We perform a detailed study of the critical behavior of the mean field diluted Ising ferromagnet by analytical and numerical tools. We obtain self-averaging for the magnetization and write down an expansion for the free energy close to the critical line. The scaling of the magnetization is also rigorously obtained and compared with extensive Monte Carlo simulations. We explain the transition from an ergodic region to a non trivial phase by commutativity breaking of the infinite volume limit and a suitable vanishing field. We find full agreement among theory, simulations and previous results.
We present a method derived by cavity arguments to compute the spin-glass and higher-order susceptibilities in diluted mean-field spin-glass models. The divergence of the spin-glass susceptibility is associated to the existence of a non-zero solution of a homogeneous linear integral equation. Higher order susceptibilities, relevant for critical dynamics through the parameter exponent $λ$, can be expressed at criticality as integrals involving the critical eigenvector. The numerical evaluation of the corresponding analytic expressions is discussed. The method is illustrated in the context of th…
The universality class of thermally diluted Ising systems, in which the realization of the disposition of magnetic atoms and vacancies is taken from the local distribution of spins in the pure original Ising model at criticality, is investigated by finite size scaling techniques using the Monte Carlo method. We find that the critical temperature, the critical exponents and therefore the universality class of these thermally diluted Ising systems depart markedly from the ones of short range correlated disordered systems. Our results agree fairly well with theoretical predictions previously made…
We study the diluted Ising ferromagnet on the Bethe lattice as a case study for the application of the cavity method to problems with Griffiths-McCoy singularities. Specifically, we are able to make much progress at infinite coupling where we compute, from the cavity method, the density of Lee-Yang zeroes in the paramagnetic Griffiths region as well as the properties of the phase transition to the ferromagnet. This phase transition is itself of a Griffiths-McCoy character albeit with a power law distribution of cluster sizes.
We report on large-scale Monte Carlo simulations of the three-dimensional 4-state Potts ferromagnet subject to quenched, random bond dilution. For small dilutions the rather strong first-order phase transition of the pure system is found to persist, whereas for larger dilutions the theoretically expected softening to a continuous transition is confirmed. The properties of the underlying disorder distributions of thermal observables are discussed and illustrated with a few selected examples.
We discuss analytical approximation schemes for the dynamics of diluted spin models. The original dynamics of the complete set of degrees of freedom is replaced by a hierarchy of equations including an increasing number of global observables, which can be closed approximately at different levels of the hierarchy. We illustrate this method on the simple example of the Ising ferromagnet on a Bethe lattice, investigating the first three possible closures, which are all exact in the long time limit, and which yield more and more accurate predictions for the finite-time behavior. We also investigat…
In the paper the Pair Approximation (PA) method for studies of the site-diluted spin-1/2 systems of arbitrary dimensionality with the long-range ferromagnetic interactions is adopted. The method allows to take into account arbitrary anisotropy of the interactions in the spin space, so it is not limited to purely Ising couplings. Within this approach, the Gibbs free energy is obtained, which allows to derive all the further interesting thermodynamic properties. In particular, we obtain an equation for the critical temperature of the second-order phase transitions for the model in question. In t…
We show that the interplay between geometric criticality and dynamical fluctuations leads to a novel universality class of the contact process on a randomly diluted lattice. The nonequilibrium phase transition across the percolation threshold of the lattice is characterized by unconventional activated (exponential) dynamical scaling and strong Griffiths effects. We calculate the critical behavior in two and three space dimensions, and we also relate our results to the recently found infinite-randomness fixed point in the disordered one-dimensional contact process.
Frequency evolutions of hysteresis loop area and hysteresis tools such as remanence and coercivity of a kinetic Ising model in the presence of quenched bond dilution are investigated in detail. The kinetic equation describing the time dependence of the magnetization is derived by means of effective-field theory with single-site correlations. It is found that the frequency dispersions of hysteresis loop area, remanence and coercivity strongly depend on the quenched bond randomness, as well as applied field amplitude and oscillation frequency. In addition, the shape of the hysteresis curves for …
We comprehensively study non-equilibrium relaxation and aging processes in the two-dimensional random-site Ising model through numerical simulations. We discuss the dynamical correlation length as well as scaling functions of various two-times quantities as a function of temperature and of the degree of dilution. For already modest values of the dynamical correlation length $L$ deviations from a simple algebraic growth, $L(t) \sim t^{1/z}$, are observed. When taking this non-algebraic growth properly into account, a simple aging behavior of the autocorrelation function is found. This is in sta…
In this report we give an overview on recent results obtained from extensive Monte Carlo (MC) computer simulations of the 3D 2-state (Ising) and 4-state Potts models with bond-dilution. The motivation to study the 4-state Potts model derives from the fact that, in the pure case, this model is known to exhibit a fairly strong first-order transition, such that a disorder-induced softening to a second-order transition would give clear support of the theoretical expectations. Modeling the disorder by bond-dilution enables in the Ising case a test of the expected universality with respect to the ty…
In this paper we develop the interpolating cavity field technique for the mean field ferromagnetic p-spin. The model we introduce is a natural extension of the diluted Curie-Weiss model to p>2 spin interactions. Several properties of the free energy are analyzed and, in particular, we show that it recovers the expressions already known for p=2 models and for p>2 fully connected models. Further, as the model lacks criticality, we present extensive numerical simulations to evidence the presence of a first order phase transition and deepen the behavior at the transition line. Overall, a good agre…
