Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals
| Autor: | Mitscherling, Johannes, Avdoshkin, Alexander, Moore, Joel E. |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 13 + 4 pages, 1 table. Based on SM of arXiv:2409.16358. Any comments are welcome! |
| Databáze: | arXiv |
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