Exact solution of the dimer model on the generalized finite checkerboard lattice

We present the exact closed-form expression for the partition function of a dimer model on a generalized finite checkerboard rectangular lattice under periodic boundary conditions. We investigate three different sets of dimer weights, each with different critical behaviors. We then consider different limits for the model on the three lattices. In one limit, the model for each of the three lattices is reduced to the dimer model on a rectangular lattice, which belongs to the $c=-2$ universality class. In another limit, two of the lattices reduce to the anisotropic Kasteleyn model on a honeycomb lattice, the universality class of which is given by $c=1$. The result that the dimer model on a generalized checkerboard rectangular lattice can manifest different critical behaviors is consistent with early studies in the thermodynamic limit and also provides insight into corrections to scaling arising from the finite-size versions of the model.

Figure 1

The unit cell for dimer models on the checkerboard lattices A, B, and C in solid lines. The choice of arrows satisfies Kasteleyn's clockwise-odd theorem [43].

Figure 2

$K_1$ and $K_2$ dimer model on the brick lattice, which is topologically equivalent to the honeycomb lattice.