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 $\ifmmode\pm\else\textpm\fi{}K$ valleys of twisted bilayer transition metal dichalcogenides, in particular, bilayer ${\mathrm{MoTe}}_{2}$. Because of the large spin-orbit splitting at the monolayer valence band maxima, the low energy valence states of the twisted bilayer ${\mathrm{MoTe}}_{2}$ at the $+K$ ($\ensuremath{-}K$) valley can be described using a two-band model with a layer-pseudospin magnetic field $\mathbf{\ensuremath{\Delta}}(\mathbit{r})$ that has the moir\'e period. We show that $\mathbf{\ensuremath{\Delta}}(\mathbit{r})$ has a topologically nontrivial 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.