Hartree-Fock study of the moiré Hubbard model for twisted bilayer transition metal dichalcogenides

Twisted bilayer transition metal dichalcogenides have emerged as important model systems for the investigation of correlated electron physics because their interaction strength, carrier concentration, band structure, and inversion symmetry breaking are controllable by device fabrication, twist angle, and, most importantly, gate voltage, which can be varied in situ. The low-energy physics of some of these materials has been shown to be described by a “moiré Hubbard model” generalized from the usual Hubbard model by the addition of strong, tunable spin-orbit coupling and inversion symmetry breaking. In this work, we use a Hartree-Fock approximation to reach a comprehensive understanding of the moiré Hubbard model on the mean-field level. We determine the magnetic and metal-insulator phase diagrams, and assess the effects of spin-orbit coupling, inversion symmetry breaking, and the tunable van Hove singularity. We also consider the spin and orbital effects of applied magnetic fields. This work provides guidance for experiments and sets the stage for beyond-mean-field calculations.

Figure 1

(a) Brillouin zones of the top (solid line) and bottom (dashed line) layer components of a twisted $\mathrm{W}{\mathrm{Se}}_2$ bilayer. Blue ${\mathrm{K}}_{0,\uparrow }$ (red ${\mathrm{K}}_{0,\downarrow }^{\prime }$) represents one valley with spin-up (spin-down) band at the valence-band edges. Small hexagons indicate moiré Brillouin zones. (b) An illustrative band structure based on the continuum model of ${\mathrm{tWSe}}_2$. We highlight the topmost valence bands that can be matched to the Hubbard model. Blue solid (dashed) arrows represent the dominant spin of the top (bottom) layer at the ${\vec{K}}_0$ valley. Green dotted lines indicate the energy level of half filling of the topmost valence bands.

Figure 2

(a) Sketch of the phase ${\varphi }_{i,j}$ between a given site $i$ and its neighbor site $j$ on a triangular lattice, which is chosen based on symmetry. (b) Possible magnetic ordering patterns. For the tetrahedral order, the magnetic order on site $i$ is defined as $\frac{S}{\sqrt{3}}(cos{\vec{Q}}_0·{\vec{R}}_i,cos{\vec{Q}}_1·{\vec{R}}_i,cos{\vec{Q}}_{-1}·{\vec{R}}_i)$, with ${\vec{Q}}_0=(2\pi ,0), {\vec{Q}}_{\pm 1}=(-\pi ,\pm \sqrt{3}\pi )$. The magnetic moment directions on the lattice are specified by the corresponding arrows shown in the tetrahedral sketch. For ${120}^{\circ }$ order, there are two chiralities. “$+$” and “$-$” are defined by the sign of $\kappa =\frac{2}{3\sqrt{3}}({\vec{S}}_1\times {\vec{S}}_2+{\vec{S}}_2\times {\vec{S}}_3+{\vec{S}}_3\times {\vec{S}}_1)·{\widehat{\vec{e}}}_z$. We only draw the basic patterns; others can be generated by applying appropriate symmetry operations.

Figure 3

Phase diagram at half filling. The solid black line marks the transition to the indicated magnetic order. The dashed black line marks the opening of the gap. The energy gap is calculated as the energy difference $\mathrm{\Delta }E$ between the highest filled electron state and the lowest unfilled electron state. The gap opening position is defined as the position where $\mathrm{\Delta }E>0.01$. The red arrow shows the parameter space trajectory followed when the displacement field is increased in experiments, which changes $\varphi$ from 0 to $\sim \pm \pi /3$ and increases $t$, thus decreasing $U/t$.

Figure 4

Magnetization $m$ and energy gap of ${120}^{\circ }$ spiral order (120-xy-2) at $\varphi =0$ and $\pi /6$ from Hartree-Fock calculation. The transition is found to be two staged for almost all values of $\varphi$ (a magnetic transition followed by a metal-insulator transition). At $\varphi =\pi /6$, the magnetic transition coincides with the metal-insulator transition.

Figure 5

Energy bands of 120-xy-2 at $\varphi =0$ and $\pi /6$ at an intermediate interaction $U$. The band is plotted along lines in the folded zone of the ${120}^{\circ }$ three-sublattice spiral order, which is one-third of the original moiré Brillouin zone. The green zone shows the hole pocket of the lower band and the yellow zone shows the electron pockets of the upper band at $\varphi =0$. In the left panel, the dashed blue line indicates the chemical potential at half filling. In the right panel, any chemical potential in the band gap corresponds to half filling.

Figure 6

(a) Phase diagram, (b) energy gap, and (c) magnetization at half filling for $0\le \varphi \le \pi /2$ for several $U$ values. (a) In the phase diagram, gray represents the paramagnetism. The intensity of the blue and orange indicates the canting angle $\theta$ of the magnetic order. Dark orange and dark blue represent the 120-xy-2 order and ferro-xy, correspondingly, and the white regions have no xy moment. (b, c) In the energy gap and magnetic order plots, the colors represent the corresponding DM phase $\varphi$. For $g\approx 10$ in ${\mathrm{tWSe}}_2$ with bandwidth around $100\text{meV}\approx 10t, g{\mu }_{B}B/t=1$ corresponds to $B\approx 17$ T.

Figure 7

(a) Density of states versus filling calculated at $U=0$. Colors represent different choices of $\varphi$. (b) Dispersion of spin-up electrons in the moiré Brillouin zone calculated at $U=0$. The dashed white line shows the constant energy surface that intersects the energy saddle point. The labeled point in each plot indicates one of the van Hove locations.

Figure 8

An illustrative sketch of occupied states. Blue (red) area represents the occupied Brillouin zone of spin up (down). Blue (red) dots are the van Hove locations of spin up (down). The arrow indicates the wave vector that connects the perfect nesting electrons between the same spins (${\vec{Q}}_{0,\pm 1}$) and between spin up and spin down (${\vec{Q}}_{0,\pm 1}^{\varphi }$).

Figure 9

(a) $n_{vHs}\text{−}\varphi$ curve for $0<\varphi <\frac{\pi }{2}$ at $U=0$. The blue points are extracted from the numerical density-of-states calculation. $n_{vHs}$ is the density filling where the Fermi surface intersects with the van Hove singularity. The orange line is an empirical formula $n_{vHs}\approx cos\left(3\varphi \right)/2+1$ that fits the numerical calculation well. (b) Sketch of the predicted phase diagram with only nearest-neighbor hopping in the weak-coupling limit.

Figure 10

Hartree-Fock phase diagram at general fillings at weak and strong couplings with nine commensurate orders considered. Each color represents a different magnetic order. “xy” indicates that the magnetic order is in the $x\text{−}y$ plane, and “z” represents the $z$ direction. Regions filled by more than one color are viewed as degenerate regions, where the energy difference between the two magnetic orders is smaller than ${10}^{-3}$ from numerical calculations.

Figure 11

Band structure of the continuum model of ${\mathrm{tWSe}}_2$ with and without the displacement field $D$ and the high-order term $\mathcal{C}|\vec{k}-{\vec{K}}_0{(}^{\prime }){|}^{3}cos\left(3{\alpha }_{\vec{k}}\right)$. Blue and red solid lines represent ${\vec{K}}_0$ and ${\vec{K}}_0^{\prime }$ valleys. Green dashed lines indicate the energy level of half filling of the topmost valence bands.