lattice at finite hole density. Although mean-field theory is insufficient to describe two-dimensional critical systems, it is a useful tool to obtain qualitative features of the phase diagram as well as the behavior deep in the phases, away from criticality.

The details of this theory are presented in Appendix \[app:mean-field-details\]. We begin by constructing a Grassmann representation of the partition function for an interacting dimer model using the standard methods introduced by Samuel.[@samuel80; @Samuel] The resulting theory involves Grassmann (anti-commuting) variables residing on the sites of the square lattice. The action of the Grassmann integral is non-trivial and is parametrized by the hole fugacity $z$ and a coupling between dimers $V=z^2 (e^u-1)$, where $u=-2\ln w$.

Since the action of the resulting Grassmann path integral is not quadratic in the Grassmann variables, it cannot be reduced to the computation of a determinant. Thus, we use a standard mean-field approach which, in this case, involves the introduction of two Hubbard-Stratonovich (bosonic) fields $\phi_i$ and $\chi_{ij}$, defined on the sites and links of the square lattice respectively. Upon integrating out the Grassmann variables one obtains an effective theory for the fields $\phi_i$ and $\chi_{ij}$ which, as usual, is solved within a saddle-point expansion. 