Translation symmetry-enriched toric code insulator

Weak topological phases with strong electronic correlation could lead to exotic interplay between topology and translation symmetry. Here we introduce a 2D electronic insulator that possesses a toric code topological order, enriched by translation symmetry. This state can be realized from disordering a weak topological superconductor by double-vortex condensation. We term this toric code insulator, whose anyonic excitations consist of a charge-e chargon, a neutral fermion and two types of visons. In this talk, we first explain the distinction of two types of visons through the fractional Josephson effect of 1D topological superconductor. We show that these two types of visons are related by a discrete translation symmetry and have mutual semionic braiding statistics, leading to a symmetry-enrichment akin to the type in Wen's plaquette model and Kitaev's honeycomb model. We provide a microscopic construction of this state using a three-fluid coupled-wire model, from which we explore the consequences of symmetry-enrichment, as well as potential experimental realizations.