Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions

Fytas, Nikolaos G.; Martı́n-Mayor, V.; Picco, Marco; Sourlas, Nicolas

doi:10.1103/physrevlett.116.227201

Cited by 60 publications

(85 citation statements)

References 56 publications

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“…Supersymmetry predicts η = η (moreover, the D-dimensional RFIM η = η are predicted to be equal to the anomalous dimension of the pure Ising model in dimension D − 2). Extensive numerical simulations at zero temperature showed that these relations fail at D = 3 [23] and D = 4 [25], but they are valid with good accuracy at D = 5 [26]. These numerical results suggest that supersymmetry may be really at play at D = 5.…”

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confidence: 75%

“…The system has a ferromagnetic phase at small σ, that, upon increasing the disorder, becomes paramagnetic at the critical point σ c . Here, we work directly at σ c , namely at 6.02395 ≈ σ c (D = 5) [26] and at 4.17749 ≈ σ c (D = 4) [25]. We consider two correlation functions, namely the connected and disconnected propagators, C (con) xy and C (dis) xy :…”

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confidence: 99%

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Evidence for Supersymmetry in the Random-Field Ising Model at D=5

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We provide a non-trivial test of supersymmetry in the random-field Ising model at five spatial dimensions, by means of extensive zero-temperature numerical simulations. Indeed, supersymmetry relates correlation functions in a D-dimensional disordered system with some other correlation functions in a D − 2 clean system. We first show how to check these relationships in a finite-size scaling calculation, and then perform a high-accuracy test. While the supersymmetric predictions are satisfied even to our high-accuracy at D = 5, they fail to describe our results at D = 4.

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Evidence for Supersymmetry in the Random-Field Ising Model at D=5

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“…Unfortunately, in the present D = 3 case, we can not draw a definite conclusion on the validity of the two-exponent scaling scenario. Additional work is under way to tackle this problem at higher dimensions (D > 3) [80].…”

Section: Resultsmentioning

confidence: 99%

“…The most obvious generalization is of course the RFIM in higher dimensions (see, e.g., [80]). However, similar ideas can be applied to many disordered systems and should be useful when one needs to take derivatives, or to perform reweighting extrapolations, with respect to the disorder-distribution parameters.…”

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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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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“…Furthermore, there are two types of RFM/RAM spin models. The first is the Ising model, where the spins are scalars S i = ±1 and are randomly pointing either parallel or antiparallel to each other in the ground state [55][56][57][58][59][60][61][62]. The second type is the Heisenberg model, where the spins are vectors S i that in the ground state are pointing noncollinearly in random directions [63][64][65][66][67][68].…”

Section: Introductionmentioning

confidence: 99%

Scattering theory of transport through disordered magnets

2019

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We present a scattering theory of transport through noncollinear disordered magnetic insulators. For concreteness, we study and compare the random field model (RFM) and the random anisotropy model (RAM). The RFM and RAM are used to model random spin disorder systems and amorphous materials, respectively. We utilize the Landauer-Büttiker formalism to compute the transmission probability and spin conductance of one-dimensional disordered spin chains. The RFM and the RAM both exhibit Anderson localization, which means that the transmission probability and spin conductance decay exponentially with the system length. We define two localization lengths based on the transmission probability and the spin conductance, respectively. Next, we numerically determine the relationship between the localization lengths and the strength of the disorder. In the limit of weak disorder, we find that the localization lengths obey power laws and determine the critical exponents. Our results are expressed via the universal exchange length and are therefore expected to be general. arXiv:1908.00817v3 [cond-mat.mes-hall]

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Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions

Cited by 60 publications

References 56 publications

Evidence for Supersymmetry in the Random-Field Ising Model at D=5

Evidence for Supersymmetry in the Random-Field Ising Model at D=5

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

Scattering theory of transport through disordered magnets

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