Weak Random Fields Research Articles

Fast magnetoacoustic waves in a randomly structured solar corona

The propagation of fast magnetoacoustic waves in a randomly structured solar corona is considered in the linear and cold plasma limits. The random field is assumed to be static and associated with plasma density inhomogeneities only. A transcendental dispersion relation for the fast magnetoacoustic waves which propagate perpendicularly to the magnetic field is derived in the weak random field approximation. It is shown analytically that the fast magnetosonic waves experience acceleration, attenuation, and dispersion in comparison to the homogeneous case. These analytical findings are essentially confirmed by numerical simulations for a wide-spectrum pulse, except that the waves were found decelerated. It is concluded that the coronal Moreton waves can be applied to MHD seismology of the solar corona.

Nonlinear response of relaxor ferroelectrics

The third order nonlinear response χ3 of relaxor ferroelectrics has been calculated within the framework of a recently proposed spherical random bond-random field model. It is assumed that the order parameter field can be associated with the polarization of nanosized polar clusters and is thus a continuous vector of variable length subject to a global spherical constraint. It is shown that for weak random fields the scaled third order nonlinear dielectric susceptiblity a 3 = χ3/χ4 1, where χ1 is the linear susceptibility, has a narrow peak in the spherical glass phase without long range order, but there is no sharp freezing transition. In case of weak random bonds the system behaves as a random field frustrated ferroelectric, with a 3 making a jump at the critical temperature.

Spherical random-bond–random-field model of relaxor ferroelectrics

A model of relaxor ferroelectrics based on the interacting polar clusters picture has been formulated. The electric dipole moment of a nanosized polar domain is allowed a large number of discrete orientations and its length is assumed to fluctuate in a broad interval. Introducing a set of quasicontinuous order parameter fields and imposing a global spherical constraint, the spherical random-bond--random-field (SRBRF) model is written down and its static properties are investigated. It is found that for weak random fields the scaled third-order nonlinear susceptibility ${a}_{3}={\ensuremath{\chi}}_{3}/{\ensuremath{\chi}}_{1}^{4}$ shows a nearly divergent behavior in the spherical glass phase, but there is no such anomaly in a random-field frustrated ferroelectric state. The probability distribution of local cluster polarization is calculated and its relation to the quadrupole perturbed NMR line shape of ${}^{93}\mathrm{Nb}$ in PMN is discussed. The fact that the observed line shape is Gaussian at all temperatures provides strong support to the SRBRF model.

Current Noise in an Irradiated Point Contact

We propose a new approach to calculate current and current correlations in a ballistic quantum point contact interacting with a classical field. The approach is based on the concept of scattering states for a time dependent Hamiltonian neglecting electron-electron interaction. Using this approach we calculated the spectra of the current noise in a biased point contact irradiated by a weak random field. For typical radiation frequencies \nu less than the temperature T and the bias voltage V we find a narrow peak of width \nu on top of a broad background of width \max (T,eV).

Dynamic criticality in driven disordered systems: role of depinning and driving rate in Barkhausen noise

We study Barkhausen noise in a diluted two-dimensional Ising model with the extended domain wall and weak random fields occurring due to coarse graining. We report two types of scaling behavior corresponding to (a) low disorder regime where a single domain wall slips through a series of positions when the external field is increased, and (b) large disorder regime, which is characterized with nucleation of many domains. The effects of finite concentration of nonmagnetic ions and variable driving rate on the scaling exponents is discussed in both regimes. The universal scaling behavior at low disorder is shown to belong to a class of critical dynamic systems, which are described by a fixed point of the stochastic transport equation with self-consistent disorder correlations.

Mean-field calculations for the axial next-nearest-neighbor Ising model in a random field

We consider a mean-field version of an Ising model on a simple cubic lattice with competing interactions between nearest and next-nearest neighbors along the $z$ axis in the presence of a random field. We obtain phase diagrams in terms of the temperature, the ratio of the strengths of the competing interactions, and the intensity of the $\ifmmode\pm\else\textpm\fi{}H$ random fields. There are no drastic qualitative changes for weak random fields. The effect of moderate random fields consists in the change of order of the transitions to the disordered phase. Although the transition from the paramagnetic to the ordered (ferromagnetic and modulated) phases in the absence of a random field is always continuous, for random fields of moderate intensity we show the appearance of tricritical points separating the critical lines (at higher temperatures) from a line of first-order transitions (at lower temperatures). We show the presence of modulated phases with disordered layers (with zero magnetization) and the pinching of the corresponding regions in the phase diagram. The modulated structures are destroyed in the presence of strong random fields.