Dynamic behavior of a site diluted Ising ferromagnet in the presence of periodically oscillating magnetic field has been analyzed by means of the effective field theory (EFT). Dynamic equation of motion have been solved for a honeycomb lattice ($z=3$) with the help of a Glauber type stochastic process. The global phase diagrams and the variation of the corresponding dynamic order parameter as a function of the Hamiltonian parameters and temperature has been investigated in detail and it has been shown that the system exhibits reentrant phenomena, as well as a dynamic tricritical point which di…
The thermodynamics of a site-diluted ferromagnetic Heisenberg model for spin $S=1/2$ with interaction anisotropy in spin space is investigated. The study is aimed at presenting the magnetocaloric properties of such a model, including the entropy and temperature changes in magnetization/demagnetization processes, generalized Grüneisen ratio as well as the quantities characterizing the efficiency of magnetic cooling cycles. The results are obtained using Pair Approximation (PA) method and extensively compared with the Molecular Field Approximation (MFA) calculations. The importance of interactio…
Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional weakly site-diluted Ising model with spin concentrations $p=0.95$ and 0.8 at criticality. In contrast to studies of the critical behavior of the pure systems by the short-time dynamics method, our investigations of site-diluted Ising model have revealed three stages of the dynamic evolution characterizing a crossover phenomenon from the critical behavior typical for the pure systems to behavior determined by the influence of disorder. The static and dynamic critical exponents are determined with the u…
We study numerically the coarsening kinetics of a two-dimensional ferromagnetic system with aleatory bond dilution. We show that interfaces between domains of opposite magnetisation are fractal on every lengthscale, but with different properties at short or long distances. Specifically, on lengthscales larger than the typical domains' size the topology is that of critical random percolation, similarly to what observed in clean systems or models with different kinds of quenched disorder. On smaller lengthscales a dilution dependent fractal dimension emerges. The Hausdorff dimension increases wi…
We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the +- J Ising model along the paramagnetic-ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dyn…
In the paper a theoretical study of critical (Curie) temperature of diluted ferromagnet based on multilayer graphene (or graphite) with hydrogen adatoms deposited over carbon atoms belonging to single sublattice is presented. The calculations are performed within Pair Approximation (PA) for diluted ferromagnetic system with long-range interactions. The method is able to take into account the spin-space anisotropy of coupling. The results obtained within Mean Field Approximation (MFA) are also presented for comparison. The assumed interaction between hydrogen adatom spins is inversely proportio…
The dynamics and behavior of ferromagnets have a great relevance even beyond the domain of statistical physics. In this work, we propose a Monte Carlo method, based on random graphs, for modeling their dilution. In particular, we focus on ferromagnets with dimension $D \ge 4$, which can be approximated by the Curie-Weiss model. Since the latter has as graphic counterpart a complete graph, a dilution can be in this case viewed as a pruning process. Hence, in order to exploit this mapping, the proposed strategy uses a modified version of the Erdős-Renyi graph model. In doing so, we are able both…
Diffusion on a diluted hypercube has been proposed as a model for glassy relaxation and is an example of the more general class of stochastic processes on graphs. In this article we determine numerically through large scale simulations the eigenvalue spectra for this stochastic process and calculate explicitly the time evolution for the autocorrelation function and for the return probability, all at criticality, with hypercube dimensions $N$ up to N=28. We show that at long times both relaxation functions can be described by stretched exponentials with exponent 1/3 and a characteristic relaxat…
The field-induced quantum criticality of compounds with ferromagnetically coupled structural spin units (as dimers and ladders) is explored by applying Wilson's renormalization group framework to an appropriate effective action. We determine the low-temperature phase boundary and the behavior of relevant quantities decreasing the temperature with the applied magnetic field fixed at its quantum critical point value. In this context, a plausible interpretation of some recent experimental results is also suggested.
We discuss how a spin system, which is subject to quenched disorder, might exhibit multicritical behaviors at criticality if the distribution of the impurities is arbitrary. In order to provide realistic candidates for such multicritical behaviors, we discuss several generalizations of the standard randomly diluted Ising's universality class adopting the $ε$-expansion close to several upper critical dimensions. In the presentation, we spend a special effort in bridging between CFT and RG results and discuss in detail the computation of quantities, which are of prominent interest in the case of…
In real magnets the tendency towards ferromagnetism, promoted by exchange coupling, is usually frustrated by dipolar interaction. As a result, the uniformly ordered phase is replaced by modulated (multi-domain) phases, characterized by different order parameters rather than the global magnetization. The transitions occurring within those modulated phases and towards the disordered phase are generally not of second-order type. Nevertheless, strong experimental evidence indicates that a standard critical behavior is recovered when comparatively small fields are applied that stabilize the uniform…
Some results on the two-point function and on the analytic structure of the momenta of the effective fugacity at the origin for a class of diluted ferromagnetic Ising models on the Cayley tree are presented.
We consider a $ϕ^4$-theory with a position-dependent distance from the critical point. One realization of this model is a classical ferromagnet subject to non-uniform mechanical stress. We find a sharp phase transition where the envelope of the local magnetization vanishes uniformly. The first-order transition in a quantum ferromagnet also remains sharp. The universal mechanism leading to a tricritical point in an itinerant quantum ferromagnet is suppressed, and in principle one can recover a quantum critical point with mean-field exponents. Observable consequences of these results are discuss…
The magnetic properties and critical behavior of both ferromagnetic pure and metallic nanoparticles having concurrently atomic disorder, dilution and competing interactions, are studied in the framework of an Ising model. We have used both the free energy variational principle based on the Bogoliubov inequality and Monte Carlo simulation. As a case of study for random diluted nanoparticles we have considered the Fe$_{0.5}$Mn$_{0.1}$Al$_{0.4}$ alloy characterized for exhibiting, under bulk conditions, low temperature reentrant spin glass (RSG) behavior and for which experimental and simulation …
