  We use Quantum Monte Carlo method employing stochastic-series-expansion
technique to study the ground state properties of the $t_2-V_1$ model on a
square lattice. We find that, away from half-fillings, the minimal combination
of nearest-neighbor repulsion $V_1$ and next-nearest-neighbor hopping $t_2$ may
give rise to checkerboard supersolidity. The nature of the quantum phase
transition, where the superfluid changes to a checkerboard supersolid, depends
on the relative strength of $V_1/t_2$ and the average site occupancy.
Interestingly, the model exhibits a mixed-order transition near half filling;
at a higher (lower) filling, tricriticality is witnessed followed by a
second-order transition at densities even further away from half filling. Close
to half filling, the model displays the extreme Thouless effect and transits
from a superfluid to a checkerboard solid.