Dynamic Susceptibility for Ising-Spin Clusters in Random Fields

The dynamic susceptibility for a cluster of six coupled random field Ising spins in two different distributions, binary (BD) and Gaussian (GD), are calculated and exact results are obtained. The real and imaginary parts of the dynamic susceptibility display maxima when plotted versus temperature. These maxima can be described by an Arrhenius law. If the logarithm of the susceptibilities is plotted as a function of the logarithm of frequency and if the clusters are frustrated, then the real part displays a sequence of plateau regions and the imaginary part has a sequence of maxima in weak random fields. In the BD case of random field for large amplitudes there is only one plateau and one corresponding maximum as in ferromagnetic (FM) and paramagnetic (PM) cases. Our results confirm that any weak random field will turn out to destroy the ordered state and random field Ising-spin clusters behave like Ising-spin glasses.

Neutron and x-ray scattering studies of field-cooled orderingin the three-dimensional random-field Ising model

We report a magnetic neutron and x-ray scattering study of the correlation length and order parameter of the random-field Ising magnets ${\mathrm{Fe}}_{0.5}$ ${\mathrm{Zn}}_{0.5}$ ${\mathrm{F}}_{2}$ and ${\mathrm{Mn}}_{0.45}$ ${\mathrm{Zn}}_{0.55}$ ${\mathrm{F}}_{2}$ on field cooling. Fitting the inverse correlation length, \ensuremath{\kappa}, measured above the metastability temperature, ${\mathrm{T}}_{\mathrm{M}}$ (H) in ${\mathrm{Fe}}_{0.5}$ ${\mathrm{Zn}}_{0.5}$ ${\mathrm{F}}_{2}$ , at H=5 and 6 T to a power law yields the exponent \ensuremath{\nu}=1.5\ifmmode\pm\else\textpm\fi{}0.3 together with the equilibrium transition temperatures, ${\mathrm{T}}_{\mathrm{N}}$ (H), which are found to be well below ${\mathrm{T}}_{\mathrm{M}}$ (H). We also estimate \ensuremath{\gamma}=2.6\ifmmode\pm\else\textpm\fi{}0.5 and \ensuremath{\gamma}-bar=5.7\ifmmode\pm\else\textpm\fi{}1 for the connected susceptibility, \ensuremath{\chi}, and the disconnected susceptibility, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$ , exponents, respectively. The x-ray data reveal a long range ordered component coexisting with short range magnetic order in the field-cooled state, for weak random fields in both samples. The nucleation of the long range order occurs below ${\mathrm{T}}_{\mathrm{M}}$ (H). The intensity of the long range order component decreases with increasing field.

The three-dimensional three-state Potts ferromagnet exposed to random fields: evidence for a second order transition

Using large scale Monte Carlo simulations, the ordering of the three-dimensional three state Potts ferromagnet exposed to random fields is investigated. Studies of the order parameter probability distribution and of various of its moments suggest that the order of the transition depends on the strength of the random field: i.e., the first order transition of the pure ferromagnetic model persists for weak random fields, but turns into a second order transition for a range of random fields of medium strength. For large random fields the transition seems to be first order again. In this range large domains of strongly aligned Potts spins occur already in the disordered phase and the associated slow relaxation hampers significantly the Monte Carlo study of thermodynamic equilibrium phenomena. These results are discussed in the light of current theoretical concepts. Possible applications to experiments on diluted anisotropic molecular crystals and orientational glasses are briefly mentioned.

Numerical Calculations for the Surface on a Semi-Infinite Ising Model with a Random Surface Field11The project supported by National Natural Science Foundation of China and the Chinese Academy of Sciences through Grant LWTZ-1298

We study the critical behavior of the surface on a semi-infinite simple cubic lattice Ising model with a bimodal random surface field by large cell mean-field renormalization group method and Monte Carlo simulations. Our results show that the surface ferromagnetic phase exists in the weak random field range above the bulk critical temperature. The surface specific heat is not divergence and the susceptibility show a cusp singularity at the surface ferromagnetic-paramagnetic transition for a relatively large random field.

