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 $\theta$ term. At an open boundary, the $\theta$ 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 $\theta >0$) 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.

• Received 1 November 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.067205