Hyperbolic band theory through Higgs bundles

TL;DR

This paper extends hyperbolic band theory by incorporating Higgs bundles, revealing new mathematical structures and interpretations that could impact quantum matter research and deepen understanding of band theory in complex geometries.

Contribution

It introduces Higgs bundles into hyperbolic band theory, providing a novel framework that links spectral data to lattice and momentum, and offers new insights into Euclidean band theory.

Findings

Higgs bundles encode crystal lattice and momentum in hyperbolic band theory.

Spectral data of Higgs bundles acts as a complex analogue of crystal momentum.

New perspective on Euclidean band theory through hyperbolic geometric structures.

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.