1301.7675 (Xiao-Gang Wen)

Topological invariants of symmetry-protected and symmetry-enriched
topological phases of interacting bosons or fermions    [PDF]

Xiao-Gang Wen

Recently, it was realized that quantum states of matter can be classified as long-range entangled (LRE) states (ie the topologically ordered states) and short-range entangled (SRE) states. The SRE states with a symmetry SG [named as symmetry-protected topological (SPT) states] are shown to be classified by group cohomology class H^d(SG,R/Z) in d-dimensional space-time. The LRE states with a symmetry SG are named as symmetry-enriched topological (SET) states. One class of SET states are described by weak-coupling gauge theories with gauge group GG and quantized topological terms. Those SET states (ie the quantized topological terms) are classified H^d(PSG, R/Z) in d space-time dimensions, where the projective symmetry group PSG is an extension of SG by GG: SG=PSG/GG. In this paper, we study the physical properties of those SPT/SET states, such as the fractionalization of the quantum numbers of the global symmetry. Those physical properties are topological invariants of the SPT/SET states that allow us to experimentally or numerically detect those SPT/SET states, ie to measure the elements in H^d(PSG, R/Z) that label different SPT/SET states.

View original: http://arxiv.org/abs/1301.7675