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arXiv:1106.1742 · Journal of Statistical Mechanics Theory and Experiment

Random field Ising model: statistical properties of low-energy excitations and equilibrium avalanches

Cécile Monthus, Thomas Garel

Published Jul 19, 2011• 14 citations
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Abstract

With respect to the usual thermal ferromagnetic transitions, the zero-temperature finite-disorder critical point of the random field Ising model (RFIM) has the peculiarity of involving some ‘droplet’ exponent θ that enters the generalized hyperscaling relation 2 − α = ν( d − θ). In the present paper, to better understand the meaning of this droplet exponent θ beyond its role in the thermodynamics, we discuss the statistics of low-energy excitations generated by an imposed single spin-flip with respect to the ground state, as well as the statistics of equilibrium avalanches, i.e. the magnetization jumps that occur in the sequence of ground states as a function of the external magnetic field. The droplet scaling theory predicts that the distribution dl / l 1 + θ of the linear size l of low-energy excitations transforms into the distribution ds / s 1 + θ/ d f for the size s (number of spins) of excitations of fractal dimension d f ( s ∼ l d f ). In the non-mean-field region d < d c , droplets are compact d f = d , whereas in the mean-field region d > d c , droplets have a fractal dimension d f = 2θ leading to the well-known mean-field result ds / s 3/2 . Zero-field equilibrium avalanches are expected to display the same distribution ds / s 1 + θ/ d f . We also discuss the statistics of equilibrium avalanches integrated over the external field and finite-size behaviors. These expectations are checked numerically for the Dyson hierarchical version of the RFIM, where the droplet exponent θ(σ) can be varied as a function of the effective long-range interaction J ( r ) ∼ 1/ r d + σ in d = 1.

DOI
10.1088/1742-5468/2011/07/p07010

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With respect to the usual thermal ferromagnetic transitions, the zero-temperature finite-disorder critical point of the random field Ising model (RFIM) has the peculiarity of involving some ‘droplet’ exponent θ that enters the generalized hyperscaling relation 2 − α = ν( d − θ). In the present paper, to better understand the meaning of this droplet exponent θ beyond its role in the thermodynamics, we discuss the statistics of low-energy excitations generated by an imposed single spin-flip with respect to the ground state, as well as the statistics of equilibrium avalanches, i.e. the magnetization jumps that occur in the sequence of ground states as a function of the external magnetic field. The droplet scaling theory predicts that the distribution dl / l 1 + θ of the linear size l of low-energy excitations transforms into the distribution ds / s 1 + θ/ d f for the size s (number of spins) of excitations of fractal dimension d f ( s ∼ l d f ). In the non-mean-field region d < d c , droplets are compact d f = d , whereas in the mean-field region d > d c , droplets have a fractal dimension d f = 2θ leading to the well-known mean-field result ds / s 3/2 . Zero-field equilibrium avalanches are expected to display the same distribution ds / s 1 + θ/ d f . We also discuss the statistics of equilibrium avalanches integrated over the external field and finite-size behaviors. These expectations are checked numerically for the Dyson hierarchical version of the RFIM, where the droplet exponent θ(σ) can be varied as a function of the effective long-range interaction J ( r ) ∼ 1/ r d + σ in d = 1.
Domain walls and droplet-like excitation of the random-field Ising magnet are studied in d=3,4,5,6,7d={3,4,5,6,7}d=3,4,5,6,7 dimensions by means of exact numerical ground-state calculations. They are obtained using the established mapping to the graph-theoretical maximum-flow problem. This allows us to study large system sizes of more than 5 $\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins in exact thermal equilibrium. All simulations are carried out at the critical point for the strength hhh of the random fields, h=hc(d)h={h}_{c}(d)h=hc​(d). Using finite-size scaling, energetic and geometric properties like stiffness exponents and fractal dimensions are calculated. Using these results, we test (hyper)scaling relations, which seem to be fulfilled below the upper critical dimension du=6{d}_{u}=6du​=6. Also, for d<dud<{d}_{u}d<du​, the stiffness exponent can be obtained from the scaling of the ground-state energy.
The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ds=2,{d}_{s}=2,ds​=2, the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.
For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J ( r ) ∼ r − σ and distributed with the Lévy symmetric stable distribution of index 1 < μ ≤ 2 (including the usual Gaussian case μ = 2), we consider the region σ > 1/ μ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings J L ∝ L θ μ ( σ ) is found to be whenever the long-ranged couplings are relevant θ μ ( σ ) ≥ −1. For the statistics of the ground state energy over disordered samples, we obtain that the droplet exponent θ μ ( σ ) governs the leading correction to extensivity of the averaged value . The characteristic scale of the fluctuations around this average is of order , and the rescaled variable is Gaussian distributed for μ = 2, or displays the negative power-law tail in 1/(− u ) 1+ μ for u → − ∞ in the Lévy case 1 < μ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.
Abstract We analyze the nonequilibrium athermal random field Ising model (RFIM) at equilateral cubic lattices of finite size L and show that the entire range of disorder consists of three distinct domains in which the model manifests different scaling behaviour. The first domain contains the values of disorder R that are below the critical disorder R c where the spanning avalanches almost surely appear when the system is driven by the external magnetic field. The spanning avalanches become unlikely for disorders above the size-dependent effective disorder R c eff ( L ) > R c , and the system response is size-independent. Between the foregoing two lies the domain of transitional disorders R c < R < R c eff ( L ) vanishing in the thermodynamic limit. In this domain, not recognized in the literature so far, all types of spanning avalanches exist, whereas for R < R c only the avalanches spanning all three dimensions are present. Like for R < R c the data collapsing is possible only for distributions having the same value of (1 − R c / R ) L 1/ ν , however with different universal scaling functions than in R < R c case; ν is the correlation length exponent. The foregoing findings follow from the extensive simulations of L ⩽ 2048 systems enabling us to propose modified values of some of the RFIM critical exponents and nonuniversal critical parameters as well as the analytic forms of universal scaling functions and different definitions of reduced magnetization and reduced magnetic field.
Hysteresis loops are often seen in experiments at first-order phase transformations, when the system goes out of equilibrium. They may have a macroscopic jump (roughly as in the supercooling of liquids) or they may be smoothly varying (as seen in most magnets). We have studied the nonequilibrium zero-temperature random-field Ising-model as a model for hysteretic behavior at first-order phase transformations. As disorder is added, one finds a transition where the jump in the magnetization (corresponding to an infinite avalanche) decreases to zero. At this transition we find a diverging length scale, power-law distributions of noise (avalanches), and universal behavior. We expand the critical exponents about mean-field theory in 6-\ensuremath{\epsilon} dimensions. Using a mapping to the pure Ising model, we Borel sum the 6-\ensuremath{\epsilon} expansion to O(${\mathrm{\ensuremath{\epsilon}}}^{5}$) for the correlation length exponent. We have developed a method for directly calculating avalanche distribution exponents, which we perform to O(\ensuremath{\epsilon}). Our analytical predictions agree with numerical exponents in two, three, four, and five dimensions [Perkovi\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. Lett. 75, 4528 (1995)]. \textcopyright{} 1996 The American Physical Society.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.