HyperCells and HyperBloch: open-source software packages for studying hyperbolic lattices based on triangle groups

Hyperbolic lattices form the analogue of periodic structures in the hyperbolic plane and have been experimentally realized as networks in various metamaterial platforms such as coplanar-waveguide resonators and electric circuits. Naturally, the lattices possess discrete translation symmetry. However, due to the negative curvature, the resulting translation group is non-commutative which complicates not only the formulation of band theory but also the construction of periodic boundary conditions that converge to the thermodynamic limit. Both infinite lattices and finite clusters with periodic boundary conditions can be conveniently described in terms of triangle groups, which take the role of the space groups.

In this talk, I am going to introduce two recently released open source software packages, called HyperCells and HyperBloch, which provide convenient tools to construct connected and symmetric unit cells, including the associated translations, define arbitrary tight-binding models on them, and apply the recently developed supercell method for hyperbolic band theory to gain access at infinite-lattice eigenstates and -energies. The construction is based on an algebraic description of the lattice in terms of the corresponding triangle group, which facilitates a discussion of not only the translation symmetry but point-group symmetries as well. I will illustrate the scope and usage of both packages by discussing several examples and showing some recent results obtained using them.