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 λ = π 2 . A one-to-many mapping of vertex onto dimer configurations allows the freefermion solutions to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by 45 • 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 TemperleyLieb algebra with loop fugacity β = 2 cos λ = 0. At the isotropic point, the exact solution allows for the explicit counting of 45 • rotated dimer configurations on a periodic M × 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 logarithmic conformal field theory with central charge c = −2, minimal conformal weight ∆ min = −1/8 and effective central charge c eff = 1.

Multipoint correlators in the Abelian sandpile model

“…Three well-studied, inequivalent lattice models have central charge c = −2 and would be natural candidates: the dense polymer model [32], the dimer model-at least for some of its features [30,31,34]-and the Abelian sandpile model [23]. All three models have features that are very close to the symplectic fermion theory.…”

Yang–Baxter integrable dimers on a strip

“…Although dimers was first solved many years ago, there remain a number of unanswered questions concerning the CFT description of the dimer model on the square lattice. In a previous paper [19], the dimer model on a cylinder, with 45 • rotated dimers, was solved exactly. Moreover, the modular invariant conformal partition function was obtained from finite-size corrections and shown to precisely agree with the modular invariant partition function of critical dense polymers.…”

“…Traditionally, it is asserted that dimers is described [18] by a c = 1 Gaussian free field. But without full access to the various sectors and boundary conditions on the strip, it is difficult to distinguish between a c = 1 theory and a c eff = 1 theory and a number of authors [8,9] have suggested that dimers is described by a logarithmic Conformal Field Theory (CFT) with c = −2.Recently, the dimer model was shown [19] to be Yang-Baxter integrable [20] by mapping [21,17,22] it onto the free-fermion six vertex model [23][24][25][26][27]. Notably, this maps six vertex configurations onto dimer configurations where the dimers are rotated by 45 • , as shown in Figures 2, 4 and 5, compared to their usual orientation parallel to the bonds of the square lattice.…”

“…The primary characterization of logarithmic CFTs is the appearance of nontrivial Jordan blocks in the Virasoro dilation operator L 0 . Indeed, for simple dimer boundary conditions on the strip, corresponding to the U q (sl(2))-invariant XX Hamiltonian H of the free-fermion six vertex model, the preliminary results of [19] indicate that H admits nontrivial Jordan blocks for finite systems. Since the appearance of these blocks is stable, as the system size increases, these blocks are expected to persist for large sizes and appear in the Virasoro dilation operator L 0 .In this paper, we solve exactly the anisotropic square lattice dimer model with the 45 • rotated orientation on the strip in sectors labelled by the magnetization S z of the related free-fermion six vertex model.…”

Quantum integrability and quantum groups: a special issue in memory of Petr P Kulish

“…The spectrum of integrable super-spin chains is considered in [36] for various boundary conditions. The two-dimensional dimer mode is studied in [37] through its correspondence to the free-fermion point of the six-vertex model.…”