Ising model on the Apollonian network with node dependent interactions

This work considers an Ising model on the Apollonian network, where the exchange constant $J_{i,j}\sim1/(k_ik_j)^\mu$ between two neighboring spins $(i,j)$ is a function of the degree $k$ 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 $P(k)\sim k^{-\gamma}$, with node dependent interacting constants. We observe that, by increasing $\mu$, the critical behavior of the model changes, from a phase transition at $T=\infty$ for a uniform system $(\mu=0)$, to a T=0 phase transition when $\mu=1$: 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

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

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 $r^\alpha$, $1<\alpha<2$, 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 $\nu$ 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

Non-nequilibrium model on Apollonian networks

We investigate the Majority-Vote Model with two states ($-1,+1$) and a noise $q$ on Apollonian networks. The main result found here is the presence of the phase transition as a function of the noise parameter $q$. We also studies de effect of redirecting a fraction $p$ of the links of the network. By means of Monte Carlo simulations, we obtained the exponent ratio $\gamma/\nu$, $\beta/\nu$, and $1/\nu$ for several values of rewiring probability $p$. The critical noise was determined $q_{c}$ and $U^{*}$ also was calculated. The effective dimensionality of the system was observed to be independent on $p$, and the value $D_{eff} \approx1.0$ is observed for these networks. Previous results on the Ising model in Apollonian Networks have reported no presence of a phase transition. Therefore, the results present here demonstrate that the Majority-Vote Model belongs to a different universality class as the equilibrium Ising Model on Apollonian Network.Comment: 5 pages, 5 figure