Griffith singularities in the random-field Ising model

The D-dimensional ferromagnetic Ising model with weak Gaussian random fields is considered. In dimensions D<3 due to rare large-scale thermal excitations (large-spin clusters with magnetization opposite to the ferromagnetic background) in the low-temperature region h02<<T<<1 (where h0 is the characteristic value of the field) the free energy is shown to contain a non-analytic contribution of the form exp(-(const/2h02)(h02/T)(3-D/2)).

Strongly coupled Ising chain under a weak random field

We present here an analytical method based on the transfer matrix M to study quenched disorder in one-dimensional spin systems in the limit of strong couplings and weak disorder. The procedure is formulated for the random-field Ising chain of finite length L, and its properties, represented as functions of M, satisfy a differential equation of the Fokker-Planck type. This equation describes “evolution” with L, and a central-limit theorem of a novel kind provides the equation and its solutions with universal character. We obtain analytical expressions for the moments of the magnetization of the infinite length chain and study the approach to the infinite coupling limit. We find that the random-field free energy f H and the Edwards-Anderson order parameter m 2 satisfy a simple relation. We discuss our results in connection to previous work by Luck and Nieuwenhuizen.

NMR determination of order parameters in the quadrupolar glasses Na(CN)xCl1−xandNaxK1−xCN

It is shown that quadrupolar perturbed nuclear magnetic resonance (NMR) is a powerful method to investigate the quadrupolar glasses Na(CN${)}_{\mathit{x}}$${\mathrm{Cl}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$ and ${\mathrm{Na}}_{\mathit{x}}$${\mathrm{K}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$CN. In both systems at the sodium and chlorine sites, distributions of electric-field-gradient tensors occur which are restricted by the fact that the average structure of the systems under investigation is cubic. Correspondingly, inhomogeneous distributions of NMR lines result, which for I=3/2 nuclear-spin systems consist of inhomogeneously broadened central lines and broad distributions of satellite lines. Measurements of these frequency distributions and their dependences on the composition, the orientation, and the temperature of the samples are presented. The widths of the electric-field-gradient-tensor distributions are related in a general quadrupolar glass model to the quadrupolar Edwards-Anderson order parameter ${\mathit{q}}_{\mathrm{EA}}$. As a consequence, the temperature dependence of ${\mathit{q}}_{\mathrm{EA}}$ is derived, reflecting the random orientational freeze-out of the CN quadrupoles with decreasing temperature. By interpreting the results in terms of theoretical models, it is shown that in the mixed cyanides we deal with a smearing of a collective quadrupolar glass transition by weak random fields and not with a pure random-field-type freezing or a pure random-bond-type glass transition. The results are compared to those obtained from other experimental methods. In particular, the critical elastic behavior of these systems is discussed in a general context.

Random field systems

Abstract Spin models in weak random fields are good representations of a large number of impure compounds which undergo magnetic or structural transitions. From a theoretical point of view these systems exhibit frustration on a mesoscopic scale and take therefore an indermediate position between (spin) glasses and unfrustrated (diluted) disordered systems. A simple but rigorously proven domain argument shows that long range order is not destroyed by weak random fields in more than two dimensions if the order parameter has a discrete (e.g. Ising) symmetry. The characteristic interaction energy between the spins and the frozen-in defects increases as L Θ with the length scale L so that the static random-field-induced fluctuations dominate over the dynamic thermal fluctuations. Consequently, the critical behaviour is described by a T = 0 fixed point of the RNG transformation. This leads to pronounced non-classical critical exponents, a modified hyper-scaling v(d - Θ) = 2 – α and an activated critical dynamic...

Localization and the integer-quantized Hall effect: Diffusion constant for high Landau levels and white noise potential

The diffusion constant and the diagonal conductivity for non-interacting electrons in a two-dimensional, disordered system are studied. A homogeneous magnetic field perpendicular to the electron system is assumed. For weak short-range random potentials and high fields the Landau quantum numbern can be used as expansion parameter. In the limit of high Landau levels the system shows metallic behaviour. Corrections for finiten decrease the conductivity and indicate localized states in the whole energy band. A breakdown of the expansion and stronger localization are observed only for the lowest Landau levels if the typical experimental length scale of the quantized Hall effect is used.

