Surface field theories of point group symmetry protected topological phases

TL;DR

This paper identifies and analyzes surface field theories for three-dimensional bosonic point group symmetry protected topological phases, revealing their anomalous nature and potential for describing gapped surfaces with topological order.

Contribution

The paper introduces three specific surface field theories for bosonic pgSPT phases, connecting them to symmetry protection and topological order, and extends their applicability to general point group symmetries.

Findings

QED3 with four fermion flavors describes many mirror-protected bosonic pgSPT surfaces.

A variant of QED3 with charge-1 and charge-3 fermions models E8 state surfaces.

An O(4) nonlinear sigma model with a topological theta-term describes U(1)×Z2^P symmetric bosonic pgSPT surfaces.

Abstract

We identify field theories that describe the surfaces of three-dimensional bosonic point group symmetry protected topological (pgSPT) phases. The anomalous nature of the surface field theories is revealed via a dimensional reduction argument. Specifically, we study three different surface field theories. The first field theory is quantum electrodynamics in three space-time dimensions (QED3) with four flavors of fermions. We show this theory can describe the surfaces of a majority of bosonic pgSPT phases protected by a single mirror reflection, or by $C_{nv}$ point group symmetry for $n=2,3,4,6$. The second field theory is a variant of QED3 with charge-1 and charge-3 Dirac fermions. This field theory can describe the surface of a reflection symmetric pgSPT phase built by placing an $E_{8}$ state on the mirror plane. The third field theory is an ${\rm O}(4)$ non-linear sigma model with a topological theta-term at $\theta=\pi$, or, equivalently, a non-compact ${\rm CP}^1$ model. Using a coupled wire construction, we show this is a surface theory for bosonic pgSPT phases with ${\rm U}(1) \times \mathbb{Z}_{2}^{P}$ symmetry. For the latter two field theories, we discuss the connection to gapped surfaces with topological order. Moreover, we conjecture that the latter two field theories can describe surfaces of more general bosonic pgSPT phases with $C_{nv}$ point group symmetry.