Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model

Abstract

It is shown that dimers is Yang-Baxter integrable as a six-vertex model at the free-fermion point with crossing parameter $\lambda=\tfrac{\pi}{2}$. A one-to-many mapping of vertex onto dimer configurations allows the free-fermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by 45 degrees compared to their usual orientation. This dimer model is exactly solvable in geometries of arbitrary finite size. In this paper, we establish and solve inversion identities for dimers with periodic boundary conditions on the cylinder. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal (logarithmic) degrees of freedom. In a suitable gauge, the dimer model is described by the Temperley-Lieb algebra with loop fugacity $\beta=2\cos\lambda=0$. At the isotropic point, the exact solution allows for the explicit counting of 45 degrees rotated dimer configurations on a periodic $M\times N$ rectangular lattice. We show that the modular invariant partition function on the torus is the same as symplectic fermions and critical dense polymers. We also show that nontrivial Jordan cells appear for the dimer Hamiltonian on the strip with vacuum boundary conditions. We therefore argue that, in the continuum scaling limit, the dimer model gives rise to a {\em logarithmic} conformal field theory with central charge c=-2, minimal conformal weight $\Delta_{\text{min}}=-1/8$ and effective central charge $c_{\text{eff}}=1$.