Topological mass generation in gapless systems

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.