Fractonic matter in symmetry-enriched $U\left(1\right)$ gauge theory

In this work we explore the interplay between global symmetry and the mobility of quasiparticle excitations. We show that fractonic matter naturally appears in a three-dimensional $U\left(1\right)$ gauge theory, enriched by global $U\left(1\right)$ and translational symmetries, via the mechanism of anyonic spin-orbit coupling. We develop a systematic understanding of such symmetry-enforced mobility restrictions in terms of the classification of $U\left(1\right)$ gauge theories enriched by $U\left(1\right)$ and translational symmetries. We provide a unified construction of these phases by gauging layered symmetry-protected topological phases.

Figure 1

(a) Gauging $S_{\widehat{x}}$ and $S_{\widehat{y}}$ symmetry in a ${\mathbb{Z}}_N$ toric code model. Solid circles are original spins. Squares are gauge field spins, inside which the two dots represent gauge fields for $S_{\widehat{x}}$ and $S_{\widehat{y}}$ symmetries, respectively. (b) $e$ charge configuration created by Eq. (9). (c) $e$ charge configuration created by Eq. (10). In (b) and (c) the numbers denote the corresponding ${\mathbb{Z}}_N$ gauge charge.

Figure 2

Examples of the definition of $S_{\mu \nu }$ and $S_{\mu }$. The unit cell of the diamond lattice is depicted in (a) and (b). The operator $S_{\mu \nu }$ measures the electric field lines that go through the $\mu \nu$ direction. (a) As an example $S_{12}$ is shown. (b) An example $S_{\mu }$ operator which measures the electric dipole moment of the system along the ${\mathbf{e}}_{\mu }$ direction.

Figure 3

A monopole-antimonopole pair in a layered SPT phase. A Dirac string penetrating layers of 2D SPT states is illustrated.