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 topological order and a symmetry group of the form , where is an arbitrary group that may include spatial symmetries and/or time reversal. The anomalous fractionalization patterns we identify cannot occur in strictly systems but can occur at surfaces of symmetry-protected topological (SPT) phases. This observation leads to examples of bosonic topological crystalline insulators (TCIs) that, to our knowledge, have not previously been identified. In some cases, these 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 symmetry on the fluxes. For some anomalies, this can be described in terms of dimensional reduction to SPT phases. We apply our method to several different symmetry groups with nontrivial anomalies, including and , where and are time-reversal and reflection symmetry, respectively.
- Received 13 August 2015
DOI:https://doi.org/10.1103/PhysRevX.6.041006
This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
One way to identify a new quantum phase of matter is to show that another quantum phase of matter is impossible. There are theories naively describing a -dimensional phase of matter that cannot be realized in a -dimensional system but can be realized at the surface of a ()-dimensional system. Such theories are referred to as anomalous, and proving a theory to be anomalous provides a route to identifying previously unknown quantum phases of matter. Here, we develop an approach to show that certain two-dimensional theories are anomalous, and we identify new phases of matter in three dimensions.
There are a number of fascinating quantum phases of matter whose edge or surface states are known to be anomalous. One example is the quantum Hall effect, which occurs in two-dimensional electron systems. The anomalous one-dimensional edge includes electrons moving only in one direction, as if going down a one-way street.
So-called topological crystalline insulators are similar, except that the symmetry of rigid motions of the crystal lattice plays a key role. Here, by showing that certain two-dimensional theories are anomalous, we theoretically identify several examples of topological crystalline insulators of bosons in three dimensions.
We expect that our findings will pave the way for the discovery of new topological crystalline insulators.