scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Percolation fractal dimension in scattering line shapes of the random-field Ising model

TL;DR: In this article, the authors show evidence for fractal structure from spanning clusters in the d=2 and 3 random field Ising models as realized in dilute antiferromagnets.
About: This article is published in Journal of Magnetism and Magnetic Materials.The article was published on 2004-05-01 and is currently open access. It has received 6 citations till now. The article focuses on the topics: Fractal & Ising model.
Citations
More filters
Journal ArticleDOI
TL;DR: By performing a high-statistics simulation of the D=3 random-field Ising model at zero temperature for different shapes of the random- field distribution, it is shown that the model is ruled by a single universality class, and that scaling is described by two independent exponents.
Abstract: We solve a long-standing puzzle in statistical mechanics of disordered systems. By performing a high-statistics simulation of the $D=3$ random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute the complete set of critical exponents for this class, including the correction-to-scaling exponent, and we show, to high numerical accuracy, that scaling is described by two independent exponents. Discrepancies with previous works are explained in terms of strong scaling corrections.

99 citations

Journal ArticleDOI
TL;DR: In this paper, the authors performed large-scale Monte Carlo simulations using the Machta-Newman-Chayes algorithms to study the critical behavior of both the diluted antiferromagnet in a field with 30% dilution and the random-field Ising model with Gaussian random fields for different field strengths.
Abstract: We perform large-scale Monte Carlo simulations using the Machta-Newman-Chayes algorithms to study the critical behavior of both the diluted antiferromagnet in a field with 30% dilution and the random-field Ising model with Gaussian random fields for different field strengths. Analytical calculations by Cardy [Phys. Rev. B 29, 505 (1984)] predict that both models map onto each other and share the same universality class in the limit of vanishing fields. However, a detailed finite-size scaling analysis of the Binder cumulant, the two-point finite-size correlation length, and the susceptibility suggests that even in the limit of small fields, where the mapping is expected to work, both models are not in the same universality class. Based on our numerical data, we present analytical expressions for the phase boundaries of both models.

14 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present structure factor data for two systems: (a) droplet-in-droplet morphologies of double-phase separating mixtures; and (b) ground-state morphologies in dilute anti-ferromagnets.
Abstract: Many experiments yield multi-scale morphologies which are smooth on some length scales and fractal on others. Accurate statements about morphological properties, e.g., roughness exponent, fractal dimension, domain size, interfacial width, etc. are obtained from the correlation function and structure factor. In this paper, we present structure factor data for two systems: (a) droplet-in-droplet morphologies of double-phase-separating mixtures; and (b) ground-state morphologies in dilute anti-ferromagnets. An important characteristic of the scattering data is a non-Porod tail, which is associated with scattering off rough domains and interfaces.

11 citations

Journal ArticleDOI
TL;DR: In this paper, the critical exponent for the three-dimensional random-field Ising model (RFIM) order parameter upon zero-field cooling (ZFC) using extinction-free magnetic x-ray scattering techniques was determined.
Abstract: The critical exponent {beta}=0.17{+-}0.01, where the quoted statistical error is from fits to the data, has been determined for the three-dimensional random-field Ising model (RFIM) order parameter upon zero-field cooling (ZFC) using extinction-free magnetic x-ray scattering techniques for Fe{sub 0.85}Zn{sub 0.15}F{sub 2}. This result is consistent with other exponents determined for the RFIM in that Rushbrooke scaling is satisfied. Nevertheless, there is poor agreement with equilibrium computer simulations, and the ZFC results do not agree with field-cooling results. We present details of hysteresis in Bragg scattering amplitudes and line shapes that help elucidate the effects of thermal cycling in the RFIM, as realized in dilute antiferromagnets in an applied field. We show that the ZFC critical-like behavior is consistent with a second-order phase transitions, albeit quasistationary rather than truly equilibrium in nature, as evident from the large thermal hysteresis observed near the transition.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the authors extended the ground-state algorithm by various methods in order to analyze low-energy excitations for the three-dimensional RFIM with Gaussian distributed disorder that appear in the form of clusters of connected spins.
Abstract: The random-field Ising model (RFIM), one of the basic models for quenched disorder, can be studied numerically with the help of efficient ground-state algorithms. In this study, we extend these algorithm by various methods in order to analyze low-energy excitations for the three-dimensional RFIM with Gaussian distributed disorder that appear in the form of clusters of connected spins. We analyze several properties of these clusters. Our results support the validity of the droplet-model description for the RFIM.

9 citations

References
More filters
Journal ArticleDOI
TL;DR: For random-field models, this work rigorously proves uniqueness of the Gibbs state 2D Ising systems, and absence of continuous symmetry breaking in the Heisenberg model in d\ensuremath{\le}4, as predicted by Imry and Ma.
Abstract: It is shown, by a general argument, that in 2D quenched randomness results in the elimination of discontinuities in the density of the thermodynamic variable conjugate to the fluctuating parameter. Analogous results for continuous symmetry breaking extend to d\ensuremath{\le}4. In particular, for random-field models we rigorously prove uniqueness of the Gibbs state 2D Ising systems, and absence of continuous symmetry breaking in the Heisenberg model in d\ensuremath{\le}4, as predicted by Imry and Ma. Another manifestation of the general statement is found in 2D random-bond Potts models where a phase transition persists, but ceases to be first order.

