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

• URL: http://arxiv.org/abs/2412.03637v1
• Date: Wed, 04 Dec 2024 19:00:00 GMT
• Title: Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals
• Authors: Johannes Mitscherling, Alexander Avdoshkin, Joel E. Moore,
• Abstract summary: More complex aspects of geometry emerge in properties linking multiple bands, such as optical responses. We identify novel multi-state geometrical invariants using an explicitly gauge-invariant formalism based on projection operators. We provide more detail on the projector formalism and the geometrical invariants arising in the vicinity of a specific value of crystal momentum.
• Score: 44.99833362998488
• License:
• 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.

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