Automorphic Bloch theorems for hyperbolic lattices

Abstract

Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional 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:(1) whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; (2) whether a formal Bloch theorem can be rigorously established; and (3) 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 one-dimensional irreps appear, and the hyperbolic band theory previously developed becomes exact.

Hyperbolic band theory through Higgs bundles

“…The foundation is a hyperbolic version of the classical Bloch theorem, which constructs the eigenstates of the crystal Hamiltonian from crystal momenta. Hyperbolic Bloch states were introduced in [46], and a corresponding version of Bloch's theorem was formulated in [45].…”

“…This argument is essentially given in [45], though framed slightly differently. That paper dealt with a finite subset of the full hyperbolic lattice in the tight-binding model, and consequently a finite-dimensional Hilbert space.…”

“…We call this the abelian Brillouin zone, which was introduced for hyperbolic crystals in [46]. The other connected components of momentum space are the character varieties Hom irr (Γ, U (n))/U (n), which we will call the nonabelian Brillouin zones, introduced in [45]. We can express these geometrically as moduli spaces of vector bundles on Riemann surfaces.…”

“…A similar scene plays out for higher rank representations, but the details are more involved [45]. Starting from a trivial vector bundle H × C n → H, we obtain a topologically trivial 2 vector bundle E → Σ by quotienting out Γ acting by γ : (x, v) → (γ(x), ρ(γ)(v)).…”

“…In regards to the precise organization of the paper, Section 2 synthesizes the hyperbolic band theory framework of [46,45] in a self-contained and mathematically-oriented manner. It contributes a description of the abelian and nonabelian crystal Hamiltonian, with various geometric and physical interpretations.…”