Flux-fusion anomaly test and bosonic topological crystalline insulators

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.Comment: 18+13 pages, 4 figures. Significant changes to introduction, and other changes to improve presentation. Title shortene

Symmetry fractionalization and anomaly detection in three-dimensional topological phases

In a phase with fractional excitations, topological properties are enriched in the presence of global symmetry. In particular, fractional excitations can transform under symmetry in a fractionalized manner, resulting in different Symmetry Enriched Topological (SET) phases. While a good deal is now understood in $2D$ regarding what symmetry fractionalization patterns are possible, the situation in $3D$ is much more open. A new feature in $3D$ is the existence of loop excitations, so to study $3D$ SET phases, first we need to understand how to properly describe the fractionalized action of symmetry on loops. Using a dimensional reduction procedure, we show that these loop excitations exist as the boundary between two $2D$ SET phases, and the symmetry action is characterized by the corresponding difference in SET orders. Moreover, similar to the $2D$ case, we find that some seemingly possible symmetry fractionalization patterns are actually anomalous and cannot be realized strictly in $3D$. We detect such anomalies using the flux fusion method we introduced previously in $2D$. To illustrate these ideas, we use the $3D$ $Z_2$ gauge theory with $Z_2$ global symmetry as an example, and enumerate and describe the corresponding SET phases. In particular, we find four non-anomalous SET phases and one anomalous SET phase, which we show can be realized as the surface of a $4D$ system with symmetry protected topological order.Comment: 19 pages, 8 figure