From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory

TL;DR

This paper reviews recent advances in topological band theory, focusing on quantum geometry, delicate and multigap topology, and bundle gerbes, revealing new physical phenomena and guiding future research in condensed matter physics.

Contribution

It synthesizes three emerging research threads—quantum geometric tensor, delicate/multigap topology, and bundle gerbes—highlighting their interconnected roles in topological band theory.

Findings

Quantum geometric tensor components can be accessed via optical probes.

Delicate and multigap topologies have unique physical signatures.

Bundle gerbes capture higher-form topological features of energy bands.

Abstract

Topological principles constitute at present an integral component of condensed matter physics, permeating the modern characterization of electronic states while also guiding materials design. In this brief Perspective, I highlight three research threads in single-particle topological band theory that have recently gained momentum: (i) the rise of the quantum geometric tensor, whose components can at present be directly accessed with optical probes; (ii) the notions of delicate and multigap topology, which fall outside the scope of tenfold way and symmetry-based indicators yet leave robust physical fingerprints; and (iii) the consideration of bundle gerbes, which capture formerly overlooked higher-form topological aspects of energy bands. These distinct directions have been elegantly woven together: delicate and multigap topological insulators have peculiar features in quantum geometry that can be conveniently captured by bundle gerbes. This viewpoint exposes the recently identified quantization of a non-linear optical response, and it invites deeper and systematic investigations into geometric and topological aspects of band structures beyond conventional Berryology.