Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases

TL;DR

This paper develops a systematic method to construct gapped interfaces and boundaries for fermionic symmetry-protected topological states by extending their symmetry groups, with explicit formulas and examples in 2+1D and 3+1D.

Contribution

It introduces a pullback trivialization technique for constructing FSPT interfaces and boundaries, including explicit consistency formulas and a detailed example in 3+1D.

Findings

Derived general consistency formulas for (2+1)D and (3+1)D systems.

Presented explicit example of a 3+1D FSPT with Majorana chain decorations.

Developed a method involving symmetry group extension and gauge transformations.

Abstract

We use the pullback trivialization technique to systematically construct gapped interfaces and anomalous boundaries for fermionic symmetry-protected topological (FSPT) states by extending their symmetry group $G_f = \mathbb{Z}_2^f \times_{\omega_2} G$ to larger groups. These FSPT states may involve decoration layers of both Majorana chains and complex fermions. We derive general consistency formulas explicitly for (2+1)D and (3+1)D systems, where nontrivial twists arise from fermionic symmetric local unitaries or "gauge transformations" that ensure coboundaries vanish at the cochain level. Additionally, we present explicit example for a (3+1)D FSPT of symmetry group $G_f=\mathbb{Z}_2^f \times \mathbb{Z}_4 \times \mathbb{Z}_4$ with Majorana chain decorations.