The Higgs Mechanism in Higher-Rank Symmetric $U(1)$ Gauge Theories

TL;DR

This paper explores how the Higgs mechanism applies to higher-rank symmetric $U(1)$ gauge theories, revealing connections to fracton phases and novel topological orders, with detailed models and phase transition analysis.

Contribution

It introduces two classes of higher-rank $U(1)$ gauge theories, analyzes their Higgs phases, and establishes links to fracton order and conventional topological phases.

Findings

Some scalar charge theories have X-cube fracton order as Higgs phase.

Not all higher-rank theories produce fractonic phases; some yield conventional topological order.

Identifies a possible direct phase transition between multiple $ ext{Z}_2$ gauge theories and fracton order.

Abstract

We use the Higgs mechanism to investigate connections between higher-rank symmetric $U(1)$ gauge theories and gapped fracton phases. We define two classes of rank-2 symmetric $U(1)$ gauge theories: the $(m,n)$ scalar and vector charge theories, for integer $m$ and $n$, which respect the symmetry of the square (cubic) lattice in two (three) spatial dimensions. We further provide local lattice rotor models whose low energy dynamics are described by these theories. We then describe in detail the Higgs phases obtained when the $U(1)$ gauge symmetry is spontaneously broken to a discrete subgroup. A subset of the scalar charge theories indeed have X-cube fracton order as their Higgs phase, although we find that this can only occur if the continuum higher rank gauge theory breaks continuous spatial rotational symmetry. However, not all higher rank gauge theories have fractonic Higgs phases; other Higgs phases possess conventional topological order. Nevertheless, they yield interesting novel exactly solvable models of conventional topological order, somewhat reminiscent of the color code models in both two and three spatial dimensions. We also investigate phase transitions in these models and find a possible direct phase transition between four copies of $\mathbb{Z}_2$ gauge theory in three spatial dimensions and X-cube fracton order.