The GAP package HyperCells

HyperCells is a GAP package that allows constructing primitive cells and supercells of hyperbolic lattices based on triangle groups and quotients with torsionfree normal subgroups. This information can be used as an input for the HyperBloch Mathematica package to compute band structures and density of states of hyperbolic tight-binding models.

The package can be downloaded from Github, and a detailed documentation is available on a dedicated package website.

An introduction to the underlying mathematical and physics concepts can be found in the following preprint

• P. M. Lenggenhager, J. Maciejko, and T. Bzdušek, Non-Abelian hyperbolic band theory from supercells, Phys. Rev. Lett. 131, 226401 (2023) (publicly accessible preprint atarXiv:2305.04945).

and the doctoral thesis

• P. M. Lenggenhager, Emerging avenues in band theory: multigap topology and hyperbolic lattices, PhD thesis, ETH Zurich (2023)

Basic functionality is demonstrated in the Supplmentary Code and Data associated with the PRL manuscript. In addition, we are planning to upload tutorials that showcase the implementation of several standard constructions by June 2024.

If you use this package, please cite at least one of the above references in addition to the package itself:

• P. M. Lenggenhager, J. Maciejko, and T. Bzdušek, HyperCells: A GAP package for constructing primitive cells and supercells of hyperbolic lattices, https://github.com/patrick-lenggenhager/HyperCells (2023)

and the list of quotient groups:

• M. Conder, Quotients of triangle groups acting on surfaces of genus 2 to 101, https://www.math.auckland.ac.nz/~conder/TriangleGroupQuotients101.txt (2007)