Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases

Abstract

Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state(or atomic insulator) as long as certain global symmetry $G$ is unbroken. At low energies, most of two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of multi-component Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., $\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\cdots$). In addition to the usual multi-component $BF$ topological term at level-$1$, we find that there are new topological terms with quantized coefficients (e.g., $a^1\wedge a^2\wedge d a^2$ and $a^1\wedge a^2\wedge a^3\wedge a^4$) in TQFT actions, where $a^{1},a^2,\cdots$ are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as $G$ is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.