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 normal subgroups.

:book: We have a new website with detailed instructions on the installation process, a getting-started guide and hands-on tutorials: https://www.hypercells.net.

An introduction to the underlying concepts can be found in the following publication

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/HyperCells/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)

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

The GAP package HyperCells - Table of Contents

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)

Code contributers:
Marcelo Looser (marcelo.looser@outlook.com)

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");

Extensions (optional)

The HyperCells package has an integrated word simplification procedure for a selection of functions. Two methods are available: a default brute-force method, and a method based on the Knuth-Bendix completion algorithm. The latter can only be used provided the kbmag package (version 1.5.10+) is available.

The default configuration of the kbmag package allows HyperCells to simplify words in groups with a maximal number of generators of 127. However, this limit can manually be extended up to

The corresponding adjustments are laid out in a README file in the kbmag package and can be found in the folder containing GAP: “…/gap/gap-< version >/pkg/kbmag/standalone”, with the following instructions:

NEW in Version 2.3: It is now possible to use kbmag with more than the
previous default number of 127 generators. To use up to 65535 generators,
before making the package, edit the file "defs.h" in the lib directory,
and change the two lines:

#define MAXGEN MAXCHAR /* maximum number of generators */
typedef char gen; /* for generators of monoids and groups */

to

#define MAXGEN MAXUSHORT /* maximum number of generators */
typedef unsigned short gen; /* for generators of monoids and groups */

Once these changes are made, kbmag needs to be recompiled. This can be done in the terminal, where in the kbmag directory one needs to execute the command make clean and afterwards make.

If these changes are not made while using the Knuth-Bendix completion algorithm based simplification and unit cells compactified on Riemann surfaces with genus exceeding 63 are used, the procedure will not be executed and a warning will be printed in GAP:

#WARNING: maximal number of generators have been exceeded; non-simplified words 
will be used. Please follow the instructions in the chapter Introduction section
Simplify extension (optional) in the HyperCells reference manual.

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.

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/HyperCells/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/HyperCells/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/HyperCells/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.