Selberg trace formula and hyperbolic band theory

Adil, Attar,; Boettcher, Igor

doi:10.48550/arxiv.2201.06587

Cited by 1 publication

(1 citation statement)

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“…A subsequent experiment realizes a hyperbolic drum as an electric circuit [44], culminating in the experimental detection of negative curvature via the detection of the eigenmodes of the Laplace-Beltrami operator and of signal propagation along hyperbolic geodesics. Further theoretical works emerging recently in hyperbolic matter and band theory include [37,1,8,64,4,6].…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic band theory through Higgs bundles

Kienzle,

Rayan

2022

Preprint

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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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Section: Introductionmentioning

confidence: 99%