Abelian and non-Abelian fractionalized states in twisted MoTe$_2$: A generalized Landau-level theory

TL;DR

This paper introduces a universal variational framework to map Chern bands onto generalized Landau levels, enabling the analysis of fractionalized states, including non-Abelian phases, in twisted bilayer MoTe$_2$.

Contribution

The authors develop a quantitative method to decompose Bloch bands into generalized Landau levels and apply it to twisted MoTe$_2$, revealing potential for realizing non-Abelian fractional Chern insulators.

Findings

The first moiré valence band is dominated by the zeroth LL, supporting Abelian fractional Chern insulators.

The second moiré band at certain twist angles is dominated by the first LL, enabling non-Abelian Moore-Read states.

Numerical evidence suggests a non-Abelian Moore-Read state at $ u_h=5/2$ at specific twist angles.

Abstract

Fractional Chern insulators are lattice analogs of fractional quantum Hall states that realize fractionalized quasiparticles without an external magnetic field. A key strategy to understand and design these phases is to map Chern bands onto Landau levels (LLs). Here, we introduce a universal framework that variationally decomposes Bloch bands into generalized LLs, providing a controlled and quantitative characterization of their effective LL nature. Applying this approach to twisted bilayer MoTe$_2$ modeled by first-principles-derived moir\'e Hamiltonians, we find that the first moir\'e valence band is dominated by the generalized zeroth LL across a broad range of twist angles, facilitating the formation of Abelian fractional Chern insulators in the Jain sequences. The second moir\'e band, renormalized via Hartree-Fock calculations at hole filling $\nu_h = 2$, is dominated by the generalized first LL at twist angles $\theta = 2.45^\circ$ and $2.13^\circ$. At $\theta = 2.45^\circ$, we find numerical evidence for a non-Abelian Moore--Read (MR) state at $\nu_h = 5/2$, with consistent signatures in both the energy spectrum and the particle entanglement spectrum. Interpolation studies further demonstrate an adiabatic connection between this state and the MR state in the conventional first LL. In contrast, at $\theta = 2.13^\circ$, a charge-density-wave state prevails in the competition with the MR state due to the larger bandwidth. Our variational mapping provides a theoretical framework for exploring exotic fractionalized phases, including non-Abelian states, in realistic systems.