Review of Recent Developments in the Random-Field Ising Model

Nikolaos G Fytas,Nicolas Sourlas,Víctor Martín-Mayor,Marco Picco

doi:10.1007/s10955-018-1955-7

Abstract

A lot of progress has been made recently in our understanding of the random-field Ising model thanks to large-scale numerical simulations. In particular, it has been shown that, contrary to previous statements: the critical exponents for different probability distributions of the random fields and for diluted antiferromagnets in a field are the same. Therefore, critical universality, which is a perturbative renormalization-group prediction, holds beyond the validity regime of perturbation theory. Most notably, dimensional reduction is restored at five dimensions, i.e., the exponents of the random-field Ising model at five dimensions and those of the pure Ising ferromagnet at three dimensions are the same.

Highlights

The random–field Ising model (RFIM) is the simplest disordered system
Dimensional reduction is restored at five dimensions, i.e., the exponents of the random–field Ising model at five dimensions and those of the pure Ising ferromagnet at three dimensions are the same
The perturbative renormalization group (PRG) can be carried out to all orders of perturbation theory

Summary

INTRODUCTION

The random–field Ising model (RFIM) is the simplest disordered system. Its Hamiltonian is. The perturbative renormalization group (PRG) can be carried out to all orders of perturbation theory ( this is the only known case where PRG can be carried to all orders) It predicts dimensional reduction and supersymmetry [1,2,3]. This means that the critical exponents of the RFIM in d dimensions are the same as the exponents of the pure Ising model in d − 2 dimensions. It has been proven that dimensional reduction is not true in three dimensions [4]. Why PRG breaks down at three dimensions? PRG and dimensional reduction are restored at five dimensions. There is a maximum violation of self–averaging in the distribution of low lying excited states near the critical point

UNIVERSALITY

DISCUSSION

CONCLUSIONS