Topological mass generation in gapless systems

TL;DR

This paper explores how topological mass generation occurs in gapless systems, specifically in axion electrodynamics, revealing spontaneous higher-form symmetry breaking without topological order and deriving effective theories consistent with anomalies.

Contribution

It demonstrates topological mass generation in gapless systems, linking gapped and gapless modes via dimensional reduction and identifying symmetry breaking without topological order.

Findings

Gapped modes relate to lower-dimensional gapped systems via dimensional reduction

Spontaneous breaking of higher-form $ ext{Z}_k$ symmetry occurs without topological order

Effective theory of gapless mode satisfies chiral anomaly matching

Abstract

Mass generation of gauge fields can be universally described by topological couplings in gapped systems, such as the Abelian Higgs model in $(3+1)$ dimensions and the Maxwell-Chern-Simons theory in $(2+1)$ dimensions. These systems also exhibit the spontaneous breaking of higher-form $\mathbb{Z}_k$ symmetries and topological orders for level $k \geq 2$. In this paper, we consider topological mass generation in gapless systems. As a paradigmatic example, we study the axion electrodynamics with level $k$ in $(3+1)$ dimensions in background fields that hosts both gapped and gapless modes. We argue that the gapped mode is related to those in fully gapped systems in lower dimensions via dimensional reduction. We show that this system exhibits the spontaneous breaking of a higher-form $\mathbb{Z}_k$ symmetry despite the absence of the conventional topological order. In the case of the background magnetic field, we also derive the low-energy effective theory of the gapless mode with the quadratic dispersion relation and show that it satisfies the chiral anomaly matching.