Theoretical Study on Bulk-boundary Correspondences for 3D Symmetry-protected Topological Phases

Symmetry-protected topological (SPT) phases are a class of systems with an energy gap in the bulk, just like the usual insulators, however the boundary is guaranteed to have nontrivial properties. The nontrivial boundary properties are protected by the symmetries of the system, which will be absent if the symmetries are removed.For three-dimensional (3D) SPT phases, the surface may have three different forms: (1) a gapless and fully symmetric state, (2) a state with one of the symmetries broken spontaneously, and (3) a gapped, symmetric and topologically ordered surface. Whichever it is, the non-triviality of the surface is reflected by the fact that it carries a quantum anomaly of the symmetries, i.e., the surface cannot be realized in strict 2D.In this proposal, we focus on the third type of surfaces —- topologically ordered surfaces. A topologically ordered system hosts exotic quasiparticles, named anyons. Anyons obey fractional braiding statistics and are currently under extensive search, as they can be used for fault-tolerant topological quantum computation. In the presence of symmetries, anyons can intertwine with symmetries in various ways, dubbed different symmetry-enriched topological (SET) phases. Some SET phases carry quantum anomalies of the symmetries.The purpose of this project is to carry out a comprehensive theoretical study on the correspondences between 2D anomalous SETs and 3D SPTs, namely bulk-boundary correspondences. That is, we would like to study the question of which 3D SPT can support a given 2D SET on its surface. We will emphasize the physical pictures behind these correspondences.More specifically, we will study bulk-boundary correspondences in (1) bosonic systems with both mirror-reflection and onsite unitary symmetries; (2) systems with time-reversal/ mirror-reflection and Lie group symmetries; (3) fermionic systems with discrete onsite unitary symmetries; (4) systems with general crystalline symmetries; and (5) study some extensions to the similar problem of quantum anomalies in 3D SET systems.This study is of fundamental importance to the general principles of the interplay between topology and symmetry in 3D quantum many-body systems. The bulk-boundary relations are also practically useful for the ongoing search of anyons, as the anomaly-free condition or that a system must carry certain anomaly puts very strong constraints on which topological order can occur in specific systems or materials.