Symmetry-protected topological phases with charge and spin symmetries: response theory and dynamical gauge theory in 2D, 3D and the surface of 3D

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 SPT states at least with charge symmetry (U(1) or Z$_N$) or spin $S^z$ rotation symmetry (U(1) or Z$_N$) in 2D, 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)$\rtimes$Z$^T_2$, U$_s$(1)$\times$Z$^T_2$, U$_c$(1)$\times$[U$_s$(1)$\rtimes$Z$_2$]; subscripts $c$ and $s$ are short for charge and spin; Z$^T_2$ 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 BF term and axionic $\Theta$-term in 3D bulk. And the surface of SET is described by the chiral boson theory with quantum anomaly.Comment: 23 pages, 1 figure, REVTeX; Table II and Table III for summary of part of key result

Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries

We propose a general approach to construct symmetry protected topological (SPT) states i.e the short-range entangled states with symmetry) in 2D spin/boson systems on lattice. In our approach, we fractionalize spins/bosons into different fermions, which occupy nontrivial Chern bands. After the Gutzwiller projection of the free fermion state obtained by filling the Chern bands, we can obtain SPT states on lattice. In particular, we constructed a U(1) SPT state of a spin-1 model, a SO(3) SPT state of a boson system with spin-1 bosons and spinless bosons, and a SU(2) SPT state of a spin-1/2 boson system. By applying the "spin gauge field" which directly couples to the spin density and spin current of $S^z$ components, we also calculate the quantum spin Hall conductance in each SPT state. The projective ground states can be further studied numerically in the future by variational Monte Carlo etc.Comment: 7+ pages, accepted by Phys. Rev.

Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulators

Bosonic topological insulators (BTI) in three dimensions are symmetry-protected topological phases (SPT) protected by time-reversal and boson number conservation {symmetries}. BTI in three dimensions were first proposed and classified by the group cohomology theory which suggests two distinct root states, each carrying a $\mathbb{Z}_2$ index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTI, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via \textit{vortex-line condensation} {from} a 3d superfluid to achieve all {three} root states. It naturally produces bulk topological quantum field theory (TQFT) description for each root state. Topologically ordered states on the surface are \textit{rigorously} derived by placing TQFT on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to $Z_N$ symmetries and discuss potential SPT phases beyond the group cohomology classification.Comment: ReVTeX 4.1 (published version