Symmetry-protected quantum spin Hall phases in two dimensions.

Symmetry-protected topological (SPT) states are short-range entangled states with symmetry. Nontrivial SPT states have symmetry-protected gapless edge excitations. In 2 dimension (2D), there are an infinite number of nontrivial SPT phases with SU(2) or SO(3) symmetry. These phases can be described by SU(2) or SO(3) nonlinear-sigma models with a quantized topological θ term. At an open boundary, the θ term becomes the Wess-Zumino-Witten term and consequently the boundary excitations are decoupled gapless left movers and right movers. Only the left movers (if θ>0) carry the SU(2) or SO(3) quantum numbers. As a result, the SU(2) SPT phases have a half-integer quantized spin Hall conductance and the SO(3) SPT phases have an even-integer quantized spin Hall conductance. Both the SU(2) and SO(3) SPT phases are symmetric under their U(1) subgroup and can be viewed as U(1) SPT phases with even-integer quantized Hall conductance.