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

TL;DR

The paper develops a comprehensive framework for symmetry-protected topological (SPT) phases with charge and spin symmetries in 2D, 3D, and on 3D surfaces. It juxtaposes a response-theory approach, using non-dynamical U(1) gauge fields and a Theta-matrix to classify and quantify charge, spin, and cross responses (Witten, quantum Hall, and mutual variants), with a dynamical gauge-theory approach that gauges discrete subgroups to obtain symmetry-enriched topological (SET) phases described by BF terms and axionic Theta terms. It demonstrates how bulk SPTs induce anomalous surface theories that require an extra dimension for realization, and it provides explicit constructions and quantization rules for various symmetry groups (U(1), Z_N, and mixed cases), including concrete K_G-matrix descriptions. The work connects SPTs to SET via gauging, discusses surface chiral boson theories that cancel bulk anomalies, and outlines potential numerical and lattice implementations. Overall, it offers a systematic, cohomology-informed route to understanding electromagnetic and spin responses, Witten effects, and the bulk–boundary correspondence for 3D SPTs and their 2D surfaces.

Abstract

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 $π$-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 $Θ$-term in 3D bulk. And the surface of SET is described by the chiral boson theory with quantum anomaly.

Figures (1)

• Figure 1: (Color online). Illustration of the experimental setup to realize the mutual-Witten effect. $x$-, $y$-, and $z$-axises form the three spatial directions. In both (a) and (b), a ferromagnetic thin film (FM) is located between a trivial U$_c$(1)$\times$[U$_s$(1)$\rtimes$Z$_2$] state with $\theta_0=0$ and the nontrivial state with $\theta_0=\pi$. The width of the film is sufficiently small. In (a), the red and blue balls stand for a spin impurity and an image magnetic monopole of electromagnetic $A^c_\mu$ gauge field, respectively. The solid blue lines in the region $(z_2, z_4)\bigcup(z_4,\infty)$ represent the magnetic field $\mathbf{B}^c$ induced by the spin impurity. In (b), the blue and red balls stand for an electric charge impurity and an image magnetic monopole of $A^s_\mu$ gauge field (i.e., the "spin gauge field"), respectively. The solid red lines in the region $(z_2,z_4)$ represent the magnetic field $\mathbf{B}^s$ induced by the electric charge impurity (see texts).