Hyperbolic band theory through Higgs bundles

• URL: http://arxiv.org/abs/2201.12689v2
• Date: Sat, 14 May 2022 15:44:04 GMT
• Title: Hyperbolic band theory through Higgs bundles
• Authors: Elliot Kienzle, Steven Rayan
• Abstract summary: Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation. A corresponding hyperbolic band theory has emerged, extending 2-dimensional Euclidean band theory in a natural way to higher-genus configuration spaces. Higgs bundles enjoy natural interpretations in the context of band theory.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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