villain variables, and applies it to simulate the 2D U(1) gauge Higgs model at non-zero theta.

Findings

Validated the topological charge properties in 2D and 4D

Analyzed the index theorem in two dimensions

Simulated the 2D U(1) gauge Higgs model at θ=π

Abstract

In this contribution we revisit the lattice discretization of the topological charge for abelian lattice field theories. The construction departs from an initially non-compact discretization of the gauge fields and after absorbing $2\pi$ shifts of the gauge fields leads to a generalized Villain action that also includes the topological term. The topological charge in two, as well as in four dimensions can be expressed in terms of only the integer-valued Villain variables. We test various properties of the topological charge and in particular analyze the index theorem in two dimensions and discuss the Witten effect in 4-d. As an application of our formulation we present results from a simulation of the 2-d U(1) gauge Higgs model at vacuum angle $\theta = \pi$, where we use a suitable worldline/worldsheet representation to overcome the complex action problem at non-zero $\theta$.