We study the interplay between topology and electron-electron interactions in the moiré ${\mathrm{MoTe}}_2/{\mathrm{WSe}}_2$ 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 nontrivial topology after the application of perpendicular electric field (displacement field). The model is supplemented by onsite and intersite 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.

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• Edge states
• Flat bands
• Hall effect
• Kane-Mele model
• Magnetic order
• Spin-orbit coupling
• Topological insulators
• Topological materials
• Valley degrees of freedom