Multiple Phase Transitions in Extended Hard Core Lattice Gas Models in Two Dimensions

We study the $k$-NN hard core lattice gas model in which the first $k$ next nearest neighbor sites of a particle are excluded from occupation by other particles on a two dimensional square lattice. This model is the lattice version of the hard disc system with increasing $k$ corresponding to decreasing lattice spacing. While the hard disc system is known to undergo a two step freezing process with increasing density, the lattice model has been known to show only one transition. Here, based on Monte Carlo simulations and high density expansions of the free energy and density, we argue that for $k=4,10,11,14...$, the lattice model undergoes multiple transitions with increasing density. Using Monte Carlo simulations, we confirm the same for $k=4,\ldots, 11$. This, in turn, resolves an existing puzzle as to why the 4-NN model has a continuous transition against the expectation of a first order transition.