  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.
