Tessellations of the hyperbolic spaces by regular polygons support discrete quantum and classical models with 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 Letter delivers a practical and rigorous solution for this open problem on generic $\{p,q\}$-tessellations. This enables us to identify the true spectral gaps of bulk Hamiltonians and 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 prove the emergence of topological channels whenever a soft physical interface is created between different topological classes of Hamiltonians.

• Received 31 March 2023
• Revised 17 July 2023
• Accepted 28 September 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.176603