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

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Fytas, Nikolaos G.
Picco, Marco
Sourlas, Nicolas
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American Physical Society
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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 ≤ 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.
© 2017 American Physical Society. We thank the staff of the BIFI supercomputing center for their assistance and for computing time at the cluster Memento. This work was partially supported by MINECO (Spain) through Grant No. FIS2015-65078-C2-1-P. We thank the Royal Society International Exchanges Scheme 2016/R1 for supporting N.G.F. and M.P. N.G.F. is grateful to Coventry University for providing support through a Research Sabbatical Fellowship during which part of this work has been completed.
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