275 research outputs found
Disorder induced rounding of the phase transition in the large q-state Potts model
The phase transition in the q-state Potts model with homogeneous
ferromagnetic couplings is strongly first order for large q, while is rounded
in the presence of quenched disorder. Here we study this phenomenon on
different two-dimensional lattices by using the fact that the partition
function of the model is dominated by a single diagram of the high-temperature
expansion, which is calculated by an efficient combinatorial optimization
algorithm. For a given finite sample with discrete randomness the free energy
is a pice-wise linear function of the temperature, which is rounded after
averaging, however the discontinuity of the internal energy at the transition
point (i.e. the latent heat) stays finite even in the thermodynamic limit. For
a continuous disorder, instead, the latent heat vanishes. At the phase
transition point the dominant diagram percolates and the total magnetic moment
is related to the size of the percolating cluster. Its fractal dimension is
found d_f=(5+\sqrt{5})/4 and it is independent of the type of the lattice and
the form of disorder. We argue that the critical behavior is exclusively
determined by disorder and the corresponding fixed point is the isotropic
version of the so called infinite randomness fixed point, which is realized in
random quantum spin chains. From this mapping we conjecture the values of the
critical exponents as \beta=2-d_f, \beta_s=1/2 and \nu=1.Comment: 12 pages, 12 figures, version as publishe
Crossing bonds in the random-cluster model
We derive the scaling dimension associated with crossing bonds in the
random-cluster representation of the two-dimensional Potts model, by means of a
mapping on the Coulomb gas. The scaling field associated with crossing bonds
appears to be irrelevant, on the critical as well as on the tricritical branch.
The latter result stands in a remarkable contrast with the existing result for
the tricritical O(n) model that crossing bonds are relevant. In order to obtain
independent confirmation of the Coulomb gas result for the crossing-bond
exponent, we perform a finite-size-scaling analysis based on numerical
transfer-matrix calculations.Comment: 2 figure
Potts and percolation models on bowtie lattices
We give the exact critical frontier of the Potts model on bowtie lattices.
For the case of , the critical frontier yields the thresholds of bond
percolation on these lattices, which are exactly consistent with the results
given by Ziff et al [J. Phys. A 39, 15083 (2006)]. For the Potts model on
the bowtie-A lattice, the critical point is in agreement with that of the Ising
model on this lattice, which has been exactly solved. Furthermore, we do
extensive Monte Carlo simulations of Potts model on the bowtie-A lattice with
noninteger . Our numerical results, which are accurate up to 7 significant
digits, are consistent with the theoretical predictions. We also simulate the
site percolation on the bowtie-A lattice, and the threshold is
. In the simulations of bond percolation and site
percolation, we find that the shape-dependent properties of the percolation
model on the bowtie-A lattice are somewhat different from those of an isotropic
lattice, which may be caused by the anisotropy of the lattice.Comment: 18 pages, 9 figures and 3 table
Rounding of first-order phase transitions and optimal cooperation in scale-free networks
We consider the ferromagnetic large- state Potts model in complex evolving
networks, which is equivalent to an optimal cooperation problem, in which the
agents try to optimize the total sum of pair cooperation benefits and the
supports of independent projects. The agents are found to be typically of two
kinds: a fraction of (being the magnetization of the Potts model) belongs
to a large cooperating cluster, whereas the others are isolated one man's
projects. It is shown rigorously that the homogeneous model has a strongly
first-order phase transition, which turns to second-order for random
interactions (benefits), the properties of which are studied numerically on the
Barab\'asi-Albert network. The distribution of finite-size transition points is
characterized by a shift exponent, , and by a different
width exponent, , whereas the magnetization at the transition
point scales with the size of the network, , as: , with
.Comment: 8 pages, 6 figure
Local Statistics of Realizable Vertex Models
We study planar "vertex" models, which are probability measures on edge
subsets of a planar graph, satisfying certain constraints at each vertex,
examples including dimer model, and 1-2 model, which we will define. We express
the local statistics of a large class of vertex models on a finite hexagonal
lattice as a linear combination of the local statistics of dimers on the
corresponding Fisher graph, with the help of a generalized holographic
algorithm. Using an torus to approximate the periodic infinite
graph, we give an explicit integral formula for the free energy and local
statistics for configurations of the vertex model on an infinite bi-periodic
graph. As an example, we simulate the 1-2 model by the technique of Glauber
dynamics
Density of critical clusters in strips of strongly disordered systems
We consider two models with disorder dominated critical points and study the
distribution of clusters which are confined in strips and touch one or both
boundaries. For the classical random bond Potts model in the large-q limit we
study optimal Fortuin-Kasteleyn clusters by combinatorial optimization
algorithm. For the random transverse-field Ising chain clusters are defined and
calculated through the strong disorder renormalization group method. The
numerically calculated density profiles close to the boundaries are shown to
follow scaling predictions. For the random bond Potts model we have obtained
accurate numerical estimates for the critical exponents and demonstrated that
the density profiles are well described by conformal formulae.Comment: 9 pages, 9 figure
Generalized Geometric Cluster Algorithm for Fluid Simulation
We present a detailed description of the generalized geometric cluster
algorithm for the efficient simulation of continuum fluids. The connection with
well-known cluster algorithms for lattice spin models is discussed, and an
explicit full cluster decomposition is derived for a particle configuration in
a fluid. We investigate a number of basic properties of the geometric cluster
algorithm, including the dependence of the cluster-size distribution on density
and temperature. Practical aspects of its implementation and possible
extensions are discussed. The capabilities and efficiency of our approach are
illustrated by means of two example studies.Comment: Accepted for publication in Phys. Rev. E. Follow-up to
cond-mat/041274
Logarithmic conformal field theory with boundary
This lecture note covers topics on boundary conformal field theory, modular
transformations and the Verlinde formula, and boundary logarithmic CFT. An
introductory review on CFT with boundary and a discussion of its applications
to logarithmic cases are given. LCFT at is mainly discussed.Comment: 38 pages, 4 figures, LaTeX. Notes of lectures at the International
Summer School on Logarithmic Conformal Field Theory and its Applications,
Sept. 2001, IPM, Tehran. Typos fixe
Coulomb and Liquid Dimer Models in Three Dimensions
We study classical hard-core dimer models on three-dimensional lattices using
analytical approaches and Monte Carlo simulations. On the bipartite cubic
lattice, a local gauge field generalization of the height representation used
on the square lattice predicts that the dimers are in a critical Coulomb phase
with algebraic, dipolar, correlations, in excellent agreement with our
large-scale Monte Carlo simulations. The non-bipartite FCC and Fisher lattices
lack such a representation, and we find that these models have both confined
and exponentially deconfined but no critical phases. We conjecture that
extended critical phases are realized only on bipartite lattices, even in
higher dimensions.Comment: 4 pages with corrections and update
Critical line of an n-component cubic model
We consider a special case of the n-component cubic model on the square
lattice, for which an expansion exists in Ising-like graphs. We construct a
transfer matrix and perform a finite-size-scaling analysis to determine the
critical points for several values of n. Furthermore we determine several
universal quantities, including three critical exponents. For n<2, these
results agree well with the theoretical predictions for the critical O(n)
branch. This model is also a special case of the () model of
Domany and Riedel. It appears that the self-dual plane of the latter model
contains the exactly known critical points of the n=1 and 2 cubic models. For
this reason we have checked whether this is also the case for 1<n<2. However,
this possibility is excluded by our numerical results
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