We study the Mott metal-insulator transition in the two-band Hubbard model with different hopping amplitudes $t_1$ and $t_2$ for the two orbitals on the two-dimensional square lattice by using {it non-magnetic} variational wave functions, similarly to what has been considered in the limit of infinite dimensions by dynamical mean-field theory. We work out the phase diagram at half filling (i.e., two electrons per site) as a function of $R=t_2/t_1$ and the on-site Coulomb repulsion $U$, for two values of the Hunds coupling $J=0$ and $J/U=0.1$. Our results are in good agreement with previous dynamical mean-field theory calculations, demonstrating that the non-magnetic phase diagram is only slightly modified from infinite to two spatial dimensions. Three phases are present: a metallic one, for small values of $U$, where both orbitals are itinerant; a Mott insulator, for large values of $U$, where both orbitals are localized because of the Coulomb repulsion; and the so-called orbital-selective Mott insulator (OSMI), for small values of $R$ and intermediate $U$s, where one orbital is localized while the other one is still itinerant. The effect of the Hunds coupling is two-fold: on one side, it favors the full Mott phase over the OSMI; on the other side, it stabilizes the OSMI at larger values of $R$.
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