Topological insulators in twisted transition metal dichalcogenide homobilayers

We show that moir'e bands of twisted homobilayers can be topologically nontrivial, and illustrate the tendency by studying valence band states in $\pm K$ valleys of twisted bilayer transition metal dichalcogenides, in particular, bilayer MoTe${}_2$. Because of the large spin-orbit splitting at the monolayer valence band maxima, the low energy valence states of the twisted bilayer MoTe${}_2$ at $+K$ ($-K$) valley can be described using a two-band model with a layer-pseudospin magnetic field $\mathbf{\Delta }(\mathit{r})$ that has the moir'e period. We show that $\mathbf{\Delta }(\mathit{r})$ has a topologically non-trivial skyrmion lattice texture in real space, and that the topmost moir'e valence bands provide a realization of the Kane-Mele quantum spin-Hall model, i.e., the two-dimensional time-reversal-invariant topological insulator. Because the bands narrow at small twist angles, a rich set of broken symmetry insulating states can occur at integer numbers of electrons per moir'e cell.