202 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
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
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
SLE in self-dual critical Z(N) spin systems: CFT predictions
The Schramm-Loewner evolution (SLE) describes the continuum limit of domain
walls at phase transitions in two dimensional statistical systems. We consider
here the SLEs in the self-dual Z(N) spin models at the critical point. For N=2
and N=3 these models correspond to the Ising and three-state Potts model. For
N>5 the critical self-dual Z(N) spin models are described in the continuum
limit by non-minimal conformal field theories with central charge c>=1. By
studying the representations of the corresponding chiral algebra, we show that
two particular operators satisfy a two level null vector condition which, for
N>=4, presents an additional term coming from the extra symmetry currents
action. For N=2,3 these operators correspond to the boundary conditions
changing operators associated to the SLE_{16/3} (Ising model) and to the
SLE_{24/5} and SLE_{10/3} (three-state Potts model). We suggest a definition of
the interfaces within the Z(N) lattice models. The scaling limit of these
interfaces is expected to be described at the self-dual critical point and for
N>=4 by the SLE_{4(N+1)/(N+2)} and SLE_{4(N+2)/(N+1)} processes.Comment: 22 pages, 6 figures. v2: Nuclear Physics B Published versio
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
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
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