Kavli Affiliate: Joel E. Moore
| First 5 Authors: Johannes Mitscherling, Alexander Avdoshkin, Joel E. Moore, ,
| Summary:
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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