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

TL;DR

The paper develops a realistic moiré real-space Hubbard model for twisted MoTe$_2$ and applies state-of-the-art tensor-network methods to map the ground-state, finite-temperature, and dynamical properties of correlated topological phases at various fillings. It reveals a rich phase diagram including Chern insulators, fractional Chern insulators, quantum anomalous Hall crystals, and charge orders, with ferromagnetic order underpinning many of these states. Dynamical spectra show gapped charge excitations in FCIs and band-folded features in QAHCs, while finite-temperature analysis exposes multiple energy scales: a finite magnetic transition temperature $T_c$, a charge-activation temperature $T^*$, and zero-temperature charge gaps $\Delta_{\nu}$, aligning with and clarifying experimental observations. The work also predicts robust QAHC states at fractional fillings and highlights how band folding and hole-crystal mechanisms can stabilize integer Hall conductivities in fractional regimes, offering a comprehensive framework for correlated topological phases in MoTe$_2$-based moiré systems and related materials.

Abstract

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.

Figures (17)

• Figure 1: Real-space model on the honeycomb lattice and its global phase diagram. Panels (a) and (b) show real-space electronic density profile of exponentially localized Wannier orbitals, normalized to the peak density. $\rho^{(l)}_{i\uparrow} = | w^{(l)}_{i\uparrow} |^2$ for spin-$\uparrow$ is centered on site $i=(A,B)$ in layer $l=(b,t)$. The Wannier orbitals reside mostly at the honeycomb lattice sites, corresponding to the XM/MX stacking regions of the twisted bilayer MoTe$_2$. (c) Schematic representation of the model with hopping parameters ($t_1$, $t_2$, $t_3$) and interactions ($U$, $V_1$, $V_2$, $V_3$) up to the third nearest neighbor on a honeycomb lattice. $\boldsymbol{a}_1=(\sqrt{3}/2, 1/2)$ and $\boldsymbol{a}_2=(0,1)$ denote the primitive vectors of the honeycomb lattice. Numerical simulations employ a X-cylinder (XC) geometry with periodic boundary conditions along $\boldsymbol{a}_2$ direction. (d) Magnetic phase diagram in the $\epsilon_\mathrm{r}$-$\nu$ plane, where the dashed line (obtained with $N_y=5$ cylinder) and dash-dotted boundary lines ($N_y=4$ data) separate the FM$_z$ and non-FM$_z$ phases. (e) Phase diagrams with representative fillings ($\nu = 1, 2/3, 1/2$, and $1/3$) contains various phases, including the CI, FCI, QAHC, and CO phases. The phase diagram is constructed using width-6 ($N_y=6$) simulations, a system size that accommodates the root configurations of both FCIs and composite Fermi liquids Haldane1991statisticsScott2016The.
• Figure 2: GWC and FCI phases for $\nu=1/3$ filling. (a) Phase diagram containing the GWC and FCI states tuned by the relative dielectric constant $\epsilon_\mathrm{r}$. The two phases are characterized by $\rho({\boldsymbol{q}})$ and the Hall conductivity $\sigma_{xy}$, obtained on $N_y=5$ and $6$ cylinders.Charge distribution (b) $n(\boldsymbol{r})$ in real-space and (c) $\rho(\boldsymbol{q})$ in momentum space for the GWC state (with $\epsilon_\mathrm{r}=14$), showcasing the triangular lattice pattern of electron density modulation. (d) demonstrates $n(\boldsymbol{r})$ for the FCI state ($\epsilon_\mathrm{r}=18$), where a uniform charge distribution is observed. (e) Charge-charge correlation function $g^{nn}(\boldsymbol{r})$ in the FCI phase (with $\epsilon_\mathrm{r}=18$), displaying the charge distribution relative to the reference site $n_0$ marked by a black star. (f) Charge structure factor $S^A_n(\boldsymbol{q})$ in momentum space of the FCI state. (g) Charge pumping results $|\Delta Q|$ for various $\epsilon_\mathrm{r}$ for $N_y=6$, measured after inserting a flux $\Phi_y=6\pi$. The inset illustrates the charge pumping simulation process. For this and subsequent plots, the dashed hexagon [c.f., panels (c) and (f)] outlines the first Brillouin zones of the honeycomb lattice.
• Figure 3: Emergence of QAHC states for $\nu=2/3$ filling. (a) Phase diagram containing CO, FCI, and QAHC phases versus $\epsilon_\mathrm{r}$, characterized by $\rho(\boldsymbol{q})$ and $\sigma_{xy}$. (b) Charge density $n(\boldsymbol{r})$ and (c) hole density $h(\boldsymbol{r})$ distributions in the QAHC state obtained with $\epsilon_\mathrm{r}=18$ and on $N_y=6$ cylinder. As we define an average density of $n=0.5$ (per site) as the "full" filling $\nu=1$ (per unit cell) of the CI phase, the hole filling is thus $h(\boldsymbol{r}) =0.5-n(\boldsymbol{r})$, as depicted by green dots. The pink lines in panel (c) show the absolute values of $K^{ij}_1$ (see definition in the main text).
• Figure 4: FCI and QAHC for $\nu=1/2$ filling. (a) Charge pumping $|\Delta Q|$ after inserting flux $\Phi_y=4\pi$, for FCI and QAHC states with different $\epsilon_\mathrm{r}$ and $N_y$. (b) The quantized $\sigma_{xy}$ results obtained from charge pumping for various $\epsilon_\mathrm{r}$ and $N_y$. Panels (c,e) show $n(\boldsymbol{r})$ and $\rho(\boldsymbol{q})$, respectively, revealing non-uniform charge distribution for the FCI state obtained with $N_y=5$ and $\epsilon_\mathrm{r}=12$. Panels (d, f) show results of the QAHC state for $N_y=6$ and $\epsilon_\mathrm{r}=20$.
• Figure 5: Charge excitation spectra in CI, FCI and QAHC phases. The band-resolved spectral functions $A_{\rm uu}(\bold{k}, \omega)$ and $A_{\rm ll}(\bold{k}, \omega)$ simulated at zero temperature are shown for (a) CI state at $\nu = 1$, (b) FCI state at $\nu = 2/3$, and (c) the QAHC state at $\nu = 1/2$. The followed path in the reciprocal space is indicated in the inset of (a). As a comparison, the bare band dispersion (without interactions) is illustrated by the dashed lines in (a). The gray shadow in (c) represents the band folding with vector $\boldsymbol{q}_{\rm CO} = (0, -\pi)$. The peaks of the two folded bands are marked by dashed lines (serving as a guide for the eye). The frequency resolution is approximately 2.67 meV, as indicated by the range bar in (a).