Charge-ordered states in twisted MoTe$_2$

Abstract

We analyze interaction-driven charge-density-wave (CDW) states in the spin-valley polarized first valence miniband of twisted MoTe$_2$ (tMoTe$_2$) using an adiabatic mapping from the continuum model to an effective Landau-level (LL) problem. When projected to the lowest LL, the leading spatial harmonic of the moiré-periodic potential changes sign at a magic twist angle $θ_c$ where the band reaches its minimum bandwidth. By solving self-consistent Hartree-Fock equations in a multi-LL Hilbert space, we find that triangular-lattice CDW states with density maxima on MX (or XM) sites or on MM sites are favored on opposite sides of the magic angle at most filling factors and that stripe order appears near $ν_h=1/2$. We show that CDW states at $ν_h >1/2$ can carry a nonzero total Chern number, providing a natural route to reentrant integer quantum Hall effects and discuss the energy competition between fractional Chern insulator and CDW states.

Figures (4)

• Figure 1: Flat bands, real-space structure and effective moiré potential of tMoTe$_2$. (a) Bandwidth (meV) versus twist angle of the Adiabatic model (first-shell approximation) and the Continuum model for tMoTe$_2$. (b) Schematic of the moiré unit cell illustrating MX, XM and MM sites, and $\sqrt{3}\times \sqrt{3}$ and $2\times 2$ commensurate supercells. (c) Amplitude of the leading (first-shell) harmonic of the effective moiré potential $V_1(\theta)$ versus twist angle; $V_1$ changes sign at $\theta_c\simeq3.7^{\circ}$, switching the potential minima from MX (or XM, respectively) on the unit cell edges to MM at the unit-cell center. (d) The Landau-level mixing parameter, $\kappa \equiv [e^2/\varepsilon\,\ell_B]/(\hbar\omega_c)$, versus twist angle for $\varepsilon = 10.0$. In (a,d), the vertical gray line marks $\theta_c\simeq3.7^\circ$. (e) Illustration of how the valence-band holes tend to localize on the MX or XM sites for $\theta<\theta_c$ and on the MM sites for $\theta>\theta_c$ in tMoTe$_2$. We use model parameters from Ref. wang2024fractional.
• Figure 2: Competing charge-ordered states over a range of hole fillings and twist angles. (a) Total energy per hole $E_{\textrm{tot}}$ versus twist angle $\theta$ for competing phases at fillings $\nu_h= 1/4,\, 1/3,\, 1/2,\, 2/3,\, 3/4$. The line style indicates the rotational symmetry of the corresponding state ($C_6$, $C_3$, $C_2$); first-order transition angles are marked by square markers. Numbered markers label configurations whose real-space densities are shown in (b). (b) Real-space hole density $\rho_h(\bm{r})$ for the six labeled configurations in (a). White hexagons mark the moiré unit cell and red hexagons mark the CDW unit cell. Single-particle energies in (a) are relative to the twist-angle-dependent moiré band energy at $\nu_h=0$.
• Figure 3: (a) Lowest-Landau-level weight $w_{\mathrm{LLL}}$ of the self-consistent CDW solutions versus twist angle $\theta$ (see SM SM for the definition of $w_{\mathrm{LLL}}$). The light-blue band shows the range across fillings $\nu_h\in\{1/4,1/3,1/2,2/3,3/4\}$ as a function of $\theta$. Representative curves are overlaid: color encodes filling $\nu_h$, and line style encodes symmetry ($C_6$, $C_3$, $C_2$). The light-gray curve traces the minimum $w_{\mathrm{LLL}}^{\mathrm{min}}(\theta)$ across all datasets. (b) Total energy per hole versus filling at representative twist angles $\theta = 3.0^\circ$, $3.7^\circ$, and $4.2^\circ$; shaded bands cover the energy range over $\theta\in[2.5^\circ,4.5^\circ]$. Marker style indicates when $C_6$ symmetry is broken. The curve labeled "$0$LL" shows the interaction energy obtained by projection to the LLL (no Landau-level mixing) for comparison MacDonald_Murray_1985. Inset: schematic energetic competition between CDW and FCI states at $\nu_h=2/3$ and $3/5$, analogous to Landau-level systems lam_liquid-solid_1984levesque_crystallization_1984.
• Figure S1: Total ground-state energy for dielectric constant values $\varepsilon=15,10,5$ (first, second, and third columns from the left) in units of $e^2/(4\pi \varepsilon \varepsilon_0 \ell_B)$ for fillings (a) $3/4$, (b) $1/3$, (c) $2/3$, and (d) $1/4$.