Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions

TL;DR

This paper identifies quantum anomalies on the surfaces of (3+1)D fermionic symmetry-protected topological phases, revealing their topological classifications and how interactions reduce these classifications from infinite to finite groups.

Contribution

It uncovers specific quantum anomalies in fermionic SPT phases with CP and reflection symmetries, linking surface anomalies to bulk topological classifications and their collapse under interactions.

Findings

CP-projected partition function exhibits a $Z_2$ anomaly.

Surface partition functions are non-invariant under large gauge/diffeomorphisms.

Interaction effects reduce classification from $Z$ to $Z_{16}$.

Abstract

Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the non-interacting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)-dimensional fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\times$ reflection) and electromagnetic $\mathrm{U}(1)$ symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large $\mathrm{U}(1)$ gauge transformations, but picks up an anomalous sign, signaling a $\mathbb{Z}_2$ topological classification. Similarly, for the second example, which is related to, by CPT-theorem, time-reversal symmetric topological superconductors, we discuss the invariance/non-invariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (3d modular transformations). The connection to the collapse of the non-interacting classification by an integer ($\mathbb{Z}$) to $\mathbb{Z}_{16}$, in the presence of interactions, is discussed.