Hyperbolic band theory through Higgs bundles

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.

Flat bands and band touching from real-space topology in hyperbolic lattices

“…Second, since the translation group Γ is non-Abelian, it also admits higher-dimensional unitary irreps, in addition to the 1D irreps described by the 4D/6D Brillouin zones. These higher-dimensional irreps live in certain moduli spaces (also known as character varieties) [32,56] that can be viewed as non-Abelian Brillouin zones. To summarize, the hyperbolic lattices we study here support two types of Bloch states: those with crystal momentum k in a 4D/6D Brillouin zone, referred to as Abelian states, and those corresponding to higherdimensional irreps, dubbed non-Abelian states.…”