Moiré flat Chern bands and correlated quantum anomalous Hall states generated by spin-orbit couplings in twisted homobilayer ${\mathrm{MoS}}_2$

We predict that in a twisted homobilayer of transition-metal dichalcogenide ${\mathrm{MoS}}_2$, spin-orbit coupling in the conduction band states from $\pm K$ valleys, can give rise to moiré flat bands with nonzero Chern numbers in each valley. The nontrivial band topology originates from a unique combination of angular twist and local mirror symmetry breaking in each individual layer, which results in unusual skyrmionic spin textures in momentum space with skyrmion number $\mathcal{S}=\pm 2$. Our Hartree-Fock analysis further suggests that density-density interactions generically drive the system at $1/2$-filling into a valley-polarized state, which realizes a correlated quantum anomalous Hall state with Chern number $\mathcal{C}=\pm 2$. Effects of displacement fields are discussed with comparison to nontrivial topology from layer-pseudospin magnetic fields.

Figure 1

Schematic crystal structure of twisted bilayer ${\mathrm{MoS}}_2$ with (a) top view and (b) side view. Electrons in different layers experience opposite effective electric fields due to local lack of mirror symmetry. (c) Moiré Brillouin zone (MBZ) formed by twisting hexagonal Brillouin zones of each layer. The in-plane spinor field exhibits opposite vorticities at $K_{m,+}$ and $K_{m,-}$ of MBZ due to layer-dependent Rashba effects.

Figure 2

Moiré bands of valley $\xi =+$ formed by CB states of twisted homobilayer ${\mathrm{MoS}}_2$ at ${\theta }_0=1.4^{\circ }$ with (a) ${\alpha }_{\mathrm{so}}=0$ and (b) ${\alpha }_{\mathrm{so}}\ne 0$. The color bars indicate the out-of-plane spin expectation value $\langle S_z\rangle$ in units of $\hslash /2$. (c) Momentum-space profile of Berry curvature ${\mathrm{\Omega }}_z$ in units of ${\AA }^2$ in the second moiré band in (b). Large ${\mathrm{\Omega }}_z$ of order ${10}^4{\AA }^2$ emerges as a result of the nontrivial gap induced by SOC. (d) Momentum-space spin texture of the second moiré band in (b). White arrows indicate the in-plane spinor field, which has opposite vorticities at $K_{m,+}$ and $K_{m,-}$ and leads to a nontrivial skyrmion number $\mathcal{S}=+2$. Parameters used are presented in Table 1.

Figure 3

(a) $J_0\text{−}{J}_1$ phase diagram of the system with the second moiré band at $1/2$-filling. Under repulsive density-density interactions, only the $J_1\le J_0$ regime can be accessed [44], where the valley-polarized (VP) state is generically favored and the system becomes a correlated QAH insulator. (b) Evolutions of the order parameters ${\mathrm{\Delta }}_{+-}$ and ${\mathrm{\Delta }}_M\equiv {\mathrm{\Delta }}_{+}-{\mathrm{\Delta }}_{-}$ as the ratio $J_0/J_1$ is tuned along the line cut [green dashed line in (a)] across the phase boundary. A first-order transition happens at the critical point $J_0\simeq J_1$.

Figure 4

(a) CB Moiré bands for valley $\xi =+$ of twisted bilayer ${\mathrm{MoS}}_2$ at ${\theta }_0=1.4^{\circ }$ and $V_z=4$ meV. Under finite $V_z$, spin-up and spin-down bands still cross, and the avoided level crossing due to SOC result in a pair of bands with $\mathcal{C}=\pm 1$. (b) Topological phase diagram of CB moiré bands as a function of ${\theta }_0$ and $V_z$, with an upper boundary $V_z^{c2}$ of the entire topological regime. Yellow dashed line: Critical $V_z^{c1}$ above which nontrivial topology generated by layer pseudospin is destroyed. Values of $V_z^{c1}$ are similar to those found in VB moiré bands [29]. Area enclosed by the red dashed lines: Regime with level crossings between spin-up and spin-down CB moiré bands when ${\alpha }_{\mathrm{so}}=0$. Area in orange: Topological regime where layer pseudospin mechanism in Ref. [29] fails.