Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases

TL;DR

The paper addresses whether the topologically ordered surface state of 3D bosonic SPT phases with symmetry $G_{1}\times G_{2}$ can be realized in purely 2D systems. It employs a four-component abelian Chern-Simons description (encoded by a $K$-matrix) and an $O(4)$ nonlinear sigma model with a $\Theta$-term to capture the surface physics, carefully distinguishing classical and quantum symmetry actions. By gauging a conventional symmetry on the surface, the authors show that the remaining symmetry becomes anomalous, as reflected in the non-invariance of the post-gauge $S$- and $T$- matrices, signaling that the surface state cannot be realized on a purely 2D lattice. They illustrate this with three concrete symmetry structures ${\mathbb Z}^{A}_{2}\times {\mathbb Z}^{B}_{2}$, ${\mathbb Z}_{2}\times {\mathbb Z}^{T}_{2}$, and ${\mathbb Z}^{A}_{2}\times {\mathbb Z}^{B}_{2}\times {\mathbb Z}^{C}_{2}$, where gauging one factor breaks the others. The results point to a surface anomaly that would be resolved only by bulk contributions in a full 3D setting, aligning with related no-go results for 2D realizations of such surface states.

Abstract

We study the Z2 topologically ordered surface state of three-dimensional bosonic SPT phases with the discrete symmetries G1 x G2. It has been argued that the topologically ordered surface state cannot be realized on a purely two-dimensional lattice model. We carefully examine the statement and show that the surface state should break G2 if the symmetry G1 is gauged. This manifests the conflict of the symmetry G1 and G2 on the surface of the three-dimensional SPT phase. Given that there is no such phenomena in the purely two-dimensional model, it signals that the symmetries are encoded anomalously on the surface of the three-dimensional SPT phases and that the surface state can never be realized on the purely two-dimensional models.