Restoration of dimensional reduction in the random-field Ising model at five dimensions

Nikolaos Fytas, Victor Martin-Mayor, Marco Picco, Nicolas Sourlas

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Abstract

The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D − 2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields.We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D = 5.We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon.We close our contribution by comparing universal quantities for the random-field problem at dimensions 3 2\leq D < 6 to their values in the pure Ising model at D − 2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.
Original languageEnglish
Article number042117
Number of pages8
JournalPhysical review E: Statistical, Nonlinear, and Soft Matter Physics
Volume95
DOIs
Publication statusPublished - 10 Apr 2017

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Dimensional Reduction
Restoration
restoration
Random Field
Ising model
Ising Model
Renormalization Group
exponents
Ferromagnet
Ising
Critical Exponents
Critical Slowing down
dimensional analysis
Disordered Systems
perturbation theory
Perturbation Theory
Universality
Ground State
Equality
Probability Distribution

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Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

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Restoration of dimensional reduction in the random-field Ising model at five dimensions. / Fytas, Nikolaos; Martin-Mayor, Victor; Picco, Marco; Sourlas, Nicolas.

In: Physical review E: Statistical, Nonlinear, and Soft Matter Physics, Vol. 95, 042117, 10.04.2017.

Research output: Contribution to journalArticle

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N2 - The random-field Ising model is one of the few disordered systems where the perturbative renormalization group can be carried out to all orders of perturbation theory. This analysis predicts dimensional reduction, i.e., that the critical properties of the random-field Ising model in D dimensions are identical to those of the pure Ising ferromagnet in D − 2 dimensions. It is well known that dimensional reduction is not true in three dimensions, thus invalidating the perturbative renormalization group prediction. Here, we report high-precision numerical simulations of the 5D random-field Ising model at zero temperature. We illustrate universality by comparing different probability distributions for the random fields.We compute all the relevant critical exponents (including the critical slowing down exponent for the ground-state finding algorithm), as well as several other renormalization-group invariants. The estimated values of the critical exponents of the 5D random-field Ising model are statistically compatible to those of the pure 3D Ising ferromagnet. These results support the restoration of dimensional reduction at D = 5.We thus conclude that the failure of the perturbative renormalization group is a low-dimensional phenomenon.We close our contribution by comparing universal quantities for the random-field problem at dimensions 3 2\leq D < 6 to their values in the pure Ising model at D − 2 dimensions, and we provide a clear verification of the Rushbrooke equality at all studied dimensions.

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