Tutorials#
The HyperCells and HyperBloch packages offer a framework for efficient modeling of tight-binding models on a broad list of hyperbolic lattices. We provide tutorials which are aimed to facilitate a smooth workflow and showcase prominent examples of interest.
The HyperCells package grants easy access to supercells of hyperbolic lattices through computational group theory. The tight-binding models on these supercells enable systematic access to higher-dimensional irreducible representations on the primitive cell through the use of the HyperBloch package by applying Abelian hyperbolic band theory. In this tutorial we will construct multiple supercell sequences and demonstrate the convergence to the thermodynamic limit of the density of states through the application of the supercell method.
The application of the supercell method relies on the identification of appropriate supercell sequences. Each supercell is associated with a triangle group quotient specified by a corresponding translation group. The sequences of these translation groups form so-called coherent sequences defined by a normal subgroup relation between consecutive translation groups. In this tutorial, we showcase how such sequences can be identified. Two approaches are presented, which entail the construction of a user defined functions and normal subgroup tree graphs which describes the normal subgroup relations between any pairwise distinct translation group of a \(\{p,q\}\)-tesselation of the hyperbolic plane.
The HyperCells package is equipped with built-in functions to model hyperbolic lattices with emergent strongly correlated systems. Among prominent examples which develop pronounced flat-bands are hyperbolic lattice variants of the Euclidean Lieb and kagome lattices. In this tutorial, we will see how hyperbolic Lieb and kagome lattices are constructed through model graphs on a primitive cell and consecutive supercell in tandem with the application of the supercell method.
Haldane models on hyperbolic lattices can be constructed by using the HyperCells and HyperBloch packages in tandem. The HyperCells package provides the necessary framework to decorate nearest-neighbor models with oriented next-nearest-neighbor terms through a minimal modification of the usual workflow. These models can be endowed with oriented coupling constants in the HyperBloch package. In this tutorial, we will construct next-nearest-neighbor tight-binding models and Haldane models. Additionally, we will construct point-group matrices in order to showcase how hyperbolic lattice symmetries can be analyzed.
The HyperBloch package provides a framework for the construction of Hermitian as well as non-Hermitian Abelian Bloch Hamiltonians for tight-binding models on hyperbolic lattices. In this tutorial, we will construct two non-Hermitian tight-binding models. Specifically, we will see how Hermiticity-breaking gains and losses can be assigned through staggered complex on-site potentials as well as how to construct a variant of the Hatano-Nelson model describing a non-reciprocal tight-binding model on a hyperbolic lattice.
In this tutorial, we will see how to construct an elementary nearest-neighbor hopping model with multiple orbitals per site for (supercell) model graphs. This will enable us to set up a variant of the Benalcazar-Bernevig-Hughes model, also known as the BBH model. Moreover, we will demonstrate the bulk-boundary correspondence by analyzing spectra computed for a BBH model on finite flakes with open boundary conditions. In particular, we will see how disclination defects can be introduced in finite size systems, through the Volterra process by specifying a negative Frank angle, which demonstrates an emergent filling anomaly through corresponding spectra.
Featured function index#
The getting started and the tutorial page feature the following HyperCells and HyperBloch functions: