Hofstadter spectrum of Chern bands in twisted transition metal dichalcogenides

TL;DR

This paper investigates the Hofstadter spectrum of topological Chern bands in twisted bilayer transition metal dichalcogenides under magnetic fields, revealing how band structures evolve with twist angle and magnetic flux, and proposing new ways to identify Landau level analogs.

Contribution

It introduces a framework to analyze magnetic field effects on topological bands in twisted TMDs, connecting Hofstadter spectra with Landau levels and non-Abelian states.

Findings

Hofstadter butterfly patterns depend on twist angle and flux.

Identification of Landau level analogs in twisted bilayer MoTe$_2$.

Proposal of magnetic field studies as a tool for topological band analysis.

Abstract

We study the topological bands in twisted bilayer transition metal dichalcogenides in an external magnetic field. We first focus on a paradigmatic model of WSe$_2$, which can be described in an adiabatic approximation as particles moving in a periodic potential and an emergent periodic magnetic field with nonzero average. We understand the magnetic-field dependent spectra of WSe$_2$ based on the point net zero flux, at which the external field cancels the average emergent field. At this point, the band structure interpolates between the tightly-bound and nearly-free (weak periodic potential) paradigms as the twist angle increases. For small twist angles, the energy levels in a magnetic field mirror the Hofstadter butterfly of the Haldane model. For larger twist angles, the isolated Chern band at zero flux evolves from nearly-free bands at the point of net zero flux. We also apply our framework to a realistic model of twisted bilayer MoTe$_2$, which has recently been suggested to feature higher Landau level analogs. We show that at negative unit flux per unit cell, the bands exhibit remarkable similarity to a backfolded parabolic dispersion, even though the adiabatic approximation is inapplicable. This backfolded parabolic dispersion naturally explains the similarity of the Chern bands at zero applied flux to the two lowest Landau levels, offering a simple picture supporting the emergence of non-Abelian states in twisted bilayer MoTe$_2$. We propose the study of magnetic field dependent band structures as a versatile method to investigate the nature of topological bands and identify Landau level analogs.