Abstract
In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways , leading to a variety of symmetry enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain `anomalous' SETs can only occur on the surface of a 3D symmetry protected topological (SPT) phase. In this paper we describe a procedure for determining whether an SET of a discrete, onsite, unitary symmetry group $G$ is anomalous or not. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes. Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions are captured by the fourth cohomology group $H^4( G, \,U(1))$, which also precisely labels the set of 3D SPT phases, with symmetry group $G$. We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3d bulk SPT whose surface termination realizes it. We illustrate this idea using the chiral spin liquid ($U(1)_2$) topological order with a reduced symmetry $\mathbb{Z}_2 \times \mathbb{Z}_2 \subset SO(3)$, which can act on the semion quasiparticle in an anomalous way. We construct exactly solved 3d SPT models realizing the anomalous surface terminations, and demonstrate that they are non-trivial by computing three loop braiding statistics. Possible extensions to anti-unitary symmetries are also discussed.
Highlights
It has been realized that gapped phases can be distinguished on the basis of symmetry even when that symmetry is unbroken
We focus on symmetry-enriched topological” (SET) with unitary discrete symmetries and discuss a general way to detect anomalies in them
For the sake of illustration, we first consider the case of the nonanomalous chiral spin liquid surface, corresponding to a trivial symmetry-protected topological (SPT), in which each auxiliary degree of freedom can be either a spin-1=2 or a spinless particle b. (We describe how to modify this construction to obtain a projective semion model at the end of this section.) We further impose the constraint that, within this enlarged Hilbert space, only states with an even number of spin1=2’s at each vertex are allowed
Summary
It has been realized that gapped phases can be distinguished on the basis of symmetry even when that symmetry is unbroken. The anomalous projective semion theories represent only the simplest examples of anomalous SETs, but the method discussed in this paper is generally applicable to any topological order and discrete unitary on-site global symmetry. The mathematics underlying this method of anomaly detection, developed by Etingof et al [23], studies the problem of G extensions of fusion categories.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
