Higher Chern bands in helical homotrilayer transition metal dichalcogenides

TL;DR

This work presents helically twisted TMD trilayers as a route to higher-Chern-number bands near charge neutrality. By exploiting a clear separation of moiré scales at small twist angles, the authors derive a local continuum model and identify parameter windows where the topmost valence band in the K valley carries Chern number C = -2, tunable to C = -1 or 0 via displacement fields. An effective three-orbital lattice model is constructed through Wannierization, clarifying the Chern sequences across twist-angle regimes and enabling a tight-binding perspective that captures the continuum results. Hartree-Fock analyses at ν = -1 demonstrate the stability of the |C| = 2 band against interactions in certain regimes, suggesting rich correlated topological physics in these systems. The framework lays the groundwork for experimental realization and exploration of interacting topological phases beyond the conventional quantum Hall paradigm.

Abstract

We propose helically twisted homotrilayer transition metal dichalcogenides as a platform for realizing correlated topological phases of matter with higher and tunable Chern numbers. We show that a clear separation of scales emerges for small twist angles, allowing us to derive a low-energy continuum model that captures the physics within moiré-scale domains. We identify regimes of twist angle and displacement field for which the highest-lying hole band is isolated from other bands and is topological with $K$-valley Chern number $C=-2$. We demonstrate that varying the displacement field can induce a transition from $C=-2$ to $C=-1$, as well as from a topologically trivial band to a $C=-1$ band. We derive an effective tight-binding description for a high-symmetry stacking domain which is valid for a wide range of twist angles, and we show that the $C=-2$ band can remain stable at filling fraction $ν=-1$ in the presence of interactions in Hartree-Fock calculations.

Figures (19)

• Figure 1: (a) Schematic of the helical trilayer TMD in real space depicting the relative twist angles between the top (green), middle (black), and bottom (yellow) layers. Filled circles indicate a metal ($M$) atom, while open circles indicate a chalcogen ($X$) atom. (b) Overlapping hexagons on the left depict the rotated Brillouin zones of the three layers, with $\mathbf{K}_m$ marking the momentum of the middle layer's $K$ valley. Right zoom-in illustrates the emergent moiré Brillouin zone (mBZ) arising from valley $K$ of the monolayer Brillouin zone. (c) Phase diagram for the continuum model for the $XMX$-stacked helical trilayer TMD, with $(V, \psi, w, m^*) = (16.5\text{ meV}, -100^\circ, -18.8\text{ meV}, 0.60m_e)$. For twist angles $\theta \gtrsim 2.3^\circ$, the top three valence bands are gapped from the other remote valence bands. There are three regimes (denoted $I, \,II, \, III$) of Chern sequences. When $5.1^\circ \lesssim \theta \lesssim 6.8^\circ$, the top valence band (in green) has $K$-valley Chern number $C=-2$. The representative band structures are computed for twist angles $\theta = 3.7^\circ, 5.6^\circ, 7.2^\circ$. Each band structure is labeled with the Chern numbers and $\mathcal{C}_{3z}$ eigenvalues at the high-symmetry points of the three topmost valence bands.
• Figure 2: Left column: Local moiré-scale stacking configurations created by the $tm$ (green) and $mb$ (black) moiré patterns. For each moiré pattern, the solid dots at the vertices indicate $\mathscr{R}^M_M$ stacking, the shaded triangles indicate areas of $\mathscr{R}^X_M$ stacking, and the unshaded triangles indicate areas of $\mathscr{R}^M_X$ stacking. The rows correspond to the three high-symmetry stacking configurations $MMM$, $MXM$, and $XMX$. Right column: Corresponding atomic-scale high-symmetry configurations created by the top (green), middle (black), and bottom (yellow) layers. The filled circles represent a transition metal ($M$) atom, while the open circles represent a chalcogen ($X$) atom.
• Figure 3: (a) Chern number phase diagram for the topmost valence band of the $XMX$-stacked helical TMD homotrilayer at twist angle $\theta = 5.6^\circ$ as a function of varying continuum model parameters $V/w$ vs. $\psi$ with fixed $w = -18.8\text{ meV}$ and $m^* = 0.60m_e$. Diamond markers indicate the parameter values obtained via DFT for homobilayer tMoTe$_2$ at $\theta = 3.89^\circ$ in Refs. Jia-PhysRevB.109.205121wang2024fractional (denoted Refs. A & B) with $m^* = 0.60m_e$, and $\theta = 4.4^\circ$ in Ref. reddy2023 (denoted Ref. C) with $m^* = 0.62m_e$, as well as for three high-symmetry displacements for the untwisted MoTe$_2$ homobilayer in Ref. wu2019topological (denoted Ref. D) with $m^* = 0.62m_e$. The orange star indicates the values we use throughout this paper: $(V, \psi, w, m^*) = (16.5\text{ meV}, -100^\circ, -18.8\text{ meV}, 0.60m_e)$. Each of these sets of parameters lie within the $C = -2$ region. (b)$K$-valley Chern number phase diagram of the topmost valence band for $XMX$ stacking as a function of varying twist angle and applied displacement field $D$. As $D$ increases, there is a region of $C = -1$ that emerges for a range of twist angles in which the topmost band has $C = 0$ or $-2$ when $D = 0$. When $D$ is sufficiently large, the topmost band is trivialized at all $\theta$.
• Figure 4: Representative band structures labeled with Chern numbers and $\mathcal{C}_{3z}$ eigenvalues at the high-symmetry points after the application of a displacement field of $D = 8\,\text{meV}$. The band inversion at $K$ occurs at a lower angle ($\theta \simeq 4.9^\circ$) than the inversion at $K'$ ($\theta \simeq 5.5^\circ$). Labels $I$ and $II$ indicate band structures with identical Chern number sequences as in Regimes $I$ and $II$ in Fig. \ref{['fig:helicalgeometry']}(c).
• Figure 5: Squared modulus of Wannier orbitals centered at $\alpha$, $W^\alpha(\mathbf{r})$, for each layer $\ell \in \{t,m,b\}$ at $\theta = 3^\circ$ plotted on a logarithmic color scale in real space, with a cut-off of $10^{-6}$. Each plot provides the weight $\mathcal{W}_\ell^\alpha$ as defined in Eq. \ref{['eq:wannierorbital_weights']}. The maroon, black, and yellow dots mark the $\alpha \in \{H_1, T, H_2 \}$ sites, respectively. Real space coordinates are measured in units of the moiré period $a_M$.