16,504 research outputs found
Ising model on the Apollonian network with node dependent interactions
This work considers an Ising model on the Apollonian network, where the
exchange constant between two neighboring spins
is a function of the degree of both spins. Using the exact
geometrical construction rule for the network, the thermodynamical and magnetic
properties are evaluated by iterating a system of discrete maps that allows for
very precise results in the thermodynamic limit. The results can be compared to
the predictions of a general framework for spins models on scale-free networks,
where the node distribution , with node dependent
interacting constants. We observe that, by increasing , the critical
behavior of the model changes, from a phase transition at for a
uniform system , to a T=0 phase transition when : in the
thermodynamic limit, the system shows no exactly critical behavior at a finite
temperature. The magnetization and magnetic susceptibility are found to present
non-critical scaling properties.Comment: 6 figures, 12 figure file
Analytical approach to directed sandpile models on the Apollonian network
We investigate a set of directed sandpile models on the Apollonian network,
which are inspired on the work by Dhar and Ramaswamy (PRL \textbf{63}, 1659
(1989)) for Euclidian lattices. They are characterized by a single parameter
, that restricts the number of neighbors receiving grains from a toppling
node. Due to the geometry of the network, two and three point correlation
functions are amenable to exact treatment, leading to analytical results for
the avalanche distributions in the limit of an infinite system, for .
The exact recurrence expressions for the correlation functions are numerically
iterated to obtain results for finite size systems, when larger values of
are considered. Finally, a detailed description of the local flux properties is
provided by a multifractal scaling analysis.Comment: 7 pages in two-column format, 10 illustrations, 5 figure
Multifractal Properties of Aperiodic Ising Model: role of geometric fluctuations
The role of the geometric fluctuations on the multifractal properties of the
local magnetization of aperiodic ferromagnetic Ising models on hierachical
lattices is investigated. The geometric fluctuations are introduced by
generalized Fibonacci sequences. The local magnetization is evaluated via an
exact recurrent procedure encompassing a real space renormalization group
decimation. The symmetries of the local magnetization patterns induced by the
aperiodic couplings is found to be strongly (weakly) different, with respect to
the ones of the corresponding homogeneous systems, when the geometric
fluctuations are relevant (irrelevant) to change the critical properties of the
system. At the criticality, the measure defined by the local magnetization is
found to exhibit a non-trivial F(alpha) spectra being shifted to higher values
of alpha when relevant geometric fluctuations are considered. The critical
exponents are found to be related with some special points of the F(alpha)
function and agree with previous results obtained by the quite distinct
transfer matrix approach.Comment: 10 pages, 7 figures, 3 Tables, 17 reference
Critical exponents for the long-range Ising chain using a transfer matrix approach
The critical behavior of the Ising chain with long-range ferromagnetic
interactions decaying with distance , , is investigated
using a numerically efficient transfer matrix (TM) method. Finite size
approximations to the infinite chain are considered, in which both the number
of spins and the number of interaction constants can be independently
increased. Systems with interactions between spins up to 18 sites apart and up
to 2500 spins in the chain are considered. We obtain data for the critical
exponents associated with the correlation length based on the Finite
Range Scaling (FRS) hypothesis. FRS expressions require the evaluation of
derivatives of the thermodynamical properties, which are obtained with the help
of analytical recurrence expressions obtained within the TM framework. The Van
den Broeck extrapolation procedure is applied in order to estimate the
convergence of the exponents. The TM procedure reduces the dimension of the
matrices and circumvents several numerical matrix operations.Comment: 10 pages, 2 figures, Conference NEXT Sigma Ph
Critical properties of an aperiodic model for interacting polymers
We investigate the effects of aperiodic interactions on the critical behavior
of an interacting two-polymer model on hierarchical lattices (equivalent to the
Migadal-Kadanoff approximation for the model on Bravais lattices), via
renormalization-group and tranfer-matrix calculations. The exact
renormalization-group recursion relations always present a symmetric fixed
point, associated with the critical behavior of the underlying uniform model.
If the aperiodic interactions, defined by s ubstitution rules, lead to relevant
geometric fluctuations, this fixed point becomes fully unstable, giving rise to
novel attractors of different nature. We present an explicit example in which
this new attractor is a two-cycle, with critical indices different from the
uniform model. In case of the four-letter Rudin-Shapiro substitution rule, we
find a surprising closed curve whose points are attractors of period two,
associated with a marginal operator. Nevertheless, a scaling analysis indicates
that this attractor may lead to a new critical universality class. In order to
provide an independent confirmation of the scaling results, we turn to a direct
thermodynamic calculation of the specific-heat exponent. The thermodynamic free
energy is obtained from a transfer matrix formalism, which had been previously
introduced for spin systems, and is now extended to the two-polymer model with
aperiodic interactions.Comment: 19 pages, 6 eps figures, to appear in J. Phys A: Math. Ge
- …