1106.4772 (Xie Chen et al.)

Symmetry protected topological orders and the group cohomology of their
symmetry group    [PDF]

Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen

Symmetry protected topological (SPT) phases are gapped short-range-entangled
quantum phases with a symmetry G, which can all be smoothly connected to the
same trivial product state if we break the symmetry. In this paper, we propose
that distinct d-dimensional bosonic SPT phases with on-site symmetry G (which
may contain anti-unitary time reversal symmetry) are classified/labeled by
elements in H^{1+d}[G,U_T(1)] -- the (1+d)-Borel-cohomology group of G over the
G-module U_T(1). Our construction is based on a new type of topological term
that generalizes the topological theta-term in continuous non-linear
sigma-model to discrete non-linear sigma-models. The boundary excitations of
the non-trivial SPT phases are described by continuous/discrete non-linear
sigma-models with a non-local Lagrangian term that generalizes the
Wess-Zumino-Witten term for continuous non-linear sigma-models. We argue that
those boundary excitations are gapless, if the symmetry is not broken on the
boundary. As an application of our classification, we find that interacting
bosonic topological insulators (with time reversal and U(1) symmetry) are
classified by H^{1+d}[U(1) x| Z_2^T,U_T(1)], which contain one non-trivial
phases in 1D or 2D, and three in 3D. We also classified interacting bosonic
topological superconductors (with time reversal symmetry only), in term of
H^{1+d}[Z_2^T,U_T(1)], which contain one non-trivial phase in odd spatial
dimensions and non for even. Our classification is much more general than the
above two examples, since it is for any symmetry group. For example, the SPT
phases of integer spin systems with time reversal and U(1) symmetry are
classified by H^{1+d}[U(1) x Z_2^T,U_T(1)], which contain three non-trivial SPT
phases in 1D, non in 2D, and seven in 3D.

View original: http://arxiv.org/abs/1106.4772