Variational Mapping of Chern Bands to Landau Levels: Application to Fractional Chern Insulators in Twisted MoTe$_2$

TL;DR

This paper develops a variational method to map Chern bands to Landau levels, enabling better understanding of fractional Chern insulators in twisted bilayer MoTe$_2$ and potentially other systems.

Contribution

It introduces a variational wavefunction approach that maps Chern bands to Landau levels, capturing fractional Chern insulator states with high accuracy.

Findings

High overlap of variational wavefunction with first moiré band

Quantitative agreement with exact diagonalization spectra

Applicable to other systems beyond twisted MoTe$_2$

Abstract

We present a theoretical study of mapping between Chern bands and generalized Landau levels in twisted bilayer MoTe$_2$ ($t$MoTe$_2$), where fractional Chern insulators down to zero magnetic fields have been observed. We construct an exact Landau-level representation of moir\'e bands, where the basis functions, characterized by a uniform quantum geometry, are derived from Landau-level wavefunctions dressed by spinors aligned or antialigned with the layer pseudospin skyrmion field. We further generalize the dressed zeroth Landau level to a variational wavefunction with an ideal yet nonuniform quantum geometry and variationally maximize its weight in the first moir\'e band. The variational wavefunction has a high overlap with the first band and quantitatively captures the exact diagonalization spectra of fractional Chern insulators at hole-filling factors $\nu_h=2/3$ and $3/5$, providing a clear theoretical mechanism for the formation and properties of the fractionalized states. Our work introduces a variational approach to studying fractional states by mapping Chern bands to Landau levels, with application to other systems beyond $t$MoTe$_2$ also demonstrated.