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arXiv:cond-mat/9505056 · Europhysics Letters (EPL)

Scaling treatment of the random-field Ising model

Robin B. Stinchcombe, E. D Moore, S. L. A. de Queiroz

Published Aug 1, 1996• 4 citations
arXivHTMLPDF

Abstract

Analytic phenomenological scaling is carried out for the random field Ising model in general dimensions d using a bar geometry. Domain wall configurations and their decorated profiles and associated wandering and other exponents (ζ,γ,δ,μ) are obtained by free-energy minimization. Scaling between different bar widths provides the renormalization group (RG) transformation. Its consequences are i) criticality at h = T = 0 in d ⩽ 2 with correlation length ξ( h , T ) diverging like ξ( h ,0) ∝ h −2/(2 − d ) for d < 2 and ξ( h ,0) ∝ exp [1/( c 1 γ h γ )] for d = 2, where c 1 is a decoration constant; ii) criticality in d = 2 + ε dimensions at T = 0, h * = (ε/2 c 1 ) 1/γ , where ξ ∝ [( s − s *)/ s ] −2ε/γ , s ≡ h γ . Finite-temperature generalizations are outlined. Numerical transfer matrix calculations and results from a ground-state algorithm adapted for strips in d = 2 confirm the ingredients which provide the RG description.

DOI
10.1209/epl/i1996-00569-0

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A simple scaling description of the ordering transition in random-field Ising systems is developed and supported by renormalization-group arguments in terms of a zero-temperature critical fixed point. The main prediction is that the characteristic relaxation time \ensuremath{\tau} will diverge extremely rapidly as the critical point is approached: \ensuremath{\tau}\ensuremath{\sim}exp(${\ensuremath{\xi}}^{\mathrm{theta}}$) with \ensuremath{\xi} the correlation length and theta the ``violation of hyperscaling'' exponent (d-theta)\ensuremath{\nu}=2-\ensuremath{\alpha}. Recent experiments which exhibit onset of hysteresis in a very narrow temperature range are discussed.
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.
The random Heisenberg model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall energies at zero temperature is calculated using a spin-quench algorithm to find the ground-state energy for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall energy distribution when the lattice parameter is changed. In the strong-coupling regime (zero temperature) the model iterates toward weak coupling and therefore exhibits a ``phase transition at zero temperature'' in both two and three dimensions. The correlation-length exponent is \ensuremath{\nu}=0.714\ifmmode\pm\else\textpm\fi{}0.015 in two dimensions and \ensuremath{\nu}=1.54\ifmmode\pm\else\textpm\fi{}0.19 in three dimensions. The lower critical dimension is four.
For the Ising model with Gaussian random coupling of average J0{J}_{0}J0​ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter J0{J}_{0}J0​ can be studied via the size-LLL-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension d=2,3,4,5,6d=2,3,4,5,6d=2,3,4,5,6. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.