Flux-fusion anomaly test and bosonic topological crystalline insulators

TL;DR

The paper introduces the flux-fusion anomaly test to identify anomalous symmetry fractionalization in 2D SET phases, revealing new 3D bosonic topological crystalline insulators with unique surface phenomena.

Contribution

It presents a novel flux-fusion anomaly test method to detect anomalies in symmetry fractionalization, leading to the discovery of previously unknown 3D bosonic TCIs with exotic surface states.

Findings

Identified anomalous fractionalization patterns in bosonic Z2 topological order.

Discovered new 3D bosonic topological crystalline insulators.

Showed some anomalies relate to dimensional reduction to 1D SPT phases.

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 symmetry group of the form G = U(1) $\rtimes$ 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 non-trivial 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 non-trivial anomalies, including G = U(1) X Z2T and G = U(1) X Z2P, where Z2T and Z2P are time-reversal and d=2 reflection symmetry, respectively.