The Z2Z_2 staggered vertex model and its applications

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 two (the $Z_2$ 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, or as an anyonic chain with a generic (non-integer) number of species. 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, 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.Comment: 38 pages, 14 figures, 3 appendice

Chiral SU(2)_k currents as local operators in vertex models and spin chains

The six-vertex model and its spin-$S$ descendants obtained from the fusion procedure are well-known lattice discretizations of the SU$(2)_k$ WZW models, with $k=2S$. It is shown that, in these models, it is possible to exhibit a local observable on the lattice that behaves as the chiral current $J^a(z)$ in the continuum limit. The observable is built out of generators of the su$(2)$ Lie algebra acting on a small (finite) number of lattice sites. The construction works also for the multi-critical quantum spin chains related to the vertex models, and is verified numerically for $S=1/2$ and $S=1$ using Bethe Ansatz and form factors techniques.Comment: 31 pages, 7 figures; published versio

Discrete holomorphicity and quantized affine algebras

We consider non-local currents in the context of quantized affine algebras, following the construction introduced by Bernard and Felder. In the case of $U_q(A_1^{(1)})$ and $U_q(A_2^{(2)})$, these currents can be identified with configurations in the six-vertex and Izergin--Korepin nineteen-vertex models. Mapping these to their corresponding Temperley--Lieb loop models, we directly identify non-local currents with discretely holomorphic loop observables. In particular, we show that the bulk discrete holomorphicity relation and its recently derived boundary analogue are equivalent to conservation laws for non-local currents

Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models

In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O($n$) loop model, any such operator is naturally associated to a standard module of the periodic Temperley-Lieb algebra. We introduce a new family of representations of this algebra, with connectivity states that have two marked points, and argue that they define the fusion of two standard modules. We obtain their decomposition on the standard modules for generic values of the parameters, which in turn yields the structure of the operator product expansion of connectivity operators

Spin interfaces in the Ashkin-Teller model and SLE

We investigate the scaling properties of the spin interfaces in the Ashkin-Teller model. These interfaces are a very simple instance of lattice curves coexisting with a fluctuating degree of freedom, which renders the analytical determination of their exponents very difficult. One of our main findings is the construction of boundary conditions which ensure that the interface still satisfies the Markov property in this case. Then, using a novel technique based on the transfer matrix, we compute numerically the left-passage probability, and our results confirm that the spin interface is described by an SLE in the scaling limit. Moreover, at a particular point of the critical line, we describe a mapping of Ashkin-Teller model onto an integrable 19-vertex model, which, in turn, relates to an integrable dilute Brauer model.Comment: 12 pages, 6 figure

Integrability as a consequence of discrete holomorphicity: the Z_N model

It has recently been established that imposing the condition of discrete holomorphicity on a lattice parafermionic observable leads to the critical Boltzmann weights in a number of lattice models. Remarkably, the solutions of these linear equations also solve the Yang-Baxter equations. We extend this analysis for the Z_N model by explicitly considering the condition of discrete holomorphicity on two and three adjacent rhombi. For two rhombi this leads to a quadratic equation in the Boltzmann weights and for three rhombi a cubic equation. The two-rhombus equation implies the inversion relations. The star-triangle relation follows from the three-rhombus equation. We also show that these weights are self-dual as a consequence of discrete holomorphicity.Comment: 11 pages, 7 figures, some clarifications and a reference adde

Local height probabilities in a composite Andrews-Baxter-Forrester model

We study the local height probabilities in a composite height model, derived from the restricted solid-on-solid model introduced by Andrews, Baxter and Forrester, and their connection with conformal field theory characters. The obtained conformal field theories also describe the critical behavior of the model at two different critical points. In addition, at criticality, the model is equivalent to a one-dimensional chain of anyons, subject to competing two- and three-body interactions. The anyonic-chain interpretation provided the original motivation to introduce the composite height model, and by obtaining the critical behaviour of the composite height model, the critical behaviour of the anyonic chains is established as well. Depending on the overall sign of the hamiltonian, this critical behaviour is described by a diagonal coset-model, generalizing the minimal models for one sign, and by Fateev-Zamolodchikov parafermions for the other.Comment: 34 pages, 5 figures; v2: expanded introduction, references added and other minor change

Discrete Holomorphicity at Two-Dimensional Critical Points

After a brief review of the historical role of analyticity in the study of critical phenomena, an account is given of recent discoveries of discretely holomorphic observables in critical two-dimensional lattice models. These are objects whose correlation functions satisfy a discrete version of the Cauchy-Riemann relations. Their existence appears to have a deep relation with the integrability of the model, and they are presumably the lattice versions of the truly holomorphic observables appearing in the conformal field theory (CFT) describing the continuum limit. This hypothesis sheds light on the connection between CFT and integrability, and, if verified, can also be used to prove that the scaling limit of certain discrete curves in these models is described by Schramm-Loewner evolution (SLE).Comment: Invited talk at the 100th Statistical Mechanics Meeting, Rutgers, December 200

Finite-size left-passage probability in percolation

We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. Our calculation is based on the q-deformed Knizhnik--Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramm's left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin--Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary terms.Comment: 21 pages, 8 figure

Critical domain walls in the Ashkin-Teller model

We study the fractal properties of interfaces in the 2d Ashkin-Teller model. The fractal dimension of the symmetric interfaces is calculated along the critical line of the model in the interval between the Ising and the four-states Potts models. Using Schramm's formula for crossing probabilities we show that such interfaces can not be related to the simple SLE$_\kappa$, except for the Ising point. The same calculation on non-symmetric interfaces is performed at the four-states Potts model: the fractal dimension is compatible with the result coming from Schramm's formula, and we expect a simple SLE$_\kappa$ in this case.Comment: Final version published in JSTAT. 13 pages, 5 figures. Substantial changes in the data production, analysis and in the conclusions. Added a section about the crossing probability. Typeset with 'iopart