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AIP conference proceedings

First Order Phase Transition in a 3D disordered system

L. A. Fernández, A. Gordillo-Guerrero, Víctor Martín-Mayor, Juan Jesus Ruiz-Lorenzo, Jesús Clemente-Gallardo, Pierpaolo Bruscolini, Francisco M. Castejón, Pablo Echenique, José Félix Sáenz-Lorenzo

Published Jan 1, 2008• 0 citations
Publisher

Abstract

We present a detailed numerical study on the effects of adding quenched impurities to a three dimensional system which in the pure case undergoes a strong first order phase transition (specifically, the ferromagnetic/paramagnetic transition of the site‐diluted four states Potts model). We can state that the transition remains first‐order in the presence of quenched disorder (a small amount of it) but it turns out to be second order as more impurities are added. A tricritical point, which is studied by means of Finite‐Size Scaling, separates the first‐order and second‐order parts of the critical line. The results were made possible by a new definition of the disorder average that avoids the diverging‐variance probability distributions that arise using the standard methodology. We also made use of a recently proposed microcanonical Monte Carlo method in which entropy, instead of free energy, is the basic quantity.

DOI
10.1063/1.3033359

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We present a detailed numerical study on the effects of adding quenched impurities to a three dimensional system which in the pure case undergoes a strong first order phase transition (specifically, the ferromagnetic/paramagnetic transition of the site‐diluted four states Potts model). We can state that the transition remains first‐order in the presence of quenched disorder (a small amount of it) but it turns out to be second order as more impurities are added. A tricritical point, which is studied by means of Finite‐Size Scaling, separates the first‐order and second‐order parts of the critical line. The results were made possible by a new definition of the disorder average that avoids the diverging‐variance probability distributions that arise using the standard methodology. We also made use of a recently proposed microcanonical Monte Carlo method in which entropy, instead of free energy, is the basic quantity.
We present the first detailed numerical study in three dimensions of a first-order phase transition that remains first order in the presence of quenched disorder (specifically, the ferromagnetic-paramagnetic transition of the site-diluted four states Potts model). A tricritical point, which lies surprisingly near the pure-system limit and is studied by means of finite-size scaling, separates the first-order and second-order parts of the critical line. This investigation has been made possible by a new definition of the disorder average that avoids the diverging-variance probability distributions that plague the standard approach. Entropy, rather than free energy, is the basic object in this approach that exploits a recently introduced microcanonical Monte Carlo method.
We study by extensive Monte Carlo simulations the effect of random bond dilution on the phase transition of the three-dimensional four-state Potts model that is known to exhibit a strong first-order transition in the pure case. The phase diagram in the dilution-temperature plane is determined from the peaks of the susceptibility for sufficiently large system sizes. In the strongly disordered regime, numerical evidence for softening to a second-order transition induced by randomness is given. Here a large-scale finite-size scaling analysis, made difficult due to strong crossover effects presumably caused by the percolation fixed point, is performed.
The effect of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model with the number of spin states q=3 is studied using the Wolff single-cluster algorithm of the Monte Carlo method. By the method of fourth-order Binder cumulants, it is demonstrated that the second-order phase transition occurs in the model under study at spin concentrations p=0.9, 0.8, 0.7, and 0.65, while the first-order phase transition is observed in the pure model (p=1.0). The static critical exponents (CEs) α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length) are calculated based on the finite-size scaling theory.
Monte Carlo simulations are performed to analyze phase transitions in three-dimensional systems described by the 3-state Potts model with nonmagnetic impurities. Numerical results are presented for systems with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 on lattices of size L varying between 20 and 44. Binder’s cumulant analysis shows that the introduction of quenched disorder in the form of non-magnetic impurities induces a crossover from first-order to second-order phase transition. The finite-size scaling method is used to calculate the static critical exponents α, γ, β, and ν for specific heat, susceptibility, magnetization, and correlation length, respectively.
The influence of quenched nonmagnetic impurities on phase transitions and critical phenomena in the 3D Potts model with the spin state number q = 4 is studied using the Monte Carlo method. Systems with the linear size L = 20–32 and spin concentrations p = 1.00, 0.90, 0.65 are considered. The fourth order Binder cumulant method is used to demonstrate that in the strongly diluted regime, a phase transition of the second kind is observed in this model for the spin concentration p = 0.65, and a phase transition of the first kind is observed for the pure ( p = 1.00) and weakly diluted ( p = 0.90) models. The theory of finite-dimensional scaling is used to calculate the static critical parameters of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν.
We face the problem of phase transitions in diluted systems both from theoretical and numerical sides. We study the effects of quenched site-dilution in classical models (Heisenberg, Ising and Potts) in 2, 3, and 4 dimensions both by using the Renormalization Group and numerical simulations in the canonical and microcanonical ensembles. We propose and check a new formulation of the Finite Size Scaling ansatz (FSS) inside the microcanonical ensemble. We use microcanonical simulations to obtain a clear fist-order behavior for the diluted Potts model in 3D, estimating the tricritical dilution. We perform large-scale simulations of the 3D diluted Heisenberg model, checking its self-averaging properties. Finally we study the 4D diluted Ising model obtaining from the FSS of the specific heat a clear differentiation between the existing conflicting theories. We also compiled a large number of appendix that we expect to be used as future reference.