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.
Highlights
Following the theoretical prediction [1,2,3,4,5,6] and experimental discovery [7,8] of time-reversal invariant topological band insulators, it has become clear that symmetry plays a rich and varied role in topological phases of matter
We focus on bosonic systems with Z2 topological order and a symmetry group of the form G 1⁄4 Uð1Þ ⋊ G0, where G0 is an arbitrary group that may include spatial symmetries and/or time reversal
Some of the more technical aspects of our results are presented in several appendixes, and, in Appendix G, the anomaly test is applied to a few more examples of symmetry groups
Summary
Following the theoretical prediction [1,2,3,4,5,6] and experimental discovery [7,8] of time-reversal invariant topological band insulators, it has become clear that symmetry plays a rich and varied role in topological phases of matter. The dual vortex field theories obtained are convenient to work with and can be used to explore surface phase diagrams and phase transitions, which may be an interesting direction for future work While it is not the focus of this paper, our approach can be used to study internal symmetries when G 1⁄4 Uð1Þ ⋊ G0, and it is complementary to existing approaches in that case. Deciding whether the bulk SPT phase is nontrivial is equivalent to determining whether the corresponding symmetry fractionalization pattern is anomalous Under these assumptions, the flux-fusion anomaly test shows that some choices of 1⁄2ωm imply that the symmetry fractionalization pattern is anomalous. Some of the more technical aspects of our results are presented in several appendixes, and, in Appendix G, the anomaly test is applied to a few more examples of symmetry groups
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