Lower critical dimension for the random-field Ising model

Bricmont, Jean; Kupiainen, Antti

doi:10.1103/physrevlett.59.1829

Cited by 436 publications

(219 citation statements)

References 14 publications

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“…At high magnetic concentrations domain walls must cut a large number of magnetic bonds and long-range order (LRO) is stable, as in the Imry-Ma domain wall energy arguments [6]. Hence, the RFIM can be studied in equilibrium for x = 0.93 and a transition to LRO is observed, consistent with theory [7]. The question remains as to the nature of the disappearance of the domain walls as the magnetic concentration increases.…”

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confidence: 52%

The random field critical concentration in dilute antiferromagnets

Barber

Belanger

2000

Journal of Applied Physics

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Monte Carlo techniques are used to investigate the equilibrium threshold concentration, xe, in the dilute anisotropic antiferromagnet F exZn1−xF2 in an applied magnetic field, considered to be an ideal random-field Ising model system. Above xe equilibrium behavior is observed whereas below xe metastability and domain formation dominate. Monte Carlo results agree very well with experimental data obtained using this system.The dilute antiferromagnet (AF) F e x Zn 1−x F 2 in an applied field is a realization [1,2] of the random-field Ising model (RFIM). Many studies [3] have been done on this system for x < 0.75. In such cases, metastability and domain formation mask the equilibrium critical behavior, particularly in scattering measurements that are by nature dominated by long-range antiferromagnetic correlations. The specific heat, C m , on the other hand, is not as greatly affected by domain formation except very close to the transition, T c (H), since it is primarily sensitive to short range correlations [4]. Only recently has it been discovered [5] that equilibrium scattering behavior can be observed for x = 0.93 with no evidence of domain formation. Domain walls form with little energy cost when vacancies are so numerous that magnetic bonds can be largely avoided. At high magnetic concentrations domain walls must cut a large number of magnetic bonds and long-range order (LRO) is stable, as in the Imry-Ma domain wall energy arguments [6]. Hence, the RFIM can be studied in equilibrium for x = 0.93 and a transition to LRO is observed, consistent with theory [7]. The question remains as to the nature of the disappearance of the domain walls as the magnetic concentration increases. Do they disappear gradually or is there a critical concentration above which they do not form?We have performed Monte Carlo (MC) simulations of the RFIM modeled as closely as possible 1

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confidence: 52%

The random field critical concentration in dilute antiferromagnets

Barber

Belanger

2000

Journal of Applied Physics

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“…Dimensional reduction claims that the critical behavior of the d-dimensional random field O(N ) spin model is the same as of the (d−2)-dimensional pure O(N ) spin model, where d is the spatial dimension. It has been shown by rigorous proofs 4,5 and numerical calculations of critical exponents 6,7,8,9 that the prediction of dimensional reduction is incorrect in the random field Ising model below four dimensions. In dimensions more than 4, however, the critical phenomena of the random field O(N ) spin model should be further studied.…”

Section: Introductionmentioning

confidence: 99%

Effect of second-rank random anisotropy on critical phenomena of a random-fieldO(N)spin model in the largeNlimit

2005

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We study the critical behavior of a random field O(N ) spin model with a second-rank random anisotropy term in spatial dimensions 4 < d < 6, by means of the replica method and the 1/N expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the 1/N expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large N limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large N . Thus, we conclude that the dimensional reduction holds for sufficiently large N .

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“…The RFIM phase transition is believed to be in the same universality class as the phase transitions in diluted antiferromagnets in a uniform field and fluids in porous media. The three dimensional RFIM is known [1][2][3] to have an ordered phase at sufficiently low temperature and for weak random fields. As the temperature or the strength of the randomness is increased, there is a transition to a disordered phase.…”

Section: Introductionmentioning

confidence: 99%

“…These simulations have been limited to system size 16 3 . The jump in the magnetization can be interpreted as a very small value of the magnetization exponent but might also signal a first-order transition.…”

Section: Introductionmentioning

confidence: 99%

Replica-exchange algorithm and results for the three-dimensional random field Ising model

2000

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The random field Ising model with Gaussian disorder is studied using a new Monte Carlo algorithm. The algorithm combines the advantanges of the replica exchange method and the two-replica cluster method and is much more efficient than the Metropolis algorithm for some disorder realizations. Three-dimensional sytems of size 24 3 are studied. Each realization of disorder is simulated at a value of temperature and uniform field that is adjusted to the phase transition region for that disorder realization. Energy and magnetization distributions show large variations from one realization of disorder to another. For some realizations of disorder there are three well separated peaks in the magnetization distribution and two well separated peaks in the energy distribution suggesting a first-order transition.

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Lower critical dimension for the random-field Ising model

Cited by 436 publications

References 14 publications

The random field critical concentration in dilute antiferromagnets

The random field critical concentration in dilute antiferromagnets

Effect of second-rank random anisotropy on critical phenomena of a random-fieldO(N)spin model in the largeNlimit

Replica-exchange algorithm and results for the three-dimensional random field Ising model

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