Density of states of tight-binding models in the hyperbolic plane

Abstract

We study the energy spectrum of tight-binding Hamiltonian for regular hyperbolic tilings. More specifically, we compute the density of states using the continued-fraction expansion of the Green function on finite-size systems with more than $10^9$ sites and open boundary conditions. The coefficients of this expansion are found to quickly converge so that the thermodynamical limit can be inferred quite accurately. This density of states is in stark contrast with the prediction stemming from the recently proposed hyperbolic band theory. Thus, we conclude that the fraction of the energy spectrum described by the hyperbolic Bloch-like wave eigenfunctions vanishes in the thermodynamical limit.

12 pages, 7 figures, published version

References in corpus (5)

• Crystallography of Hyperbolic Lattices
• Automorphic Bloch theorems for hyperbolic lattices
• Hyperbolic Matter in Electrical Circuits with Tunable Complex Phases
• Hyperbolic Topological Band Insulators
• Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices

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