Hyperbolic band theory

Abstract

Motivated by the recent realization of hyperbolic lattices in circuit quantum electrodynamics, we exploit ideas from Riemann surface theory and algebraic geometry to construct the first generalization of Bloch band theory to hyperbolic lattices, which can be formulated despite the absence of commutative translation symmetries. For an arbitrary Hamiltonian with the symmetry of a {4g, 4g} tessellation of the hyperbolic plane, we produce a continuous family of eigenstates that acquire Bloch-like phase factors under the Fuchsian group of the tessellation, which is discrete but noncommutative. Quasi-periodic Bloch wavefunctions are then generalized to automorphic functions. Naturally associated with the Fuchsian group is a compact Riemann surface of genus g 2, arising from the pairwise identification of sides of a unit cell given by a hyperbolic 4g-gon. A hyperbolic analog of crystal momentum continuously parametrizes a discrete set of energy bands and lives in a 2g-dimensional torus, which is a maximal set of independent Aharonov-Bohm phases threading the 2g noncontractible cycles of the Riemann surface.This torus is known in algebraic geometry as the Jacobian of the Riemann surface. We propose the Abel-Jacobi map -which associates to each point of the Riemann surface a point in the Jacobian -as a hyperbolic analog of particle-wave duality. Point-group symmetries form a finite group of automorphisms or self-maps of the Riemann surface and act nontrivially on the Jacobian. The tight-binding approximation and Wannier functions are also suitably generalized. In genus 1, our theory reduces to known band theory for 2-dimensional Euclidean lattices. We demonstrate our theory by explicitly computing hyperbolic Bloch wavefunctions and bandstructures numerically for a regular {8, 8} tessellation associated with a particular Riemann surface of genus 2, the Bolza surface.

Higher-dimensional Euclidean and non-Euclidean structures in planar circuit quantum electrodynamics

“…The addition of superconducting qubits to the setup can then realize fully interacting models [20]. This important achievement has stimulated some recent advances such as the formulation of a band theory in hyper-bolic lattices [21] or proposals for the realization of topological phases [22].…”

Electric-circuit realization of a hyperbolic drum

“…In particular, going beyond the presented emulation of the Laplace operator in a negatively curved space, the platform allows to emulate much more complex tight-binding models. These could, for example, be used to test the recently emerging concepts of hyperbolic band theory [12][13][14] , hyperbolic crystallography 15 , and hyperbolic topological insulators 19 . Our results also demonstrate that signatures of the hyperbolic geometry are easily accessible in terms of experimental probes.…”

“…Remarkably, several strongly correlated electronic systems have been analyzed with holographic tools as representative states that could hypothetically live on a holographic boundary [3][4][5][6][7][8][9][10][11] . More recently, major advancements in the mathematical characterization of classical and quantum states in negatively curved spaces [12][13][14][15] sparked a resurgence of interest in hyperbolic lattices [16][17][18] , ushering the research of hyperbolic topological matter 19,20 . These rapid developments call for new experimental platforms to implement tabletop simulations of hyperbolic toy-models.…”

“…Based on our results, we infer that electric circuit networks could be readily utilized to implement and to experimentally verify the predicted features of the recently studied hyperbolic models of Refs. [13][14][15][16][17][18][19].…”

Automorphic Bloch theorems for hyperbolic lattices

“…In our previous work [28], which we now briefly summarize, a band theory of hyperbolic lattices was proposed, based on ideas from Riemann surface theory and algebraic geometry. To each hyperbolic lattice, we have its associated 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.…”