Hofstadter Butterflies in Topological Insulators

TL;DR

This paper explores the Hofstadter butterfly energy spectra in a 2D system of coupled topological chains under magnetic fields, revealing topologically protected surface states and their robustness.

Contribution

It introduces a hybrid topological system of coupled SSH chains, analyzing its energy spectra and topological surface states under magnetic fields for the first time.

Findings

Hofstadter butterfly patterns emerge in the coupled chain system.

Topologically protected surface states are identified and characterized.

Surface states show resilience against symmetry-preserving perturbations.

Abstract

In this chapter, we investigate the energy spectra as well as the bulk and surface states in a two-dimensional system composed of a coupled stack of one-dimensional dimerized chains in the presence of an external magnetic field. Specifically, we analyze the Hofstadter butterfly patterns that emerge in a 2D stack of coupled 1D Su-Schrieffer-Heeger (SSH) chains subject to an external transverse magnetic field. Depending on the parameter regime, we find that the energy spectra of this hybrid topological system can exhibit topologically non-trivial bulk bands separated by energy gaps. Upon introducing boundaries into the system, we observe topologically protected in-gap surface states, which are protected either by a non-trivial Chern number or by inversion symmetry. We examine the resilience of these surface states against perturbations, confirming their expected stability against local symmetry-preserving perturbations.