RANDOM FIELD AND OTHER SYSTEMS DOMINATED BY DISORDER FLUCTUATIONS

Spin-models in random fields (RFs) are good representations of many impure materials. Their macroscopic collective behaviour is dominated by the fluctuations in the random fields which accumulate on large scales even if the local field is arbitrarily small. This feature is shared by other weakly disordered models, like flux lines or domain walls in random media. We review some of the main theoretical attempts to describe such systems. A modification of Harris’ argument demonstrates that at the critical point the RF disorder is relevant and that (hyper)scaling must be changed. A domain argument invented by Imry and Ma shows that long-range order is not destroyed by weak RFs in more than d=2 dimensions. This result is supported both by a more refined treatment of the domain argument and by considering the roughness of an isolated domain wall due to the randomness. The wall (or flux line) becomes rough due to disorder but if d&gt;2 the wall remains a well-defined object in RF systems. Different approaches are used to calculate the roughness exponent ζ for walls and lines. Some applications of ζ for the description of type-II superconductors and incommensurate systems are given. More detailed calculations are possible for one-dimensional, Bethe-lattice or the hierarchical Dyson model systems, which confirm as a rule the more approximate treatment of the other sections. In one dimension there is an interesting relation between the statistical mechanics of these models and nonlinear dynamics. Non-classical critical behaviour occurs in RF systems for d&lt;6 and is determined in general by three independent exponents which fulfil certain inequalities. The new exponent θ≡yJ&gt;0 is related to the violation of conventional hyperscaling and is determined by the energy ~H0ξ0 of a correlated region of size ξ. In a renormalization group treatment, the temperature T turns out to be a (dangerous) irrelevant variable which is the most prominent property of the systems considered in this review. The irrelevance of thermal fluctuations on large scales produces metastability and hysteresis effects both in the transition region and in the ordered phase, only briefly considered here. These features occur also in other systems with a disordered T=0 fixed point like in the ordered phase of a spin-glass.

Non-ergodic instability and the quadrupolar glass state in K(CN) x Br1?x and (NaCN) x (KCN)1?x

Random strain fields, due to substitutional disorder, couple to translational and orientational dynamic modes in plastic crystals. One distinguishes the cases of weak and strong random fields. In weak fields, the crystal comes close to a structural phase transition. Then elastic restoring forces are weak while internal friction dominates. The resulting non ergodic instability marks the onset of a structural glass state. The evolution of the glass state is discussed within a selfconsistent theory. The results of an extensive numerical study are compared with experiment. In presence of strong random fields no instability occurs, and orientational freezing is a continuous process.

Random‐field ising systems: A survey of current theoretical views

Abstract Ising or Ising-like models in random fields are good representations of a large number of impure materials. The main attempts of theoretical treatments of these models--as far as they are relevant from an experimental point of view--are reviewed. A domain argument invented by Imry and Ma shows that the long-range order is not destroyed by weak random-fields in more than D = 2 dimensions. This result is supported by considerations of the roughening of an isolated domain wall in such systems: domain walls turn out to be well defined objects for D > 2, but arbitrarily convoluted for D < 2. Different approaches for calculating the roughness exponent ζ yield ζ= (5 - D)/3 in random-field systems. The application of ζ in incommensurate-commensurate critical behaviour is discussed. Non-classical critical behaviour occurs in random-field systems below D = 6 dimensions which is determined in general by three independent exponents. The new exponent yJ = θ= D/2 - σ corresponds to random-field renormalization...

Diagrammatic expansion and metastability in the random-field Ising model

Within the perturbation diagrammatic expansion we discuss the origin of differences in determinations of the lower critical dimension of the random-field Ising model and show that below four dimensions metastability and hysteresis occur. We also explain the occurrence of a quasicritical d=2 behavior at weak random fields, which is responsible for local stability of the ordered state above two dimensions.

The Random-Field Ising Model: A Free Energy Study

Thermodynamic properties of the random-field Ising model are reviewed and compared with published experimental results on diluted antiferromagnets in a uniform field. The Landau-Ginzburg free energy is calculated in the self-consistent approximation for 3-dimensional systems treating the random field induced fluctuations and the thermodynamic fluctuations of the order parameter separately. At low temperatures and in the presence of random fields two states exist, the stable, long-range ordered state, and the metastable disordered microdomain state. In a heating process, the ordered state changes to the high-temperature phase at the well defined transition temperature. The transition is first order for arbitrarily weak random field. Critical properties are briefly examined and according to the Ginzburg criterion, the classical critical behaviour is predicted for large random fields. For weak fields, nonclassical critical behaviour is expected.