Surface Topological Order and a new 't Hooft Anomaly of Interaction Enabled 3+1D Fermion SPTs

TL;DR

This paper uncovers a new 't Hooft anomaly in 3+1D fermionic SPT phases, characterized by surface states with topological order and symmetry permutation, revealing intrinsic fermionic topological phenomena in interacting models.

Contribution

It introduces a novel 't Hooft anomaly for 3+1D fermionic SPTs, characterized by a $H^3(G, Z_2)$ class, and provides a framework unifying various anomalies in bosonic and fermionic systems.

Findings

Identifies a new 't Hooft anomaly in 3+1D fermionic SPTs.

Proposes a surface topological order described by a $Z_4$ gauge theory.

Relates the anomaly to a $H^3(G, Z_2)$ class and unifies it with other known anomalies.

Abstract

Symmetry protected topological (SPT) phases are well understood in the context of free fermions and in the context of interacting but essentially bosonic models. Recently it has been realized that intrinsically fermionic SPTs exist which only appear in interacting models. Here we show that the 3+1 dimensional realizations of these phases have surface states characterized by a new 't Hooft anomaly, captured by a $H^3(G, Z_2)$ class. This is encoded in the anomalous action of symmetry on the surface states with topological order, which must necessarily permute the anyons. We discuss in detail an example with symmetry group $G = Z_2 \times Z_4$. Using a network model of the surface we derive a candidate surface topological order given by a $Z_4$ gauge theory. We relate our findings to anomalies valued in $H^3$ with various coefficients introduced previously in both bosonic and fermionic settings, and describe a general framework that unifies these various anomalies.