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

TL;DR

This paper demonstrates that interacting bosonic symmetry protected topological phases can be systematically classified using group cohomology theory, linking physical phases to mathematical cohomology classes.

Contribution

It establishes a group cohomology framework for classifying interacting bosonic SPT phases with on-site symmetry, including anti-unitary symmetries.

Findings

Bosonic SPT phases are labeled by elements in H^{1+d}[G, U_T(1)].

Boundary excitations are gapless or degenerate.

Classification extends to symmetry breaking phases with specific group data.

Abstract

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 H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(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: (G_H, G_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.