Bosonic topological crystalline insulators and anomalous symmetry fractionalization via the flux-fusion anomaly test

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)oG′, 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 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) × Z2 and G = U(1) × Z2 , where Z2 and Z2 are time-reversal and d = 2 reflection symmetry, respectively.