70 research outputs found
SLE in the three-state Potts model - a numerical study
The scaling limit of the spin cluster boundaries of the Ising model with
domain wall boundary conditions is SLE with kappa=3. We hypothesise that the
three-state Potts model with appropriate boundary conditions has spin cluster
boundaries which are also SLE in the scaling limit, but with kappa=10/3. To
test this, we generate samples using the Wolff algorithm and test them against
predictions of SLE: we examine the statistics of the Loewner driving function,
estimate the fractal dimension and test against Schramm's formula. The results
are in support of our hypothesis.Comment: 32 pages, 41 figure
Twist operator correlation functions in O(n) loop models
Using conformal field theoretic methods we calculate correlation functions of
geometric observables in the loop representation of the O(n) model at the
critical point. We focus on correlation functions containing twist operators,
combining these with anchored loops, boundaries with SLE processes and with
double SLE processes.
We focus further upon n=0, representing self-avoiding loops, which
corresponds to a logarithmic conformal field theory (LCFT) with c=0. In this
limit the twist operator plays the role of a zero weight indicator operator,
which we verify by comparison with known examples. Using the additional
conditions imposed by the twist operator null-states, we derive a new explicit
result for the probabilities that an SLE_{8/3} wind in various ways about two
points in the upper half plane, e.g. that the SLE passes to the left of both
points.
The collection of c=0 logarithmic CFT operators that we use deriving the
winding probabilities is novel, highlighting a potential incompatibility caused
by the presence of two distinct logarithmic partners to the stress tensor
within the theory. We provide evidence that both partners do appear in the
theory, one in the bulk and one on the boundary and that the incompatibility is
resolved by restrictive bulk-boundary fusion rules.Comment: 18 pages, 8 figure
Critical exponents of domain walls in the two-dimensional Potts model
We address the geometrical critical behavior of the two-dimensional Q-state
Potts model in terms of the spin clusters (i.e., connected domains where the
spin takes a constant value). These clusters are different from the usual
Fortuin-Kasteleyn clusters, and are separated by domain walls that can cross
and branch. We develop a transfer matrix technique enabling the formulation and
numerical study of spin clusters even when Q is not an integer. We further
identify geometrically the crossing events which give rise to conformal
correlation functions. This leads to an infinite series of fundamental critical
exponents h_{l_1-l_2,2 l_1}, valid for 0 </- Q </- 4, that describe the
insertion of l_1 thin and l_2 thick domain walls.Comment: 5 pages, 3 figures, 1 tabl
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
Fluctuation force exerted by a planar self-avoiding polymer
Using results from Schramm Loewner evolution (SLE), we give the expression of
the fluctuation-induced force exerted by a polymer on a small impenetrable
disk, in various 2-dimensional domain geometries. We generalize to two polymers
and examine whether the fluctuation force can trap the object into a stable
equilibrium. We compute the force exerted on objects at the domain boundary,
and the force mediated by the polymer between such objects. The results can
straightforwardly be extended to any SLE interface, including Ising,
percolation, and loop-erased random walks. Some are relevant for extremal value
statistics.Comment: 7 pages, 22 figure
General solution of an exact correlation function factorization in conformal field theory
We discuss a correlation function factorization, which relates a three-point
function to the square root of three two-point functions. This factorization is
known to hold for certain scaling operators at the two-dimensional percolation
point and in a few other cases. The correlation functions are evaluated in the
upper half-plane (or any conformally equivalent region) with operators at two
arbitrary points on the real axis, and a third arbitrary point on either the
real axis or in the interior. This type of result is of interest because it is
both exact and universal, relates higher-order correlation functions to
lower-order ones, and has a simple interpretation in terms of cluster or loop
probabilities in several statistical models. This motivated us to use the
techniques of conformal field theory to determine the general conditions for
its validity.
Here, we discover a correlation function which factorizes in this way for any
central charge c, generalizing previous results. In particular, the
factorization holds for either FK (Fortuin-Kasteleyn) or spin clusters in the
Q-state Potts models; it also applies to either the dense or dilute phases of
the O(n) loop models. Further, only one other non-trivial set of highest-weight
operators (in an irreducible Verma module) factorizes in this way. In this case
the operators have negative dimension (for c < 1) and do not seem to have a
physical realization.Comment: 7 pages, 1 figure, v2 minor revision
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
Boundary conformal field theories and loop models
We propose a systematic method to extract conformal loop models for rational
conformal field theories (CFT). Method is based on defining an ADE model for
boundary primary operators by using the fusion matrices of these operators as
adjacency matrices. These loop models respect the conformal boundary
conditions. We discuss the loop models that can be extracted by this method for
minimal CFTs and then we will give dilute O(n) loop models on the square
lattice as examples for these loop models. We give also some proposals for WZW
SU(2) models.Comment: 23 Pages, major changes! title change
Critical interfaces of the Ashkin-Teller model at the parafermionic point
We present an extensive study of interfaces defined in the Z_4 spin lattice
representation of the Ashkin-Teller (AT) model. In particular, we numerically
compute the fractal dimensions of boundary and bulk interfaces at the
Fateev-Zamolodchikov point. This point is a special point on the self-dual
critical line of the AT model and it is described in the continuum limit by the
Z_4 parafermionic theory. Extending on previous analytical and numerical
studies [10,12], we point out the existence of three different values of
fractal dimensions which characterize different kind of interfaces. We argue
that this result may be related to the classification of primary operators of
the parafermionic algebra. The scenario emerging from the studies presented
here is expected to unveil general aspects of geometrical objects of critical
AT model, and thus of c=1 critical theories in general.Comment: 15 pages, 3 figure
Critical behavior of interfaces in disordered Potts ferromagnets : statistics of free-energy, energy and interfacial adsorption
A convenient way to study phase transitions of finite spins systems of linear
size is to fix boundary conditions that impose the presence of a
system-size interface. In this paper, we study the statistical properties of
such an interface in a disordered Potts ferromagnet in dimension within
Migdal-Kadanoff real space renormalization. We first focus on the interface
free-energy and energy to measure the singularities of the average and random
contributions, as well as the corresponding histograms, both in the
low-temperature phase and at criticality. We then consider the critical
behavior of the interfacial adsorption of non-boundary states. Our main
conclusion is that all singularities involve the correlation length
appearing in the average free-energy of the interface of dimension , except for
the free-energy width that involves
the droplet exponent and another correlation length
which diverges more rapidly than . We compare with the spin-glass
transition in , where is the 'true' correlation length, and
where the interface energy presents unconventional scaling with a chaos
critical exponent [Nifle and Hilhorst, Phys. Rev. Lett. 68,
2992 (1992)]. The common feature is that in both cases, the characteristic
length scale associated with the chaotic nature of the
low-temperature phase, diverges more slowly than the correlation length.Comment: v2 : thoroughly rewritten paper with new title, new data and new
interpretations (18 pages, 22 figures
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