Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals

TL;DR

This paper develops a gauge-invariant projector calculus for quantum state geometry, providing detailed formalism and examples to advance understanding of multi-band geometrical invariants and their applications in crystal physics.

Contribution

It introduces a novel gauge-invariant formalism based on projection operators for multi-state quantum geometry, enhancing analysis of geometrical invariants in crystals.

Findings

Detailed projector formalism for quantum geometry near specific crystal momenta.

New multi-state geometrical invariants related to optical responses.

Extensions applicable to topological and geometrical properties of materials.

Abstract

The importance of simple geometrical invariants, such as the Berry curvature and quantum metric, constructed from the Bloch states of a crystal has become well-established over four decades of research. More complex aspects of geometry emerge in properties linking multiple bands, such as optical responses. In the companion work [arXiv:2409.16358], we identified novel multi-state geometrical invariants using an explicitly gauge-invariant formalism based on projection operators, which we used to clarify the relation between the shift current and the theory of electronic polarization among other advancements for second-order non-linear optics. Here, we provide considerably more detail on the projector formalism and the geometrical invariants arising in the vicinity of a specific value of crystal momentum. We combine the introduction to multi-state quantum geometry with broadly relevant algebraic relationships and detailed example calculations, enabling extensions toward future applications to topological and geometrical properties of insulators and metals.