Symmetry protected Spin Quantum Hall phases in 2-Dimensions

TL;DR

This paper explores 2D symmetry protected topological phases with SU(2) or SO(3) symmetry, revealing their boundary excitations and quantized spin Hall conductance, and describing them via nonlinear sigma models with topological terms.

Contribution

It characterizes infinite classes of 2D SPT phases with SU(2)/SO(3) symmetry using nonlinear sigma models and topological terms, highlighting their boundary states and quantized conductance.

Findings

SU(2) SPT phases have half-integer quantized spin Hall conductance.

SO(3) SPT phases have even-integer quantized spin Hall conductance.

Boundary excitations are decoupled gapless left movers carrying quantum numbers.

Abstract

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.