Exact Solution and Correlations of a Quantum Dimer Model on the Checkerboard Lattice

For decades, constrained models with dimer degrees of freedom have been a powerhouse of statistical physics and various branches of theoretical and mathematical physics, in particular aspects of quantum magnetism. For analytic purposes, their utility is usually limited to models defined on planar lattice graphs, where, thanks to a powerful theorem by Kasteleyn, many questions can be answered exactly. Here we present analytic results on a special dimer model on a {\it nonplanar} checkerboard lattice of interacting dimers, which does not permit for parallel dimers to surround diagonal links. We report exact results on the enumeration of closed packed dimer coverings on finite checkerboard lattices under periodic boundary conditions. Furthermore, we comment on the behavior of the dimer-dimer correlations and find that the correlations between any two dimers vanish identically if their distance is larger than two unit cells. Connections with $\mathbb{Z}_{2}$ gauge theory, known from planar models, are extended to the present case.