Critical exponents of random Ising-like systems in general dimensions

Yurij V. Holovatch

Th. Jolicœur, Jean-Claude Le Guillou

We show that current estimates of the critical exponents of the three-dimensional random-field Ising model are in agreement with the exponents of the pure Ising system in dimension $3\ensuremath{-}\ensuremath{\theta}$ where \ensuremath{\theta} is the exponent that governs the hyperscaling violation in the random case.

Nikolaos G. Fytas, Víctor Martín-Mayor, Marco Picco +1 more

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.

J. C. Le Guillou, Jean Zinn‐Justin

継博 溜渕

Moshe Schwartz

Random-field ferromagnetism systems are discussed, using an equivalent annealed system. It is shown how the critical exponents of the random system are related to the exponents of the pure system. The critical exponent eta is expressed in terms of a dimensionality reduction d'=d-2+ eta (d'). The lower critical dimension for the Ising model is two and for the continuous models it is four. Very strong arguments are given for a hyperscaling relation 2- alpha =d'v and for all the exponents to be given by a dimensionality reduction.