70 research outputs found
The 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 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 , 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 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- descendants obtained from the fusion
procedure are well-known lattice discretizations of the SU WZW models,
with . It is shown that, in these models, it is possible to exhibit a
local observable on the lattice that behaves as the chiral current in
the continuum limit. The observable is built out of generators of the su
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 and 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
and , 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() 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, 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 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
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