Non-Abelian hyperbolic band theory from supercells

TL;DR

This paper introduces a supercell-based method to systematically construct and analyze non-Abelian Bloch states in hyperbolic lattices, advancing the understanding of their spectral properties.

Contribution

It adapts supercell and zone folding techniques from solid-state physics to hyperbolic lattices, enabling efficient computation of their bulk spectra and eigenstates.

Findings

Rapid convergence in computing spectra for hyperbolic lattices

Effective approximation of the thermodynamic limit

First systematic construction of non-Abelian Bloch states

Abstract

Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By adapting the solid-state-physics notions of supercells and zone folding, we devise a method for the systematic construction of non-Abelian Bloch states. The method applies Abelian band theory to sequences of supercells, recursively built as symmetric aggregates of smaller cells, and enables a rapidly convergent computation of bulk spectra and eigenstates for both gapless and gapped tight-binding models. Our supercell method provides an efficient means of approximating the thermodynamic limit and marks a pivotal step towards a complete band-theoretic characterization of hyperbolic lattices.