Title

Flux-Fusion Anomaly Test and Bosonic Topological Crystalline Insulators

Document Type

Article

Publication Date

10-13-2016

Publication Title

Physical Review X

ISSN

2160-3308

Volume

6

Issue

4

First Page

041006

DOI

https://doi.org/10.1103/PhysRevX.6.041006

Abstract

We introduce a method, dubbed the flux-fusion anomaly test, to detect certain anomalous symmetry fractionalization patterns in two-dimensional symmetry-enriched topological (SET) phases. We focus on bosonic systems with ℤ2 topological order and a symmetry group of the form G=U(1)⋊G′, where G′ is an arbitrary group that may include spatial symmetries and/or time reversal. The anomalous fractionalization patterns we identify cannot occur in strictly d=2 systems but can occur at surfaces of d=3 symmetry-protected topological (SPT) phases. This observation leads to examples of d=3 bosonic topological crystalline insulators (TCIs) that, to our knowledge, have not previously been identified. In some cases, these d=3 bosonic TCIs can have an anomalous superfluid at the surface, which is characterized by nontrivial projective transformations of the superfluid vortices under symmetry. The basic idea of our anomaly test is to introduce fluxes of the U(1) symmetry and to show that some fractionalization patterns cannot be extended to a consistent action of G′ symmetry on the fluxes. For some anomalies, this can be described in terms of dimensional reduction to d=1 SPT phases. We apply our method to several different symmetry groups with nontrivial anomalies, including G=U(1)×ℤT2 and G=U(1)×ℤP2, where ℤT2 and ℤP2 are time-reversal and d=2 reflection symmetry, respectively.