Emergent Quantum Valley Hall Insulator from Electron Interactions in Transition-Metal Dichalcogenide Heterobilayers

Abstract

We explore the emergence of topological phases in moiré MoTe$_2$/WSe$_2$ bilayer, highlighting the crucial role of spin-orbit coupling and Coulomb interactions at two holes per moiré unit cell $v = 2$. Our analysis uncovers robust Quantum Valley Hall Insulating (QVHI) phase and reveals that long-range interactions alone can mediate the interlayer electron tunneling, generating topologically nontrivial bands even in the absence of the corresponding single-particle hopping. Additionally, we show that in the case of band mixing terms originating both from the interaction and single particle physics a competition between topological states realizing $s$-$wave$ and $p\pm ip$-$wave$ symmetries can appear. Moreover, within the considered theoretical framework, we present that by introducing a small Zeeman field, one can lift the band inversion in one of the valleys. This leads to a Quantum Anomalous Hall Insulating (QAHI) state with the topological gap opening in a single valley and the other being topologically trivial.

Figures (5)

• Figure 1: (a) The direction dependent sign parameter $\nu_{ij}=\pm 1$ for the six NNN hoppings appearing in Eq. (\ref{['hopping_amp_intralayer']}) , (b) The $\eta_{ij}=0,1,2$ parameters for the three NN hoppings appearing in Eq. \ref{['hopping_amp_interlayer']}.
• Figure 2: (a),(b) The interlayer tunneling expectation value $\mathcal{P}_{\perp}$ as a function of the displacement field and Coulomb interaction for the special case of $t_{\perp}=0$. The band structures in (c), (d), and (e) correspond to the band insulator phase, the QVHI phase, and to the metallic behavior and have been obtained for displacement field values $D = 45$ meV, $D = 55$ meV, and $D = 70$ meV, respectively with the interaction strengths fixed at $V = 19$ meV. The results have been obtained for $U$ = 94.7 meV and $n=2$, which is equivalent to two holes per moiré unit cell.
• Figure 3: (a) The interlayer tunneling expectation value $\mathcal{P}_{\perp}$ as function of the displacement field ($D$) and intersite Coulomb interaction ($V$) for $t_{\perp}=0$. (b) The interlayer tunneling expectation value $\mathcal{P}_{\perp}$ as function of $V$ and $t_{\perp}$ for selected displacement field $D$ = 55 meV and $\phi_{\perp}=0$. (c) The phase diagram of the model at the ($t_{\perp}, V$)-plane for $D$ = 55 meV and $\phi_{\perp}=2\pi/3$. The two topological states seen in (c) correspond to a $p\pm ip$ and $s$-$wave$ symmetries of the gap.
• Figure 4: Absolute value of the topological gap ($|\varepsilon_{\mathbf{k}\sigma\bar{\sigma}}^{12}|$) as a function of momentum for pure $s$-$wave$ (a) and $p\pm ip$-$wave$ (b) symmetries. The magenta and green dots mark the positions of the Chern charges of the value $1$ and $-1$, respectively. (a) and (b) correspond to $t_{\perp}=0$ and $t_{\perp}=3\;$meV with $\phi_{\perp}=2\pi/3$. The remaining model parameters are $V=19$ meV, $D=55$ meV and $U=94.7$ meV.
• Figure 5: (a-d) Band structure along high symmetry directions for selected values of the Zeeman field provided in the figures and for $D$ = 73 meV, $U$ = 129 meV, $V$ = 24 meV. In (e) and (f) we show the $B_z$-dependence of interlayer mixing expectation values $|\mathcal{P}^{12}_{i\uparrow j\downarrow}|\equiv\mathcal{P}_{\perp}^{\uparrow\downarrow}$ and $|\mathcal{P}^{12}_{i\downarrow j\uparrow}|\equiv\mathcal{P}_{\perp}^{\downarrow\uparrow}$ as well as the spin magnetization $S_z$.