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

TL;DR

The paper develops a universal variational framework to decompose realistic Bloch bands into generalized Landau levels (LLs) and applies it to twisted MoTe$_2$ moiré bands. It shows the first moiré band is predominantly a generalized $0$LL, enabling Abelian FCIs in the Jain sequences, while the second band attains a dominant generalized $1$LL at select twist angles and hosts a non-Abelian Moore–Read state at $ u_h=5/2$ for $ heta=2.45^ ext{ extcircled{ extdegree}}$, with and without Hartree–Fock renormalization. An adiabatic path connects the MR state in the second band to the conventional first LL MR state, though larger bandwidth can favor competing CDW order at other angles (e.g., $ heta=2.13^ ext{ extcircled{ extdegree}}$). The framework provides design principles and quantitative diagnostics for realizing and distinguishing Abelian and non-Abelian fractionalized phases in realistic moiré materials, guiding future experiments and device engineering.

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 moiré Hamiltonians, we find that the first moiré 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 moiré band, renormalized via Hartree-Fock calculations at hole filling $ν_h = 2$, is dominated by the generalized first LL at twist angles $θ= 2.45^\circ$ and $2.13^\circ$. At $θ= 2.45^\circ$, we find numerical evidence for a non-Abelian Moore--Read (MR) state at $ν_h = 5/2$, 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 $θ= 2.13^\circ$, 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.

Figures (16)

• Figure 1: (a) Moiré band structure of tMoTe$_2$ in $+K$ valley at $\theta=2.45^\circ$ along high-symmetry path in the mBZ. (b) Bandwidths of the first moiré band $\eta_1$ and the second moiré band $\eta_2$ as functions of $\theta$. (c) Energy gaps $\Delta_{12}$ between the first and second bands, and $\Delta_{23}$ between the second and third bands, plotted as a function of $\theta$. (d) Chern number $\mathcal{C}$ and quantum weight $\mathcal{K}$ for the first moiré valence band in $+K$ valley. (e) Same as (d), but for the second moiré band. Results are shown for commensurate angles $\theta$ at $2.13^\circ, 2.45^\circ, 2.65^\circ, 2.88^\circ, 3.15^\circ, 3.48^\circ$ and $3.89^\circ$.
• Figure 2: (a-b) LL weight $W_{0,n}^{(s)}$ for the first moiré band where $n$ and $s=\pm$ label, respectively, the index and chirality of the generalized LL. (c-f) Map of $\abs{\mathcal{B}^{(+)}(\boldsymbol{r})}$ scaled by its spatial average at $\theta=2.13^\circ,2.45^\circ,2.88^\circ$, and $3.15^\circ$. The arrows indicate the data range in each plot.
• Figure 3: (a-d) Overlap $\lvert c_{1,n,\boldsymbol{k}}^{(s)}\rvert=\lvert\langle\Theta^{(s)}_{n,\boldsymbol{k}}\lvert\psi_{+,1,\boldsymbol{k}}\rangle\rvert$ at $\theta=2.45^\circ$ for $s=\pm$ and $n=0,1$ in the mBZ. The arrows indicate the data range in each plot.
• Figure 4: (a-b) LL weight $W_{2,n}^{(s)}$ for the second moiré band where $n$ and $s=\pm$ label, respectively, the index and chirality of the generalized LL. (c-d) Map of $\abs{\mathcal{B}(\boldsymbol{r})}$ scaled by its spatial average at $\theta=2.13^\circ$ and $2.45^\circ$. The arrows indicate the data range in each plot.
• Figure 5: Overlap $\lvert c_{2,n,\boldsymbol{k}}^{(+)}\rvert=\lvert\langle\Theta^{(+)}_{n,\boldsymbol{k}}\lvert\psi_{+,2,\boldsymbol{k}}\rangle\rvert$ in mBZ for (a-d) $n=0,1,2,3$ at $\theta=2.13^\circ$ and (e-h) $n=0,1,2,3$ at $\theta=3.89^\circ$. The arrows indicate the data range in each plot.