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

الأصل · EN

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H¹⁺ᵈ[G, Uₜ(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module Uₜ(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (Gₕ, GΨ, H¹⁺ᵈ[GΨ, Uₜ(1)], where Gₕ is the symmetry group of the Hamiltonian and GΨ the symmetry group of the ground states.