Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulators

TL;DR

This paper introduces a universal physical mechanism called vortex-line condensation in three dimensions to realize all known bosonic topological insulator root states, providing a unified bulk TQFT description and explicit surface states.

Contribution

It proposes vortex-line condensation as a physical mechanism to achieve all three root states of 3D bosonic topological insulators, extending to Z_N symmetries and beyond group cohomology.

Findings

Bulk TQFT descriptions for each root state are derived.

Surface topologically ordered states are explicitly constructed.

The mechanism generalizes to Z_N symmetries and beyond group cohomology.

Abstract

Bosonic topological insulators (BTI) in three dimensions are symmetry-protected topological phases (SPT) protected by time-reversal and boson number conservation {symmetries}. BTI in three dimensions were first proposed and classified by the group cohomology theory which suggests two distinct root states, each carrying a $\mathbb{Z}_2$ index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via \textit{vortex-line condensation} {from} a 3d superfluid to achieve all {three} root states. It naturally produces bulk topological quantum field theory (TQFT) description for each root state. Topologically ordered states on the surface are \textit{rigorously} derived by placing TQFT on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to $Z_N$ symmetries and discuss potential SPT phases beyond the group cohomology classification.