HyperCells is a GAP package for constructing and working with the graphs underlying infinite and compactified hyperbolic lattices and their (translation) symmetries. This is achieved through the hyperbolic lattices’ structure as Cayley graphs of triangle groups and their quotients with normal subgroups.
HyperBloch is a Mathematica package for defining tight-binding models on infinite and compactified hyperbolic lattices, computing their Bloch Hamiltonians, and obtaining the corresponding eigenstates and spectrum based on hyperbolic band theory and the supercell method. Additionally, it enables advanced visualizations of compactified hyperbolic lattices and tight-binding models defined on them in the Poincaré disk model.
Overview#
Hyperbolic lattices are symmetric tessellations of the hyperbolic plane, i.e., the space of constant negative curvature. Due to their non-Euclidean geometry, the translations of hyperbolic lattices do not commute, which poses a challenge for the analytical description of models defined on them. Already constructing lattices with periodic boundary conditions is nontrivial, and doing it in a way that ensures convergence to the thermodynamic limit is even more challenging. Furthermore, the noncommutativity of translations severely complicates a momentum-space description.
This is where HyperCells and HyperBloch come into play. HyperCells provides the necessary tools to construct the graphs underlying hyperbolic lattices and their symmetries, while HyperBloch enables the computation of Bloch Hamiltonians through the supercell method (see next page). An accessible introduction to the topic in general and underlying methods, is provided, for example, in Dr. Patrick M. Lenggenhager’s PhD thesis.
This website serves as a comprehensive guide to the installation and usage of both packages, providing tutorials with detailed explanations ranging from simple examples to more advanced applications. The software packages are open-source and available on GitHub, where contributions, both reports of bugs and code contributions, are welcome. While most tutorials focus on examples from condensed matter physics and the supercell method, HyperCells has a much broader scope and can be used anywhere where hyperbolic lattices and their symmetries are of interest, independent of a potential momentum-space description.
We also provide a Github repository with a library of graphs generated with HyperCells, which can be used without the need to install the package. Check out the documentation on the file format to use them in your own projects. Alternatively, you can import them in Mathematica using the HyperBloch package.