Hofstadter problem in higher dimensions

TL;DR

This paper explores higher-dimensional generalizations of the Hofstadter problem, revealing hierarchical energy spectra and connections between flux states and Dirac fermion formalisms through numerical analysis.

Contribution

It introduces higher-dimensional Hofstadter models with Abelian and non-Abelian fields, highlighting spectral structures and equivalences with Dirac fermion representations.

Findings

Hierarchical energy spectra observed in higher-dimensional models

Equivalence between -flux state and staggered Dirac fermion formalism

Numerical evidence supporting spectral and formal connections

Abstract

We investigate some generalizations of the Hofstadter problem to higher dimensions with Abelian and non-Abelian gauge field configurations. We numerically show the hierarchical structure in the energy spectra with several lattice models. It is also pointed out the equivalence betwee the \pi-flux state and the staggered formalism of Dirac fermion.