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

TL;DR

This paper provides high-precision numerical evidence that the dimensional reduction predicted by perturbative renormalization group for the 5D random-field Ising model is valid at five dimensions, contrasting its failure in three dimensions.

Contribution

The study offers the first high-precision numerical verification of dimensional reduction at five dimensions in the RFIM, showing universality and critical exponents match those of the pure 3D Ising model.

Findings

Critical exponents in 5D RFIM match 3D pure Ising model.

Dimensional reduction is restored at D=5, not in lower dimensions.

Universal quantities agree with theoretical predictions.

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 $3 \leq D < 6$ 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.