Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases

TL;DR

This paper establishes a bulk-boundary correspondence for 3D symmetry-protected topological phases, linking bulk and surface properties via symmetry gauging and braiding statistics, providing a potentially complete classification scheme.

Contribution

It introduces a novel bulk-boundary correspondence for 3D bosonic SPT phases with finite Abelian symmetries, using symmetry gauging and braiding statistics to distinguish phases.

Findings

Bulk data uniquely identifies each 3D SPT phase.

Surface data distinguishes each gapped, symmetric surface.

Applicable to surfaces with Abelian anyons not permuted by symmetries.

Abstract

We derive a bulk-boundary correspondence for three-dimensional (3D) symmetry-protected topological (SPT) phases with unitary symmetries. The correspondence consists of three equations that relate bulk properties of these phases to properties of their gapped, symmetry-preserving surfaces. Both the bulk and surface data appearing in our correspondence are defined via a procedure in which we gauge the symmetries of the system of interest and then study the braiding statistics of excitations of the resulting gauge theory. The bulk data is defined in terms of the statistics of bulk excitations, while the surface data is defined in terms of the statistics of surface excitations. An appealing property of this data is that it is plausibly complete in the sense that the bulk data uniquely distinguishes each 3D SPT phase, while the surface data uniquely distinguishes each gapped, symmetric surface. Our correspondence applies to any 3D bosonic SPT phase with finite Abelian unitary symmetry group. It applies to any surface that (1) supports only Abelian anyons and (2) has the property that the anyons are not permuted by the symmetries.