arXiv 2201.126892022-01math-phcond-mat.mes-hallhep-thmath.AGmath.MPquant-ph

Hyperbolic band theory through Higgs bundles

Elliot Kienzle, Steven Rayan

备注：: 38 pages, 13 figures, 1 table
发表信息：: Adv. Math. 409 (2022), paper no. 108664
DOI：: 10.1016/j.aim.2022.108664

英文摘要：

Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation and which, to date, have been engineered artificially. A corresponding hyperbolic band theory has emerged, extending 2-dimensional Euclidean band theory in a natural way to higher-genus configuration spaces. Attempts to develop the hyperbolic analogue of Bloch's theorem have revealed an intrinsic role for algebro-geometric moduli spaces, notably those of stable bundles on a curve. We expand this picture to include Higgs bundles, which enjoy natural interpretations in the context of band theory. First, their spectral data encodes a crystal lattice and momentum, providing a framework for symmetric hyperbolic crystals. Second, they act as a complex analogue of crystal momentum. As an application, we elicit a new perspective on Euclidean band theory. Finally, we speculate on potential interactions of hyperbolic band theory, facilitated by Higgs bundles, with other themes in mathematics and physics.

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