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arXiv:0911.3003 · Journal of Physics A Mathematical and Theoretical

The \mathbb {Z}_2 staggered vertex model and its applications

Yacine Ikhlef, Jesper Lykke Jacobsen, Hubert Saleur

Published May 11, 2010• 35 citations
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Abstract

New solvable vertex models can be easily obtained by staggering the spectral parameter in already known ones. This simple construction reveals some surprises: for appropriate values of the staggering, highly non-trivial continuum limits can be obtained. The simplest case of staggering with period 2 (the case) for the six-vertex model was shown to be related, in one regime of the spectral parameter, to the critical antiferromagnetic Potts model on the square lattice, and has a non-compact continuum limit. Here we study the other regime: in the very anisotropic limit, it can be viewed as a zig–zag spin chain with spin anisotropy. From the Bethe–Ansatz solution, we obtain the central charge c = 2, the conformal spectrum and the continuum partition function, corresponding to one free boson and two Majorana fermions. Finally, moving in more physical territory, we obtain a massive integrable deformation of the model on the lattice. Interestingly, its scattering theory is a massive version of the one for the flow between minimal models. The corresponding field theory is argued to be a complex version of the C (2) 2 Toda theory.

DOI
10.1088/1751-8113/43/22/225201

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New solvable vertex models can be easily obtained by staggering the spectral parameter in already known ones. This simple construction reveals some surprises: for appropriate values of the staggering, highly non-trivial continuum limits can be obtained. The simplest case of staggering with period 2 (the case) for the six-vertex model was shown to be related, in one regime of the spectral parameter, to the critical antiferromagnetic Potts model on the square lattice, and has a non-compact continuum limit. Here we study the other regime: in the very anisotropic limit, it can be viewed as a zig–zag spin chain with spin anisotropy. From the Bethe–Ansatz solution, we obtain the central charge c = 2, the conformal spectrum and the continuum partition function, corresponding to one free boson and two Majorana fermions. Finally, moving in more physical territory, we obtain a massive integrable deformation of the model on the lattice. Interestingly, its scattering theory is a massive version of the one for the flow between minimal models. The corresponding field theory is argued to be a complex version of the C (2) 2 Toda theory.
Finite size spectrum of the staggered six-vertex model with Uq(sl \mathfrak{sl} sl(2))-invariant boundary conditions
Abstract The finite size spectrum of the critical ℤ2-staggered spin-1/2 XXZ model with quantum group invariant boundary conditions is studied. For a particular (self-dual) choice of the staggering the spectrum of conformal weights of this model has been recently been shown to have a continuous component, similar as in the model with periodic boundary conditions whose continuum limit has been found to be described in terms of the non-compact SU(2, ℝ)/U(1) Euclidean black hole conformal field theory (CFT). Here we show that the same is true for a range of the staggering parameter. In addition we find that levels from the discrete part of the spectrum of this CFT emerge as the anisotropy is varied. The finite size amplitudes of both the continuous and the discrete levels are related to the corresponding eigenvalues of a quasi-momentum operator which commutes with the Hamiltonian and the transfer matrix of the model.
Abstract Non-compact Conformal Field Theories (CFTs) are central to several aspects of string theory and condensed matter physics. They are characterised, in particular, by the appearance of a continuum of conformal dimensions. Surprisingly, such CFTs have been identified as the continuum limits of lattice models with a finite number of degrees of freedom per site. However, results have so far been restricted to the case of periodic boundary conditions, precluding the exploration via lattice models of aspects of non-compact boundary CFTs and the corresponding D-brane constructions. The present paper follows a series of previous works on a ℤ2-staggered XXZ spin chain, whose continuum limit is known to be a non-compact CFT related with the Euclidian black hole sigma model. By using the relationship of this spin chain with an integrable D22 {D}_2^2 D22​ D 2 2 vertex model, we here identify integrable boundary conditions that lead to a continuous spectrum of boundary exponents, and thus correspond to non-compact branes. In the context of the Potts model on a square lattice, they correspond to wired boundary conditions at the physical antiferromagnetic critical point. The relations with the boundary parafermion theories are discussed as well. We are also able to identify a boundary renormalisation group flow from the non-compact boundary conditions to the previously studied compact ones.
We study two Q -state Potts models coupled by the product of their energy operators, in the regime 2 < Q ⩽ 4 where the coupling is relevant. A particular choice of weights for the square lattice is shown to be equivalent to the integrable vertex model. It corresponds to a selfdual system of two antiferromagnetic Potts models, coupled ferromagnetically. We derive the Bethe ansatz equations and study them numerically for two arbitrary twist angles. The continuum limit is shown to involve two compact bosons and one non compact boson, with discrete states emerging from the continuum at appropriate twists. The non compact boson entails strong logarithmic corrections to the finite-size behaviour of the scaling levels, an understanding of which allows us to correct an earlier proposal for some of the critical exponents. In particular, we infer the full set of magnetic scaling dimensions (watermelon operators) of the Potts model.
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 special family of solvable five-vertex model is introduced on a square lattice. In addition to the usual nearest-neighbor interactions, the vertices defining the model also interact along one of the diagonals of the lattice. This family of models includes in a special limit the standard six-vertex model. The exact solution of these models is an application of the matrix product ansatz introduced recently and applied successfully in the solution of quantum chains. The phase diagram and the free energy of the models are calculated in the thermodynamic limit. The models exhibit massless phases, and our analytical and numerical analyses indicate that such phases are governed by a conformal field theory with central charge c=1 and continuously varying critical exponents.
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 .