Probing Tensor Monopoles and Gerbe Invariants in Three-Dimensional Topological Matter

TL;DR

This paper explores tensor monopoles and gerbe invariants in 3D topological matter, revealing their role in quantized electromagnetic responses and extending topological classifications beyond known frameworks.

Contribution

It introduces a universal construction of tensor Berry connections and links gerbe invariants to observable quantum phenomena in topological phases.

Findings

Tensor monopoles correspond to nontrivial bundle gerbes in 3D topological matter.

Obstructions in tensor Berry connections lead to quantized magnetoelectric effects.

Gerbe invariants can be probed via higher-dimensional charge fractionalizations.

Abstract

We show that momentum-space tensor monopoles corresponding to nontrivial vector bundle generalizations, known as bundle gerbes, can be realized in bands of three-dimensional topological matter with nontrivial Hopf invariants. We provide a universal construction of tensor Berry connections in these topological phases, demonstrating how obstructions therein lead to $\mathbb{Z}$-quantized bulk magnetoelectric and nonlinear optical phenomena. We then pinpoint that these quantum effects are supported by intraband and interband torsion leading to nontrivial Dixmier-Douady classes in most known Hopf phases and in more general topological insulators realizing gerbe invariants falling beyond the tenfold classification of topological phases of matter. We furthermore provide an interacting generalization upon introducing many-body gerbe invariants by employing twisted boundary conditions. This opens an avenue to study gerbe invariants realized through higher-dimensional charge fractionalizations that can be electromagnetically probed.