  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 infinite number of nontrivial SPT
phases with SU(2) or SO(3) symmetry. These phases can be described by
SU(2)/SO(3) nonlinear-sigma models with a quantized topological \theta-term. At
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)/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 an even-integer quantized spin Hall
conductance. Both the SU(2)/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.
