Tetramers with vacancies on the square lattice

Two-dimensional dimer models were introduced in statistical mechanics in the 1930s to study the adsorption of diatomic molecules on a surface [1]. In its basic formulation, a dimer on a two-dimensional lattice is an object that lives on a bond b between two nearest-neighbour sites with some associated activity zb = exp−εb/kT where εb may be thought of as the adsorption energy of the dimer. The general monomer-dimer problem, which allows dimers of some finite density to coexist with unoccupied sites on the lattice, has not been solved exactly, although some rigorous results on the nature of these systems, as well as numerical, series expansion and Monte Carlo studies, are available in the literature (see [2] for a review). In comparison, the problem of completely covering a lattice with dimers so that every site has exactly one dimer residing on it has been solved exactly for a variety of two-dimensional lattices. Henceforth, we will refer to this specific situation as the close-packed dimer model. Interest in the dimer problem was renewed by the introduction of the quantum dimer model (QDM) [3], and with the recent work on resonating valence bond states (cite) in models of magnetism. In the limit when kinetic terms vanish, the QDM reduces to a classical dimer model with interactions that favour neighbouring dimers being aligned parallel to each other. The focus of this paper will be the work by Alet et al on the interacting dimer model on the square lattice reported in [4] and [5], which overlaps with the study of doped QDMs by Papanikolau et al in [6]. In particular, several features of the model can be explained in the context of the two-dimensional Coulomb gas within the framework of renormalization group.