Lattice relaxation, electronic structure and continuum model for twisted bilayer MoTe$_2$

We investigate the lattice relaxation effect on moir\'e band structures in twisted bilayer MoTe$_2$ with two approaches: (a) large-scale plane-wave basis first principle calculation down to $2.88^{\circ}$, (b) transfer learning structure relaxation + local-basis first principles calculation down to $1.1^{\circ}$. Two types of van der Waals corrections have been examined: the D2 method of Grimme and the density-dependent energy correction. We note the density-dependent energy correction yields a continuous evolution of bandwidth with twist angles. Including second harmonic of intralayer potential/interlayer tunneling and the strain induced gauge field, we develop a more complete continuum model with a single set of parameters for a wide range of twist angles, providing a useful starting point for many body simulation.