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 vertices 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 ∘ 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 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 that of 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 Δ m i n = − 1 / 8 and effective central charge c e f f = 1 .
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 vertices 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 ∘ 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 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 that of 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 Δ m i n = − 1 / 8 and effective central charge c e f f = 1 .
Abstract The dimer model on a strip is considered as a Yang–Baxter integrable six vertex model at the free-fermion point with crossing parameter and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width N . In the continuum scaling limit, in sectors with magnetization S z , we obtain the conformal weights where . We further show that the corresponding finitized characters decompose into sums of q -Narayana numbers or, equivalently, skew q -binomials. 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 degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator L 0 . This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a logarithmic conformal field theory with central charge c = −2, minimal conformal weight and effective central charge . Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered c = −2 modules appear as submodules.
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity α that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N, this involves a non-diagonalizable braid operator J and an involution R = − (J3 − 12J)/16 = (−1)F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacity α = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler–Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...;s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity α that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N , this involves a non-diagonalizable braid operator J and an involution R = − ( J 3 − 12 J )/16 = (−1) F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacity α = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler–Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...; s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.
In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model via the so-called height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predicts particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms.
A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin–Kasteleyn random cluster models. Starting with a cylinder, the commuting periodic single-row transfer matrices are built from the periodic Temperley–Lieb algebra extended by the shift operators Ω±1. In this enlarged algebra, the non-contractible loop fugacity is α and the contractible loop fugacity is β. The torus is formed by gluing the top and bottom of the cylinder. This gives rise to a variety of non-contractible loops winding around the torus. Because of their nonlocal nature, the standard matrix trace does not produce the proper geometric torus. Instead, we introduce a modified matrix trace for this purpose. This is achieved by using a representation of the enlarged periodic Temperley–Lieb algebra with a parameter v that keeps track of the winding of defects on the cylinder. The transfer matrix representatives and their eigenvalues thus depend on v. The modified trace is constructed as a linear functional on planar connectivity diagrams in terms of matrix traces Trd (with a fixed number of defects d) and Chebyshev polynomials of the first kind. For critical dense polymers, where β=0, the transfer matrix eigenvalues are obtained by solving a functional equation in the form of an inversion identity. The solution depends on d and is subject to selection rules which we prove. Simplifications occur if all non-contractible loop fugacities are set to α=2 in which case the traces are evaluated at v=1. In the continuum scaling limit, the corresponding conformal torus partition function obtained from finite-size corrections agrees with the known modular invariant partition function of symplectic fermions.
In this present work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model ıt via the so-called height mapping which will be extensively used to interpret exact results. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predict particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms.