Interplay between topology and electron-electron interactions in the moiré MoTe$_{\mathrm{2}}$/WSe$_{\mathrm{2}}$ heterobilayer

TL;DR

This work addresses how topology and strong electron correlations emerge and interact in the moiré MoTe$_2$/WSe$_2$ heterobilayer under a perpendicular displacement field. It develops an effective two-band extended Hubbard model with complex hoppings and Ising SOC, and analyzes it using Hartree-Fock and Gutzwiller approaches, focusing on the one-hole-per-moiré-unit-cell regime ($n_{tot}=3$). The results reveal a progression from a $120^{\circ}$ in-plane antiferromagnetic insulator to a quantum anomalous Hall insulator with $|C|=1$, and finally to a ferrimagnetic metal as the field increases, with holes distributed across both layers. Inter-site Coulomb terms modulate phase boundaries and can induce charge-density-wave tendencies at fractional fillings, with qualitative agreement to experimental observations; topology is characterized via non-Abelian Berry curvature analyses and Chern-number calculations, including a second-Chern-number framework in the folded Brillouin zone.

Abstract

We study, the interplay between topology and electron-electron interactions in the moiré MoTe\(_2\)/WSe\(_2\) heterobilayer. In our analysis we apply an effective two-band model with complex hoppings that incorporates the Ising-type spin-orbit coupling and lead to a non-trivial topology after the application of perpendicular electric field (displacement field). The model is supplemented by on-site and inter-site Coulomb repulsion terms and treated by both Hartree-Fock and Gutzwiller methods. According to our analysis, for the case of one hole per moiré unit cell, the system undergoes two phase transitions with increasing displacement field. The first one is from an in-plane 120$^\circ$ antiferromagnetic charge transfer insulator to a topological insulator. At the second transition, the system becomes topologically trivial and an out-of-plane ferrimagnetic metallic phase becomes stable. In the topological region a spontaneous spin-polarization appears and the holes are distributed in both layers. Additionally, we analyze the influence of the intersite Coulomb repulsion terms on the appearance of the topological phase as well as on the formation of the charge density wave state. We discuss the obtained results in the context of available experimental data.

Figures (10)

• Figure 1: The bare band structure defined by Eq. (\ref{['ham1']}) for two selected values of the displacement field: (a) $V=0$ eV and (b) $V=0.04$ eV. The colored scale indicates the contribution of the MoTe$_2$ and WSe$_2$ states to the resulting bands. Note that by increasing the displacement field one pushes the bottom band into the upper one, initiating band inversion.
• Figure 2: (a) The two-sublattices of the resulting honeycomb lattice are marked in red (blue) and correspond to the MoTe$_2$ (WSe$_2$) layers. The schematic representation of the $120^{\circ}$ degree AF alignment at the MoTe$_2$ layer is provided by arrows. (b) In order to take into account the possible $120^{\circ}$ degree AF alignment in both layers we consider a supercell containing 3 lattice sites per each layer (marked by $A,B$, and $C$).
• Figure 3: (a) The in-plane magnetization corresponding to 120$^\circ$ AF ordering at the MoTe$_2$ layer ($S^{xy}_M$), together with the out-of-plane magnetizations in both layers ($S^z_M$ and $-S^z_W$), all as functions of the displacement field. The corresponding in-plane magnetization at the WSe$_2$ layer is zero in the whole considered range of $V$. The sequence of states that appear with increasing $V$ is: antiferromagnetic charge transfer insulator (AFI); quantum anomalous Hall insulator (QAHI) with non-zero Chern number $C=-1$; ferrimagnetic metal (FMM). (b) The number of electrons per moiré lattice site in both layers as a function of displacement field. (c) The band gap between the WSe$_2$ band and the upper MoTe$_2$ subband. In (d), (e), and (f) we show the graphical representation of the magnetic ordering in the AFI, QAHI, and FMM phases. The blue and red surfaces correspond to the WSe$_2$ and MoTe$_2$ layers. The results have been obtained for $U/|t_1|=21$ and $n_{tot}=3$, which is equivalent to one hole per moiré unit cell.
• Figure 4: The band structure for the three representative values of the displacement field which correspond to the AF insulator (a,b), quantum anomalous Hall insulator (c,d), and ferrimagnetic metal (e,f). The colored scale in the first (second) column indicates the layer (spin) contribution to the resulting state. The results have been obtained for the same model parameters as in Fig. \ref{['mean_n']}.
• Figure 5: The out-of-plane magnetization (a) and in-plane magnetization (b) in the MoTe$_2$ layer as well as $\delta n=n_M-n_{WS}$ (c), all as functions of the displacement field ($V$) and onsite Coulomb repulsion ($U$) for half-filling of the upper band ($n_\text{tot}=3$). In (d) we show the out-of-plane magnetization for selected values of $U/|t_1|=10.7$ as a function of displacement field and total number of electrons.