We present an exact solution of a one-dimensional Ising chain with both nearest-neighbor and random long-range interactions. Not surprisingly, the solution confirms the mean-field character of the transition. This solution also predicts the finite-size scaling that we observe in numerical simulations.

arXiv:cond-mat/0402138Physical Review EAug 25, 200448 citations

First-order phase transition in a 2D random-field Ising model with conflicting dynamics

Nuno Crokidakis

Nov 20, 201210 citations

AIP conference proceedings

Random field Ising chain and neutral networks with synchronous dynamics

N S Skantzos

Published Jan 1, 2001• 1 citation

Publisher

Abstract

DOI: 10.1063/1.1358170

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Nuno Crokidakis

The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables {h} that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at +ho and −ho, with the same width σ. This distribution is very general, and can recover in appropriate limits the bimodal distribution (σ → 0) and the single gaussian one (ho = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32 − 512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities ho, for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.De, 75.10Hk, 75.40.MgFirst-order phase transition in a 2D random-field Ising model with conflicting dynamics2 1.The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables {h} that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at +ho and −ho, with the same width σ. This distribution is very general, and can recover in appropriate limits the bimodal distribution (σ → 0) and the single gaussian one (ho = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32 − 512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities ho, for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.De, 75.10Hk, 75.40.MgFirst-order phase transition in a 2D random-field Ising model with conflicting dynamics2 1.

Nov 20, 201210 citations

We present an exact solution of the long-time relaxational behavior of the magnetization in the Ising and

\mathrm{XY}

chains in a quenched random field. The random field is assumed to be of infinite strength but present at only a fraction of the spin sites. We use Glauber dynamics for the Ising case, and a suitably generalized master equation for the

\mathrm{XY}

case. In both models we find that for

T=0

, as time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ the magnetization decays as exp (${\ensuremath{-}Ct}^{\frac{1}{3}}$), where

C

is a constant. At finite temperatures the ultimate asymptotic behavior is purely exponential.