Electric-magnetic duality and $\mathbb{Z}_2$ symmetry enriched Abelian lattice gauge theory

TL;DR

This paper constructs a lattice model realizing $ ext{Z}_2$ symmetry enriched topological phases, exploring electric-magnetic duality, symmetry defects, and boundary phenomena within a categorical framework.

Contribution

It provides an explicit Hamiltonian construction of $ ext{Z}_2$ symmetry enriched Abelian lattice gauge theories and analyzes their categorical symmetries and dualities.

Findings

Explicit lattice realization of EM duality in SET phases

Analysis of symmetry defects using G-crossed UBFC

Understanding of gapped boundaries and boundary-bulk duality

Abstract

Kitaev's quantum double model is a lattice gauge theoretic realization of Dijkgraaf-Witten topological quantum field theory (TQFT), its topologically protected ground state space has broad applications for topological quantum computation and topological quantum memory. We investigate the $\mathbb{Z}_2$ symmetry enriched generalization of the model for the cyclic Abelian group in a categorical framework and present an explicit Hamiltonian construction. This model provides a lattice realization of the $\mathbb{Z}_2$ symmetry enriched topological (SET) phase. We discuss in detail the categorical symmetry of the phase, for which the electric-magnetic (EM) duality symmetry is a special case. The aspects of symmetry defects are investigated using the $G$-crossed unitary braided fusion category (UBFC). By determining the corresponding anyon condensation, the gapped boundaries and boundary-bulk duality are also investigated. Then we carefully construct the explicit lattice realization of EM duality for these SET phases.