10 research outputs found
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 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
Geometrical properties of parafermionic spin models
We present measurements of the fractal dimensions associated to the
geometrical clusters for Z_4 and Z_5 spin models. We also attempted to measure
similar fractal dimensions for the generalised Fortuyin Kastelyn (FK) clusters
in these models but we discovered that these clusters do not percolate at the
critical point of the model under consideration. These results clearly mark a
difference in the behaviour of these non local objects compared to the Ising
model or the 3-state Potts model which corresponds to the simplest cases of Z_N
spin models with N=2 and N=3 respectively. We compare these fractal dimensions
with the ones obtained for SLE interfaces.Comment: 18 pages, 10 figures. v2: published versio
Spin clusters and conformal field theory
We study numerically the fractal dimensions and the bulk three-point connectivity for spin clusters of the Q-state Potts model in two dimensions with 1 <= Q <= 4. We check that the usually invoked correspondence between FK clusters and spin clusters works at the level of fractal dimensions. However, the fine structure of the conformal field theories describing critical clusters first manifests at the level of the three-point connectivities. Contrary to what was recently found for FK clusters, no obvious relation emerges for generic Q between the spin cluster connectivity and the structure constants obtained from analytic continuation of the minimal model constants. The numerical results strongly suggest that spin and FK clusters are described by conformal field theories with different realizations of the color symmetry of the Potts model
Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator
Recently, two of us argued that the probability that an FK cluster in the Q-state Potts model connects three given points is related to the time-like Liouville three-point correlation function (Delfino and Viti, 2011) [1]. Moreover, they predicted that the FK three-point connectivity has a prefactor which unveils the effects of a discrete symmetry, reminiscent of the S-Q permutation symmetry of the Q = 2,3,4 Potts model. We revisit the derivation of the time-like Liouville correlator (Zamolodchikov, 2005) [2] and show that this is the only consistent analytic continuation of the minimal model structure constants. We then present strong numerical tests of the relation between the time-like Liouville correlator and percolative properties of the FK clusters for real values of Q. (C) 2013 Elsevier B.V. All rights reserved