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Physical review. B, Condensed matter

Dilute antiferromagnetism and random fields in two-dimensional Ising systems

B. Boechat, M. A. Contínentino

Published Dec 1, 1991• 8 citations
Publisher

Abstract

We study the two-dimensional (2D) diluted antiferromagnet in a uniform magnetic field using the real-space renormalization group in a hierarchical cell. We follow separately the probability distributions of random-exchange couplings and of magnetic fields along the renormalization process. Due to the random couplings, the initial uniform magnetic field becomes a random variable with a mean value that continuously decreases to zero. In this 2D case the flow of the probability distributions is always toward the attractor of the paramagnetic phase characterized by 〈T/J〉=\ensuremath{\infty} and 〈(h/T${)}^{2}$〉=\ensuremath{\infty} in agreement with ${\mathit{d}}_{\mathit{c}}$=2. We discuss the connections of our results with those obtained previously for the random-field problem.

DOI
10.1103/physrevb.44.11767

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We study the two-dimensional (2D) diluted antiferromagnet in a uniform magnetic field using the real-space renormalization group in a hierarchical cell. We follow separately the probability distributions of random-exchange couplings and of magnetic fields along the renormalization process. Due to the random couplings, the initial uniform magnetic field becomes a random variable with a mean value that continuously decreases to zero. In this 2D case the flow of the probability distributions is always toward the attractor of the paramagnetic phase characterized by 〈T/J〉=\ensuremath{\infty} and 〈(h/T)2{)}^{2})2〉=\ensuremath{\infty} in agreement with dc{\mathit{d}}_{\mathit{c}}dc​=2. We discuss the connections of our results with those obtained previously for the random-field problem.
The bond-diluted Ising antiferromagnet in a uniform field (DAFF) on a square lattice is studied and comparison with the random-field problem is made. Position-space renormalisation group calculations for the DAFF give rise to a staircase structure in the zero-temperature phase boundary H (critical field) versus p (bond concentration), separating ordered and disordered phases. Since there is no stable long-range order in the equivalent random-field system, the authors discuss ways in which a consistent picture can be obtained, incorporating both our results and the correspondence between the two problems.
The phase diagram of the two-dimensional Blume-Capel model with a random crystal field is investigated within the framework of a real-space renormalization-group approximation. Our results suggest that, for any amount of randomness, the model exhibits a line of Ising-like continuous transitions, as in the pure model, but no first-order transition. At zero temperature the transition is also continuous, but not in the same universality class as the Ising model. In this limit, the attractor (in the renormalization-group sense) is the percolation fixed point of the site diluted spin-1/2 Ising model. The results we found are in qualitative agreement with general predictions made by Berker and Hui on the critical behavior of random models.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbor 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 + h o and − h o , with the same width σ. This distribution is very general and can recover in appropriate limits the bimodal distribution ( ) and the single-Gaussian one ( h o = 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 occurrence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities h o 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.
The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function <S0(t)Sx(t)><S0(t('))Sx(t('))> which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.