  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.
