The GAP package HyperCells

Warning: Note the potential breaking changes in version 0.9.1-beta if edge tags are explicitly referred to. The edge tag format for nearest-neighbor edges constructed using TGCellModelGraph, TessellationModelGraph, and KagomeModelGraph has been changed from [ 1, [ gi, s1, s2 ] ] to [ 1, [ [ w, gi ], s1, s2 ] ].

HyperCells is a GAP package that allows constructing primitive cells and supercells of hyperbolic lattices based on triangle groups and quotients with normal subgroups. An introduction to the underlying 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)

and the doctoral thesis

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

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, 10.5281/zenodo.10222598 (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)

A getting-started guide for this and its sister package (HyperBloch) can be found here.

Please refer to CONTRIBUTING.md on how you can contribute to the development of this package.

Authors and developers
Installation
Documentation
Limitations
HyperBloch package
How to cite
Contact
License and copyright

Authors and developers

Main developer:
Patrick M. Lenggenhager
Email: plengg@pks.mpg.de
Website: https://patrick-lenggenhager.github.io

Coauthors:
Joseph Maciejko (maciejko@ualberta.ca)
Tomáš Bzdušek (tomas.bzdusek@uzh.ch)

Installation

To install the HyperCells package, clone this repository in the ~/.gap/pkg/ directory. GAP should automatically detect the package and make it available for loading using the command

LoadPackage("HyperCells");

Documentation

The documentation is available on the accompanying Github pages website.

Limitations

Note that at this point HyperCells is still under development and, because the limitations have not yet been fully determined, released only as a beta version. Further testing will be required before an official release. However, the package is already fully functional and documented and can be used to reproduce the results of the publication mentioned above.

Known limitations

TGCellGraph objects and symmetric TGCell objects cannot be produced for cells without mirror symmetries, i.e., for quotients that act “chirally” on the surface.
Faces of TGCellGraph objects have only been tested in the context of TGTessellationGraph and TGKagomeGraph and might not work in general.

HyperBloch package

The HyperBloch package is a companion package to HyperCells for Mathematica. It allows to study the band structure of hyperbolic lattices by applying hyperbolic band theory and the supercell method. Additionally, it provides many functions to visualize the output of HyperCells. It is available on Github at

https://github.com/patrick-lenggenhager/HyperBloch

How to cite

If you use this package, please cite the package itself

@misc{HyperCells,
  title           = {Hyper{C}ells: {A} {GAP} package for constructing primitive cells and supercells of hyperbolic lattices},
  author          = {Lenggenhager, Patrick M. and Maciejko, Joseph and Bzdu\v{s}ek, Tom\'{a}\v{s}},
  year            = {2023},
  doi             = {10.5281/zenodo.10222598},
  note            = {\url{https://github.com/patrick-lenggenhager/HyperCells}}
}

and at least one of the following references:

@article{Lenggenhager:2023,
  title               = {Non-{A}belian hyperbolic band theory from supercells}, 
  author              = {Lenggenhager, Patrick M. and Maciejko, Joseph and Bzdu\v{s}ek, Tom\'{a}\v{s}},
  journal             = {Phys. Rev. Lett.},
  volume              = {131},
  issue               = {22},
  pages               = {226401},
  numpages            = {7},
  year                = {2023},
  month               = {Dec},
  publisher           = {American Physical Society},
  doi                 = {10.1103/PhysRevLett.131.226401}
}

@phdthesis{Lenggenhager:PhDThesis,
  title           = {Emerging avenues in band theory: multigap topology and hyperbolic lattices},
  author          = {Lenggenhager, Patrick M.}, 
  year            = {2023},
  school          = {ETH Zurich},
  doi             = {10.3929/ethz-b-000645370}
}

as well as the following reference for the database of quotients of triangle groups with normal subgroups:

@misc{Conder:2007,
  title           = {Quotients of triangle groups acting on surfaces of genus 2 to 101},
  author          = {Conder, Marston},
  year            = {2007},
  howpublished    = {\url{https://www.math.auckland.ac.nz/~conder/TriangleGroupQuotients101.txt}}
}

Contact

To report issues, please use the issue tracker at https://github.com/patrick-lenggenhager/HyperCells/issues.

Maintainer:
Patrick M. Lenggenhager
Email: plengg@pks.mpg.de
Homepage: https://patrick-lenggenhager.github.io

License and copyright

HyperCells is free software; you can redistribute and/or modify it under the terms of the CC BY-SA 4.0 license as described below. HyperCells is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY, see the CC BY-SA 4.0 license for more details.

This is a human-readable summary of (and not a substitute for) the license, see the attached LICENSE for the full legal text.

You are free to: Share — copy and redistribute the material in any medium or format for any purpose. Adapt — remix, transform, and build upon the material for any purpose. The licensor cannot revoke these freedoms as long as you follow the license terms.

Under the following terms: Attribution - You must give appropriate credit (see AUTHORS and How to cite above), provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. ShareAlike - If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original. No additional restrictions - You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.