Fractional Chern Insulator and Quantum Anomalous Hall Crystal in Twisted MoTe$_2$

Recent experimental advances have uncovered fractional Chern insulator (FCI) states in twisted MoTe$_2$ (tMoTe$_2$) systems under zero magnetic field. Understanding the interaction effects on topological phases within realistic model presents a significant theoretical challenge. Here, we construct a moiré superlattice model tailored for tMoTe$_2$ and conduct investigations using state-of-the-art tensor-network methods. Our ground-state calculations reveal a rich variety of interaction-driven and filling-dependent topological phases, including FCIs, Chern insulators, and generalized Wigner crystals, which are revealed in recent experiments. For FCI state, dynamical simulations uncover a single-particle excitation continuum with a finite charge gap, reflecting the fractionalized charge excitations. Finite-temperature calculations further determine characteristic charge activation and ferromagnetic transition temperatures, reconciling existing experimental discrepancies. Furthermore, using this realistic lattice model, we predict the presence of quantum anomalous Hall crystals exhibiting integer Hall conductivity at fractional fillings in tMoTe$_2$. By integrating ground-state, finite-temperature, and dynamical analyses, our work establishes a comprehensive framework for understanding correlated topological phases in tMoTe$_2$ and related moiré systems.

💡 Research Summary

This paper addresses the recent experimental discovery of fractional Chern insulator (FCI) states in twisted bilayer MoTe₂ (tMoTe₂) at zero magnetic field by constructing a realistic real‑space Hubbard model and solving it with state‑of‑the‑art tensor‑network techniques. The authors first derive a tight‑binding description from density‑functional theory and a continuum moiré model, selecting exponentially localized Wannier functions that sit on the two sublattices (A, B) of a honeycomb lattice and are layer‑resolved (bottom/top). The resulting model contains three hopping amplitudes (t₁ = –3.225 meV, t₂ = 2.120 e^{i2π/3} meV, t₃ = 0.947 meV) and long‑range Coulomb interactions parameterized by a relative dielectric constant ε_r (10–20) and a screened potential with a fixed screening length. On‑site repulsion is U = 1192.71/ε_r meV, while nearest‑, next‑nearest‑, and third‑nearest‑neighbor density‑density terms (V₁, V₂, V₃) are retained.

Using infinite‑density‑matrix‑renormalization‑group (iDMRG) on cylindrical geometries (X‑cylinders) with widths up to Ny = 6, the authors explore ground‑state properties across electron fillings ν (0 ≤ ν ≤ 1, corresponding to hole fillings –1 ≤ ν_h ≤ 0 after a particle‑hole transformation). Ferromagnetic order polarized along the spin‑z direction (FM_z) is identified by comparing energies in different S_z sectors; FM_z is robust for ν ≳ 0.2, matching experimental observations of spin polarization preceding correlated topological phases.

To probe Hall response, a charge‑pumping protocol is employed: a flux Φ_y = 2qπ is threaded along the periodic direction, and the transferred charge ΔQ is measured. The Hall conductivity follows σ_xy = ΔQ/q (in units of e²/h). The simulations reproduce integer Hall conductance σ_xy = ν for conventional Chern insulators (ν = 1) and fractional values σ_xy = ν for a series of FCIs (ν = 2/3, 4/7, 5/9, 4/9, 3/5). Notably, a robust ν = 1/3 FCI is predicted, awaiting experimental confirmation.

At ν = 1/3 the phase diagram as a function of ε_r displays a transition from a generalized Wigner crystal (GWC) at low ε_r to an FCI at higher ε_r. In the GWC, charge density concentrates on the A sublattice forming a triangular lattice, evident in both real‑space density maps and sharp peaks of the structure factor at the corresponding reciprocal vectors. Increasing ε_r suppresses this modulation, yielding a uniform density characteristic of the FCI; the charge‑charge correlation function g_nn(r) becomes featureless, and the structure factor loses Bragg peaks. The transition occurs sharply around ε_r ≈ 15.5.

Dynamic properties are investigated by computing the single‑particle spectral function A_loc(ω) at zero temperature. For ν = 2/3, 1, and 1/2, charge gaps of roughly 18.2 meV, 6.2 meV, and 8.4 meV are extracted, respectively. These gaps are significantly larger than the thermal activation energies measured experimentally, a discrepancy that the authors resolve by finite‑temperature iDMRG calculations. By evaluating A_loc(ω = 0) as a function of temperature, they identify an activation temperature T* ≈ 30 K for ν = 2/3 and 1, consistent with transport measurements. The ferromagnetic transition temperature T_c shows a non‑monotonic dependence on ν, reproducing the experimentally observed trend.

A striking theoretical prediction is the emergence of quantum anomalous Hall crystals (QAHCs). At several fractional fillings (ν = 1/2, 2/3, 3/5, 4/5, and an incommensurate ν ≈ 0.63) the system exhibits an integer Hall conductance σ_xy = 1 despite the fractional electron density. These QAHC states arise from charge‑density-wave order that folds the Brillouin zone, allowing a Chern number of one to persist while breaking time‑reversal, translational, and spin symmetries. The authors argue that by tuning ε_r or temperature, the charge order can melt, driving the QAHC into an FCI, offering a controllable pathway between distinct topological phases.

Overall, the work delivers a comprehensive, quantitative framework for tMoTe₂: it captures ferromagnetism, fractional Chern insulating behavior, charge‑ordered Wigner crystals, and predicts novel anomalous Hall crystals, all within a single realistic lattice model. The agreement with experimental charge gaps, activation temperatures, and magnetic transition trends validates the approach, while the prediction of QAHCs opens a new direction for experimental exploration of interaction‑driven topological order without external magnetic fields.