# Symmetry Fractionalization

Quantum phases with intrinsic topological order host anyonic excitations which possess fractional statistics. In addition to that, when the system has global symmetries, the anyonic excitations can also transform in a fractional way under the symmetry action. For example, they can carry fractional charges of charge conservation symmetry or be Kramer doubles under time reversal symmetry. This is the phenomenon of symmetry fractionalization. Our work tries to understand what symmetry fractionalization patterns are possible and how they can be realized. In particular, it was realized that there exists a set of consistent looking symmetry fractionalization patterns which are actually anomalous and cannot be realized on their own. We proposed various methods to for their detection and demonstrated that they can be realized as the surface state of a system in one higher dimension.

- "Symmetry fractionalization in two dimensional topological phases", Xie Chen,
*Reviews in Physics*, 2, 3, (2017).

(A review about symmetry fractionalization in 2D.) - "Flux-Fusion Anomaly Test and Bosonic Topological Crystalline Insulators", Michael Hermele , Xie Chen,
*Phys. Rev. X***6**, 041006, (2016)(The flux fusion anomaly test and its application to 2D topological phases with spatial symmetry.) - "Anomalous Symmetry Fractionalization and Surface Topological Order", Xie Chen, Fiona J. Burnell, Ashvin Vishwanath, Lukasz Fidkowski,
*Phys. Rev. X***5**, 041013 (2015)(A general method for detecting anomalies in 2D symmetry fractionalization patterns with discrete unitary internal symmetries. The procedure is demonstrated explicitly using the projective semion model.) - "Symmetry fractionalization and anomaly detection in three-dimensional topological phases", Xie Chen, Michael Hermele,
*Phys. Rev. B*94, 195120, (2016).

(Symmetry fractionalization and anomaly detection in the simplest 3D example -- Z2 topological order with Z2 symmetry.) - "Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model", Lukasz Fidkowski, Xie Chen, Ashvin Vishwanath,
*Phys. Rev. X*3, 041016 (2013).

(Symmetry fractionalization on the surface of 3D Topological Superconductor.) - "Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator", Xie Chen, Lukasz Fidkowski, Ashvin Vishwanath,
*Phys. Rev. B*89, 165132 (2014).

(Symmetry fractionalization on the surface of 3D Topological Insulator.) - "Exactly Soluble Model of a 3D Symmetry Protected Topological Phase of Bosons with Surface Topological Order", F. J. Burnell, Xie Chen, Lukasz Fidkowski, Ashvin Vishwanath,
*Phys. Rev. B*90, 245122 (2014).

(Symmetry fractionalization on the surface of 3D bosontic Time Reversal Symmetric SPT.)