Finite-size scaling in Ising-like systems with quenched random fields: Evidence of hyperscaling violation

Vink, R. L. C.; Fischer, Timo; Binder, Kurt

doi:10.1103/physreve.82.051134

Cited by 45 publications

(70 citation statements)

References 67 publications

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“…Currently, despite the huge efforts recorded in the literature, a clear picture of the model's critical behavior is still lacking. Although the view that the phase transition of the RFIM is of second order is well established [48][49][50]64], the extremely small value of the exponent β continues to cast some doubts. Moreover, a rather strong debate exists with regards to the role of disorder: the available simulations are not able to settle the question of whether the critical exponents depend on the particular choice of the distribution for the random fields, analogous to the mean-field theory predictions [37].…”

Section: Introductionmentioning

confidence: 99%

“…However, typical Monte Carlo schemes get trapped into local minima with escape time exponential in ξ θ , where ξ denotes the correlation length. Although sophisticated improvements have appeared [48][49][50][51][52], these simulations produced low-accuracy data because they were limited to linear sizes of the order of L max 32. Larger sizes can be achieved at T = 0, through the wellknown mapping of the ground state to the maximum-flow optimization problem [53][54][55][56][57][58][59][60][61][62][63][64][65][66].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

Fytas

Martı́n-Mayor²

2016

Phys. Rev. E

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It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero-and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent α of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

Fytas

Martı́n-Mayor²

2016

Phys. Rev. E

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“…This is the hyperscaling violations exponent θ , which can be related to the free-energy barrier between the ordered and the disordered phase: F ∝ L θ . 14 The computation of these barriers is very difficult with traditional methods, but straightforward with TMC. Indeed, we can identify F with the ¯ N between the two saddle points (disordered and antiferromagnetic) defined by the critical h c .…”

mentioning

confidence: 99%

Critical behavior of the dilute antiferromagnet in a magnetic field

Fernández¹,

Martı́n-Mayor²,

Yllanes³

2011

Phys. Rev. B

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We study the critical behavior of the diluted antiferromagnet in a field with the tethered Monte Carlo formalism. We compute the critical exponents (including the elusive hyperscaling violations exponent θ). Our results provide a comprehensive description of the phase transition and clarify the inconsistencies between previous experimental and theoretical work. To do so, our method addresses the usual problems of numerical work (large tunneling barriers and self-averaging violations). Understanding collective behavior in the presence of quenched disorder has long been one of the most challenging and interesting problems in statistical mechanics. One of its simplest representatives is the random field Ising model (RFIM), which has been extensively studied both theoretically and experimentally. DOI1 The RFIM is physically realized by a diluted antiferromagnet in an applied magnetic field (DAFF).It is known that the D = 3 DAFF (or RFIM) undergoes a phase transition, but the details remain controversial, with severe inconsistencies between analytical, experimental, and numerical work. A scaling theory is generally accepted, where the dimension D of the system is replaced by D − θ in the hyperscaling relation. This third independent critical exponent, believed to be θ ≈ 1.5, is inaccessible both to a direct experimental measurement and to traditional Monte Carlo methods.The values of the remaining critical exponents, seemingly more straightforward, are also controversial. On the experimental front, different ansätze for the scattering line shape yield mutually incompatible estimates of the thermal critical exponent, namely, ν = 0.87 (7) On the other hand, the numerical determination of ν has steadily shifted, the most precise estimate being 1.37(9), 5 inconsistent with the experimental values and barely compatible with α ≈ 0. The value of α itself is very hard to measure in a numerical simulation. 6More fundamentally, the smallness of the magnetic exponent β, combined with the numerical observation of metastability, 7 has led some authors to suggest that the transition in the DAFF may be of first order.Ultimately, the physical reasons for this confusion betray the fact that the traditional tools of statistical mechanics are ill suited to systems with rugged free-energy landscapes. Both experimentally and numerically, the system gets trapped in local minima, with escape times that grow as log τ ∼ ξ θ (ξ is the correlation length). This not only makes it exceedingly hard to thermalize the system, but also generates a rare-events statistics, causing self-averaging violations. 8 In this Rapid Communication we study the DAFF with the tethered Monte Carlo (TMC) formalism.9 Our approach restores self-averaging and is able to negotiate the free-energy barriers of the DAFF to equilibrate large systems safely. It also provides direct access to the key parameter θ . We thus obtain a comprehensive picture of the phase transition, consistent both with analytical results for the RFIM and with experiments on the DAFF, and shed lig...

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“…Although the view that the phase transition of the RFIM is nowadays considered to be of second order [19][20][21][22][23][24], the extremely small value of the exponent β casts some doubt on the interpretation of numerical and experimental results. Moreover, a rather strong debate with regards to the role of disorder, i.e., the dependence, or not, of the critical exponents on the particular choice of the distribution for the random fields and the value of the disorder strength, analogously to the mean-field theory predictions [15], was only recently put on a different basis [25].…”

Section: Introductionmentioning

confidence: 99%

Universality in four-dimensional random-field magnets

Fytas

Theodorakis

2015

Eur. Phys. J. B

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Abstract. We investigate the universality aspects of the four-dimensional random-field Ising model (RFIM) using numerical simulations at zero temperature. We consider two different, in terms of the field distribution, versions of the model, namely a Gaussian RFIM and an equal-weight trimodal RFIM. By implementing a computational approach that maps the ground-state of the system to the maximum-flow optimization problem of a network, we employ the most up-to-date version of the push-relabel algorithm and simulate large ensembles of disorder realizations of both models for a broad range of random-field values and system sizes. Using as finite-size measures the sample-to-sample fluctuations of the order parameter of the system, we propose, for both types of distributions, estimates of the critical field hc and the critical exponent ν of the correlation length, the latter suggesting that the two models in four dimensions share the same universality class. PACS

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Finite-size scaling in Ising-like systems with quenched random fields: Evidence of hyperscaling violation

Cited by 45 publications

References 67 publications

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

Critical behavior of the dilute antiferromagnet in a magnetic field

Universality in four-dimensional random-field magnets

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