Hyperbolic band theory

Abstract

The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit quantum electrodynamics, we exploit ideas from algebraic geometry to construct a hyperbolic generalization of Bloch theory, despite the absence of commutative translation symmetries. For a quantum particle propagating in a hyperbolic lattice potential, we construct a continuous family of eigenstates that acquire Bloch-like phase factors under a discrete but noncommutative group of hyperbolic translations, the Fuchsian group of the lattice. A hyperbolic analog of crystal momentum arises as the set of Aharonov-Bohm phases threading the cycles of a higher-genus Riemann surface associated with this group. This crystal momentum lives in a higher-dimensional Brillouin zone torus, the Jacobian of the Riemann surface, over which a discrete set of continuous energy bands can be computed.

Automorphic Bloch theorems for hyperbolic lattices

“…In our previous work ( 30 ), a band theory of hyperbolic lattices was proposed, based on ideas from Riemann surface theory and algebraic geometry. For each integer g > 1, the lattice admits a Fuchsian group Γ—a discrete but nonabelian group that, for negatively curved surfaces, plays the role of the discrete, abelian translation group of Euclidean lattices.…”

“…Ref. 30 left several important questions unanswered. First, while an infinite family of solutions to the Schrödinger equation was constructed, no proof was given that such solutions form a complete set.…”

“…In other words, ref. 30 constructed hyperbolic Bloch eigenstates but did not prove a hyperbolic Bloch theorem stating that all eigenstates are necessarily of hyperbolic Bloch form. Second, and as remarked in ref.…”

“…The hyperbolic Bloch eigenstates of ref. 30 acquire a U (1) phase factor under Γ-translations and thus belong to a 1D irrep of Γ. It is conceivable that other irreps would appear in the spectrum of a generic hyperbolic lattice Hamiltonian.…”

Tensor network models of AdS/qCFT

“…Specifically, we consider regular {n, k} tilings (using the Schläfli notation) that are composed of regular n-gons, k of which are adjacent at each corner. Symmetry groups of physical models on these tilings have previously been identified and discussed in the context of tensor networks [20][21][22] as well as physical bulk models and their band theory [23,24]. In any regular tiling -hyperbolic or not -the reflection along edges of the tiling leaves the tiling invariant, and general mappings of the tiling onto itself can be composed from such reflections, defining the symmetries of a Coxeter group.…”