A model of a mixture of spinless fermions and spin-zero hardcore bosons, with filling fractions $rho_F$ and $rho_B$, respectively, on a two-dimensional square lattice with {em composite} hopping $t$ is presented. In this model, hopping swaps the locations of a fermion and a boson at nearest-neighbor sites. When $rho_F+rho_B=1$, the fermion hopping amplitude $phi$ and boson superfluid amplitude $psi$ are calculated in the ground state within a mean-field approximation. The Fermi sector is insulating ($phi=0$) and the Bose sector is normal ($psi=0$) for $0 le rho_F < rho_c$. The model has {em coupled first-order} transitions at $rho_F = rho_c simeq 0.3$ where both $phi$ and $psi$ are discontinuous. The Fermi sector is metallic ($phi>0$) and the Bose sector is superfluid ($psi>0$) for $rho_c < rho_F < 1$. At $rho_F=1/2$, fermion density of states $rho$ has a van Hove singularity, the bulk modulus $kappa$ displays a cusp-like singularity, the system has a density wave (DW) order, and $phi$ and $psi$ are maximum. At $rho_F=rho_{kappa} simeq 0.81$, $kappa$ vanishes, becoming {em negative} for $rho_{kappa}<rho_F<1$. The role of composite hopping in the evolution of Fermi band dispersions and Fermi surfaces as a function of $rho_F$ is highlighted. The estimate for BEC critical temperature is in the subkelvin range for ultracold atom systems and several hundred kelvins for possible solid-state examples of the model.
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