Effective K valley Hamiltonian for TMD bilayers under pressure and application to twisted bilayers with pressure-induced topological phase transitions

Motivated by recent studies on topologically non-trivial moiré bands in twisted bilayer transition metal dichalcogenides (TMDs), we study MoTe${}_2$ bilayer systems subject to pressure, which is applied perpendicular to the material surface. We start our investigation by first considering an untwisted bilayer system with an arbitrary relative shift between layers; a symmetry analysis for this case permits us to obtain a simplified effective low-energy Hamiltonian valid near the important $\mathbf{K}$ valley region of the Brillouin zone. Ab initio density functional theory (DFT) was then employed to obtain relaxed geometric structures for pressures within the range of 0.0 - 3.5 GPa and corresponding band structures. The DFT data were then fitted to the low-energy Hamiltonian to obtain a pressure-dependent Hamiltonian. We then apply our model to a twisted system by treating the twist as a position-dependent shift between layers - here, we assume rigid layers, which is a crucial simplification. In summary, this approach allowed us to obtain the explicit analytical expressions for a Hamiltonian that describes a twisted MoTe\textsubscript{2} bilayer under pressure. Our Hamiltonian then permitted us to study the impact of pressure on the band topology of the twisted system. As a result, we identified many pressure-induced topological phase transitions as indicated by changes in valley Chern numbers. Moreover, we found that pressure could be employed to flatten bands in some of the cases we considered.