Hyperbolic band theory through Higgs bundles (2201.12689v2)

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.

Hyperbolic Band Theory through Higgs Bundles

The paper "Hyperbolic Band Theory through Higgs Bundles" by Elliot Kienzle and Steven Rayan explores the emergent field of hyperbolic quantum matter through a mathematical lens. The authors propose that hyperbolic band theory, which extends traditional two-dimensional Euclidean band theory to configuration spaces of higher genus, can be effectively studied using the framework of Higgs bundles. Higgs bundles enjoy natural interpretations within the context of band theory, providing a fresh perspective on the underlying algebraic and geometric structures.

Overview of Hyperbolic Band Theory

The paper builds on the concept of hyperbolic lattices, which have been artificially engineered and hold promise for quantum computing and simulation applications. The corresponding hyperbolic band theory incorporates the intrinsic properties of algebro-geometric moduli spaces, especially stable bundles on curves. The authors primarily focus on packaging spectral data from Higgs bundles, a higher-dimensional generalization of Euclidean band theory. This spectral data encodes crystal lattices and momentum states, allowing for the analysis of symmetric hyperbolic crystals and their comparison to conventional Euclidean counterparts.

Numerical Results and Claims

A significant portion of the paper is devoted to deriving the hyperbolic analogue of Bloch's theorem, crucial for understanding band theory in hyperbolic spaces. The authors propose numerical techniques to calculate the spectrum of Hamiltonians within the framework, equating crystal Hamiltonians with differential geometry problems, such as finding the spectrum of flat connection operators. In the Euclidean case, the analyzed band structures show periodic behavior and degeneracies at high-symmetry points in momentum space.

Implications and Future Directions

The research paves a theoretical foundation for future exploration into the interactions between hyperbolic band theory and other mathematical and physical frameworks. Framed as a moduli problem, the universal band structure defined by Higgs bundles invites potential applications in areas like topological materials, supersymmetric field theories, and the geometric Langlands correspondence. For instance, the paper suggests connections with high-energy physics communities frequently using Higgs bundles to paper spectral networks or BPS states.

Speculations on AI Developments

There is speculation that the paper of hyperbolic band theory could impact AI developments by influencing models that utilize geometric frameworks. Deep learning algorithms, which often benefit from succinct representations of complex systems, could potentially adopt principles outlined in this paper, improving model accuracy and prediction capabilities in quantum simulations and computations. Furthermore, the integration of symmetrical properties and geometric considerations might inform ensemble learning approaches.

The research presented in "Hyperbolic Band Theory through Higgs Bundles" contributes to an evolving narrative within quantum matter studies, positing fresh mathematical tools to understand the rich landscape presented by hyperbolic configuration spaces. The mathematical elegance and potential for future in-depth studies position this paper as a valuable resource for researchers interested in the theoretic and practical implications of hyperbolic quantum matter.