Twisted bilayers of transition metal dichalcogenides (TMD) have emerged as a novel platform for the realisation of strongly correlated phases of matter. Among other reasons, there is a great interest in TMDs because they are good candidates to host the so long sought excitonic insulating phase, that is stabilized by condensation of excitons, i.e. bound pairs of electrons and holes. The forbidden hybridization between two different AB stacked homobilayers could facilitate the condensation of inter-layer excitons, that could explain the insulating phases observed in twisted homobilayers of WSe2. Also a Curie-Weiss 1/T behavior of the magnetic susceptibility has been reported for this material pointing to the formation of local moments, a hallmark of strong correlations. However, the precise nature of such an insulating phase has not yet been determined for the excitonic order could coexist with a charge-transfer insulator, Mott insulator or it could lead to a supersolid phase.Here we present a study of the SU(4) Heisneberg and Hubbard models in the triangular lattice, that should capture the salient features of AB-stacked TMD homo-bilayers, using different many-body theoretical techniques, such as spin-wave theory and Dynamical Mean Field Theory (DMFT).We show how the system in presence of a finite layer imbalance, that is experimentally obtained using a bias voltage between the two layers, naturally hosts multi-Q order states which are inter-layer excitonic insulators with a spin texture which modulation (i.e. Q-vector) is layer resolved. Furthemore, the order phase passes from commensurate to incommensurate at a critical value of the inter-layer polarisation which marks a quantum critciality.We studied also the metal-to-insulator transition occuring in the SU(4) Hubbard model in a broken symmetric phase which is determined by the three M points in the Brillouin zone, and we sketched a phase diagram as a function of the Coulomb repulsion and filling.
