Automorphic Bloch theorems for hyperbolic lattices

TL;DR

This paper rigorously establishes a generalized Bloch theorem for hyperbolic lattices, demonstrating that hyperbolic band theory can accurately describe finite lattices and exploring the role of nonabelian translation symmetries.

Contribution

It proves a formal Bloch theorem for hyperbolic lattices, clarifies the role of higher-dimensional irreducible representations, and connects the theory to finite lattice realizations.

Findings

A generalized Bloch theorem is rigorously proved for hyperbolic lattices.

Hyperbolic band theory becomes exact for many finite lattice cases.

Higher-dimensional irreducible representations relate to moduli space of vector bundles.

Abstract

Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of 2D hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved, but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps corrrespond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact.