Abstract
Deriving Abelian as well as non-Abelian Bloch theorems requires the construction of finite clusters of lattice sites and the imposition of periodic boundary conditions (PBC), called PBC clusters. However, for hyperbolic lattices the construction of PBC clusters is highly non-trivial and requires advanced notions from group theory. Besides one-dimensional irreducible representations the elements in the hyperbolic translation groups admit higher-dimensional irreducible representations, for which there is no explicit parameterization, making analytical descriptions challenging. Previous studies have used Abelian hyperbolic band theory (AHBT) on the primitive cell which do in general not capture the thermodynamic limit. Alternatively, they have used brute force exact digitalization, which is computationally costly. Therefore, these circumstances motivate the application of approximative methods.
P. M. Lenggenhager et al. have very recently developed a method that computationally outperforms the previous approaches. Through the application of the supercell method they were able to access non-Abelian Bloch states by considering one-dimensional irreducible representations of reduced translation groups. Specifically, they suggested to apply AHBT to a appropriately constructed sequence of supercells, assembled as increasingly large aggregates of the primitive cells to supercells, and showcased the convergence of this method by selected hyperbolic tight-binding models. In conjunction, P. M. Lenggenhager has implemented the supercell method within a Mathematica and GAP package.
As such, the main objective of this thesis is the study of diverse aspects of topology and band theory in hyperbolic lattices accessed through the use of the supercell method. We aim to explore and broaden some of the recent studies in these systems such as higher-order topological phases, anomalous quantum Hall effects, flat bands in realizations of Lieb lattices, and non-Hermiticity driven topological phenomena.
