Getting started#
The task of modeling tight-binding models on hyperbolic lattices is split into two steps:
Construction of lattice/model graphs for finite systems, this includes the construction of appropriate periodic boundary conditions, while for infinite systems, the corresponding primitive cell and successive supercells are constructed in order to apply the supercell method for hyperbolic band theory. This part is implemented in the GAP package HyperCells. Additionally, the package determines maximally-symmetric Wyckoff positions and simplifies dealing with translation and point group symmetries on the hyperbolic lattice. The package also allows the definition of the graph underlying a specific model, i.e., selecting specific Wyckoff positions, nearest or next-nearest neighbors etc.
Defining tight-binding models and using hyperbolic band theory implemented in the second package, a Mathematica package called HyperBloch. It provides functions for importing and easily visualizing the clusters, supercells, and model graphs. Additionally, it allows the user to easily define tight-binding models on top of the model graph, by placing orbitals at vertices and defining hopping matrices on the edges. Finally, the Abelian Bloch Hamiltonian for the defined model is constructed and thus allows to study eigenstates and eigenvalues.
Note that the output of the HyperCells package, i.e., the cell, model and supercell model graphs can be read by any programming language, such that using the HyperBloch package is not strictly necessary. The file format definition can be found in the documentation.