453 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the Ising model with a small random magnetic field has two phases at low temperatures, i.e., its lower critical dimension is at most two.
Abstract: We show that the Ising model in three dimensions with a small random magnetic field has two phases at low temperatures, i.e., that its lower critical dimension is at most two. This is shown by our devising an exact renormalization-group flow which takes the theory to the zero-temperature, zero-field fixed point.

217 citations

Journal ArticleDOI
TL;DR: In this paper, the authors employed scaling techniques to extract the universal equilibrium scattering line shape, critical exponents {nu}=087{plus_minus}007 and {eta}=020{plus-minus}005, and amplitude ratios of this RFIM system.
Abstract: It has long been believed that equilibrium random-field Ising model (RFIM) critical scattering studies are not feasible in dilute antiferromagnets close to and below T{sub c}(H) because of severe nonequilibrium effects The high magnetic concentration Ising antiferromagnet Fe{sub 093} Zn{sub 007} F{sub 2} , however, does provide equilibrium behavior We have employed scaling techniques to extract the universal equilibrium scattering line shape, critical exponents {nu}=087{plus_minus}007 and {eta}=020{plus_minus}005 , and amplitude ratios of this RFIM system {copyright} {ital 1999} {ital The American Physical Society }

44 citations

Journal ArticleDOI
TL;DR: The ground-state structure of the two-dimensional random field Ising magnet is studied using exact numerical calculations and it is shown that even for H=0 above L(b) the domains can be fractal at low random fields, such that the largest domain spans the system at lowrandom field strength values and its mass has the fractal dimension of standard percolation D(f)=91/48.
Abstract: The ground-state structure of the two-dimensional random field Ising magnet is studied using exact numerical calculations. First we show that the ferromagnetism, which exists for small system sizes, vanishes with a large excitation at a random field strength-dependent length scale. This breakup length scale L{sub b} scales exponentially with the squared random field, exp(A/{Delta}{sup 2}). By adding an external field H, we then study the susceptibility in the ground state. If L{gt}L{sub b}, domains melt continuously and the magnetization has a smooth behavior, independent of system size, and the susceptibility decays as L{sup {minus}2}. We define a random field strength-dependent critical external field value {+-}H{sub c}({Delta}) for the up and down spins to form a percolation type of spanning cluster. The percolation transition is in the standard short-range correlated percolation universality class. The mass of the spanning cluster increases with decreasing {Delta} and the critical external field approaches zero for vanishing random field strength, implying the critical field scaling (for Gaussian disorder) H{sub c}{similar_to}({Delta}{minus}{Delta}{sub c}){sup {delta}}, where {Delta}{sub c}=1.65{+-}0.05 and {delta}=2.05{+-}0.10. Below {Delta}{sub c} the systems should percolate even when H=0. This implies that even for H=0 above L{sub b} the domains can be fractal at low random fields, suchmore » that the largest domain spans the system at low random field strength values and its mass has the fractal dimension of standard percolation D{sub f}=91/48. The structure of the spanning clusters is studied by defining red clusters, in analogy to the {open_quotes}red sites{close_quotes} of ordinary site percolation. The sizes of red clusters define an extra length scale, independent of L.« less

34 citations

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the percolation of the minority-spin orientation in the paramagnetic phase above the bulk phase transition, located at $[\ensuremath{\Delta}/J{]}_{c}(1.31\ifmmode\pm\else\textpm\fi{}0.43
Abstract: The structure of the three-dimensional (3D) random field Ising magnet is studied by ground-state calculations. We investigate the percolation of the minority-spin orientation in the paramagnetic phase above the bulk phase transition, located at $[\ensuremath{\Delta}/J{]}_{c}\ensuremath{\simeq}2.27,$ where $\ensuremath{\Delta}$ is the standard deviation of the Gaussian random fields $(J=1).$ With an external field H there is a disorder-strength-dependent critical field $\ifmmode\pm\else\textpm\fi{}{H}_{c}(\ensuremath{\Delta})$ for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. ${H}_{c}\ensuremath{\sim}(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{p}{)}^{\ensuremath{\delta}},$ where ${\ensuremath{\Delta}}_{p}=2.43\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\delta}=1.31\ifmmode\pm\else\textpm\fi{}0.03,$ implying a critical line for ${\ensuremath{\Delta}}_{c}l\ensuremath{\Delta}l~{\ensuremath{\Delta}}_{p}.$ When, with zero external field, $\ensuremath{\Delta}$ is decreased from a large value there is a transition from the simultaneous up- and down-spin spanning, with probability ${\ensuremath{\Pi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=1.00$ to ${\ensuremath{\Pi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}=0.$ This is located at $\ensuremath{\Delta}=2.32\ifmmode\pm\else\textpm\fi{}0.01,$ i.e., above ${\ensuremath{\Delta}}_{c}.$ The spanning cluster has the fractal dimension of standard percolation, ${D}_{f}=2.53$ at ${H=H}_{c}(\ensuremath{\Delta}).$ We provide evidence that this is asymptotically true even at $H=0$ for ${\ensuremath{\Delta}}_{c}l\ensuremath{\Delta}l~{\ensuremath{\Delta}}_{p}$ beyond a crossover scale that diverges as ${\ensuremath{\Delta}}_{c}$ is approached from above. Percolation implies extra finite-size effects in the ground states of the 3D random field Ising model.

19 citations

Frequently Asked Questions (1)
Q1. What have the authors contributed in "Percolation fractal dimension in scattering line shapes of the random-field ising model" ?

In this paper, the authors show evidence for fractal structure from spanning clusters in the d = 2 and d = 3 random field Ising models as realized in dilute antiferromagnets.