Symmetry protected topological orders and the group cohomology of their symmetry group

Details Symmetry protected topological orders and the group cohomology of their symmetry group Symmetry protected topological orders and the group cohomology of their symmetry group

2011-06-23

arxiv.org/abs/1106.4772v3

Physics

Condensed Matter

Strongly Correlated Electrons

53 pages, 41 figures, RevTeX4

Scientific paper

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.

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