A large class of symmetry-protected topological phases (SPT) in boson/spin systems have been recently predicted by the group cohomology theory. In this work, we consider bosonic SPT states at least with charge symmetry [U(1) or Z${}_N$] or spin-$S^z$ rotation symmetry [U(1) or Z${}_N$] in two (2D) and three dimensions (3D) and the surface of 3D. If both are U(1), we apply external electromagnetic field/“spin gauge field” to study the charge/spin response. For the SPT examples we consider {i.e., U${}_c$(1)$⋊$Z${}_2^T$, U${}_s$(1)$\times$Z${}_2^T$, U${}_c$(1)$\times$[U${}_s$(1)$⋊$Z${}_2$]; subscripts $c$ and $s$ are short for charge and spin; Z${}_2^T$ and Z${}_2$ are time-reversal symmetry and $\pi$ rotation about $S^y$, respectively}, many variants of Witten effect in the 3D SPT bulk and various versions of anomalous surface quantum Hall effect are defined and systematically investigated. If charge or spin symmetry reduces to Z${}_N$ by considering charge-$N$ or spin-$N$ condensate, instead of the linear response approach, we gauge the charge/spin symmetry, leading to a dynamical gauge theory with some remaining global symmetry. The 3D dynamical gauge theory describes a symmetry-enriched topological phase (SET), i.e., a topologically ordered state with global symmetry which admits nontrivial ground-state degeneracy depending on spatial manifold topology. For the SPT examples we consider, the corresponding SET states are described by dynamical topological gauge theory with topological $\mathit{BF}$ term and axionic $\Theta$ term in 3D bulk. In addition, the surface of SET is described by the chiral boson theory with quantum anomaly.

• Received 9 July 2013

DOI:https://doi.org/10.1103/PhysRevB.88.235109