2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons

TL;DR

This paper explores lattice regularizations of 2+1d Lifshitz theories, revealing different symmetries and dualities, and connects them to tensor gauge theories with fracton excitations, providing exact duality mappings.

Contribution

It introduces two distinct lattice models of the Lifshitz theory, analyzes their global symmetries and anomalies, and establishes their dualities with tensor gauge theories featuring fractons.

Findings

Different global symmetries and anomalies in the two lattice models

One model exhibits self-duality

Exact lattice dualities between Lifshitz, scalar, and vector charge gauge theories

Abstract

The 2+1d continuum Lifshitz theory of a free compact scalar field plays a prominent role in a variety of quantum systems in condensed matter physics and high energy physics. It is known that in compact space, it has an infinite ground state degeneracy. In order to understand this theory better, we consider two candidate lattice regularizations of it using the modified Villain formalism. We show that these two lattice theories have significantly different global symmetries (including a dipole global symmetry), anomalies, ground state degeneracies, and dualities. In particular, one of them is self-dual. Given these theories and their global symmetries, we can couple them to corresponding gauge theories. These are two different $U(1)$ tensor gauge theories. The resulting models have excitations with restricted mobility, i.e., fractons. Finally, we give an exact lattice realization of the fracton/lineon-elasticity dualities for the Lifshitz theory, scalar and vector charge gauge theories.