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Mixed-order transition and tricritical point associated with checkerboard supersolidity in the two-dimensional $t_2-V_1$ model

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.