Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations

Abstract

Tessellations of the hyperbolic spaces by regular polygons are becoming popular because they support discrete quantum and classical models displaying unique spectral and topological characteristics. Resolving the true bulk spectra and the thermodynamic response functions of these models requires converging periodic boundary conditions and our work delivers a practical solution for this open problem on generic {p,q}-tessellations. This enables us to identify the true spectral gaps of bulk Hamiltonians and, as an application, we construct all but one topological models that deliver the topological gaps predicted by the K-theory of the lattices. We demonstrate the emergence of the expected topological spectral flows whenever two such bulk models are deformed into each other and, additionally, we prove the emergence of topological channels whenever a soft physical interface is created between different topological classes of Hamiltonians.

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• Hyperbolic lattices and two-dimensional Yang-Mills theory
• Hyperbolic Spin Liquids
• Non-Hermitian Skin Effect Along Hyperbolic Geodesics
• Unveiling Non-Hermitian Spectral Topology in Hyperbolic Lattices with Non-Abelian Translation Symmetry
• Uncovering Exceptional Contours in non-Hermitian Hyperbolic Matter
• Entanglement scaling behaviors of free fermions on hyperbolic lattices
• Classical spin liquids from frustrated Ising models in hyperbolic space
• Discrete holography and density of states in the crossover from hyperbolic to Euclidean lattices
• Topological states and flat bands in exactly solvable decorated Cayley trees