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We consider an effective Hubbard model with spin- and direction-dependentcomplex hoppings $t$, applied to twisted homobilayer WSe$_2$ using avariational Monte Carlo approach. The electronic correlations are taken intoaccount by applying the Gutzwiller on-site correlator as well as long-rangeJastrow correlators subjected to noninteracting part being of Pfaffian form.Our analysis shows the emergence of Mott insulating state at critical value ofHubbard interaction $U_{c1}\approx 6.5|t|\div7|t|$ estimated by extrapolatingthe density-density equal-time two-particle Green's functions. The signaturesof an intermediate insulating phase between $U_{c1}$ and$U_{c2}\approx9.5|t|\div10|t|$ are also discussed. Furthermore, we report theformation of the $120^{\circ}$ in-plane N\'eel state indicated by the detailedanalysis of the spin-spin correlation functions. As shown, switching betweenantiferromagnetic phases characterized by opposite chirality could beexperimentally realized by the change of perpendicular electric field. Inproper range of electric fields also a transition to in-plane ferromagneticstate appears.

Variational Monte-Carlo Approach for Hubbard Model Applied to Twisted Bilayer WSe$_2$ at Half-Filling