Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands

Flexible long period moiré superlattices form in two-dimensional van der Waals crystals containing layers that differ slightly in lattice constant or orientation. In this Letter we show theoretically that isolated flat moiré bands described by generalized triangular lattice Hubbard models are present in twisted transition metal dichalcogenide heterobilayers. The hopping and interaction strength parameters of the Hubbard model can be tuned by varying the twist angle and the three-dimensional dielectric environment. When the flat moiré bands are partially filled, candidate many-body ground states at some special filling factors include spin-liquid states, quantum anomalous Hall insulators, and chiral $d$-wave superconductors.

Figure 1

(a) Schematic band structure of monolayer ${\mathrm{WSe}}_2$ with a large (small) spin splitting at valence (conduction) band extrema located at the $\pm K$ valleys. (b) $AA$ stacked $\mathrm{W}{X}_2/\mathrm{Mo}{X}_2$ bilayers with an additional in-plane displacement $\mathit{d}$, and a twist angle $\theta$. ${\mathit{a}}_1$ and ${\mathit{a}}_2$ are primitive translation vectors of $\mathrm{W}{X}_2$. (c) Dependence of the ${\mathrm{WSe}}_2$ valence band maximum $\mathrm{\Delta }(\mathit{d})$ on displacement $\mathit{d}$ in $AA$ stacked ${\mathrm{WSe}}_2/{\mathrm{MoSe}}_2$ with zero twist angle. $\mathrm{\Delta }(\mathit{d})$ has triangular lattice periodicity and one maximum per triangular lattice unit cell. (d) When a moiré pattern is formed the band maximum variation is magnified from the atomic scale to the moiré pattern scale. The color scales in (c) and (d) are identical and the orange dashed lines in (d) are near-neighbor links that connect $\mathrm{\Delta }(\mathit{r})$ maxima.

Figure 2

(a) Moiré bands at twist angle $\theta =2.0°$. The red dashed line is a tight-binding-model fit to the highest valence band that includes hopping up to the third nearest neighbor. (b) Density of states as a function of the hole filling factor $n_h/n_M$ (bottom) and the hole density $n_h$ (top). (c) The Wannier function $w(\mathit{r})$ associated with the highest-energy band in (a). $w(\mathit{r})$ is centered on one of the moiré potential maxima positions. The orange lines are the links of the moiré triangular lattice. (d) Hopping parameters $t_n$ vs neighbor number $n$ as a function of moiré period $a_M$ (bottom) and twist angle $\theta$ (top).

Figure 3

(a) Hubbard model repulsive interaction parameters $\epsilon U_n$, and (b) spin exchange interactions $J_n$ as a function of moiré period $a_M$ and twist angle $\theta$. The insets of (a) and (b), respectively, show the ratios $U_0/t_1$ and $J_2/J_1$. The semiconductor background dielectric constant $\epsilon$ was set to 10 for the evaluation of $U_n$. Smaller effective values of $\epsilon$ are applicable when the dielectric environment is engineered to maximize interaction strength. For example, $\epsilon$ is about 5 when hexagonal boron nitride is used as the dielectric layer [18].

Figure 4

(a) Energy contours in the moiré Brillouin zone for the highest energy band in Fig. 2. (b) Magnetic order on a triangular lattice with four-sublattice tetrahedral antiferromagnetic order. The magnetic moment directions on the four sublattices are specified by the corresponding arrows in (c).