200 research outputs found
Operator content of the critical Potts model in d dimensions and logarithmic correlations
Using the symmetric group symmetry of the -state Potts model, we
classify the (scalar) operator content of its underlying field theory in
arbitrary dimension. In addition to the usual identity, energy and
magnetization operators, we find fields that generalize the -cluster
operators well-known in two dimensions, together with their subleading
counterparts. We give the explicit form of all these operators -- up to
non-universal constants -- both on the lattice and in the continuum limit for
the Landau theory. We compute exactly their two- and three-point correlation
functions on an arbitrary graph in terms of simple probabilities, and give the
general form of these correlation functions in the continuum limit at the
critical point. Specializing to integer values of the parameter , we argue
that the analytic continuation of the symmetry yields logarithmic
correlations at the critical point in arbitrary dimension, thus implying a
mixing of some scaling fields by the scale transformation generator. All these
logarithmic correlation functions are given a clear geometrical meaning, which
can be checked in numerical simulations. Several physical examples are
discussed, including bond percolation, spanning trees and forests, resistor
networks and the Ising model. We also briefly address the generalization of our
approach to the model.Comment: 35 pages, 6 figure
The semiflexible fully-packed loop model and interacting rhombus tilings
Motivated by a recent adsorption experiment [M.O. Blunt et al., Science 322,
1077 (2008)], we study tilings of the plane with three different types of
rhombi. An interaction disfavors pairs of adjacent rhombi of the same type.
This is shown to be a special case of a model of fully-packed loops with
interactions between monomers at distance two along a loop. We solve the latter
model using Coulomb gas techniques and show that its critical exponents vary
continuously with the interaction strenght. At low temperature it undergoes a
Kosterlitz-Thouless transition to an ordered phase, which is predicted from
numerics to occur at a temperature T \sim 110K in the experiments.Comment: 4 pages, 4 figures, v2: corrected typo, v3: minor modifications,
published versio
The arboreal gas and the supersphere sigma model
We discuss the relationship between the phase diagram of the Q=0 state Potts
model, the arboreal gas model, and the supersphere sigma model S^{0,2} =
OSP(1/2) / OSP(0/2). We identify the Potts antiferromagnetic critical point
with the critical point of the arboreal gas (at negative tree fugacity), and
with a critical point of the sigma model. We show that the corresponding
conformal theory on the square lattice has a non-linearly realized OSP(2/2) =
SL(1/2) symmetry, and involves non-compact degrees of freedom, with a
continuous spectrum of critical exponents. The role of global topological
properties in the sigma model transition is discussed in terms of a generalized
arboreal gas model.Comment: 23 pages, 4 figure
Boundary chromatic polynomial
We consider proper colorings of planar graphs embedded in the annulus, such
that vertices on one rim can take Q_s colors, while all remaining vertices can
take Q colors. The corresponding chromatic polynomial is related to the
partition function of a boundary loop model. Using results for the latter, the
phase diagram of the coloring problem (with real Q and Q_s) is inferred, in the
limits of two-dimensional or quasi one-dimensional infinite graphs. We find in
particular that the special role played by Beraha numbers Q=4 cos^2(pi/n) for
the usual chromatic polynomial does not extend to the case Q different from
Q_s. The agreement with (scarce) existing numerical results is perfect; further
numerical checks are presented here.Comment: 20 pages, 7 figure
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