Cross-dimensional valley excitons from F\"{o}rster coupling in arbitrarily twisted stacks of monolayer semiconductors

In stacks of transition metal dichalcogenide monolayers with arbitrary twisting angles, we explore a new class of bright excitons arising from the pronounced F\"{o}rster coupling, whose dimensionality is tuned by its in-plane momentum. The low energy sector at small momenta is two-dimensional, featuring a Mexican Hat dispersion, while the high energy sector at larger momenta becomes three-dimensional (3D) with sizable group velocity both in-plane and out-of-plane. By choices of the spacer thickness, interface exciton mode strongly localized at designated layers can emerge out of the cross-dimensional bulk dispersion for a topological origin. Step-edges in spacers can be exploited for engineering lateral interfaces to enable interlayer communication of the topological interface exciton. Combined with the polarization selection rule inherited from the monolayer building block, these exotic exciton properties open up new opportunities for multilayer design towards 3D integration of valley exciton optoelectronics.