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Thermodynamic collapse in a lattice-gas model for a two-component system of penetrable particles

We study a lattice-gas model of penetrable particles on a square-lattice substrate with same-site and nearest-neighbor interactions. Penetrability implies that the number of particles occupying a single lattice site is unlimited and the model itself is intended as a simple representation of penetrable particles encountered in realistic soft-matter systems. Our specific focus is on a binary mixture, where particles of the same species repel and those of the opposite species attract each other. As a consequence of penetrability and the unlimited occupation of each site, the system exhibits thermodynamic collapse, which in simulations is manifested by an emergence of extremely dense clusters scattered throughout the system with energy of a cluster $E\propto -n^2$ where $n$ is the number of particles in a cluster. After transforming a particle system into a spin system, in the large density limit the Hamiltonian recovers a simple harmonic form, resulting in the discrete Gaussian model used in the past to model the roughening transition of interfaces. For finite densities, due to the presence of a non-harmonic term, the system is approximated using a variational Gaussian model.