Long-range random-field Ising model: Phase transition threshold and equivalence of short and long ranges

The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field à la Imry-Ma, a threshold value for the power ρ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ρc=3/2, at a difference with the zero field model in which ρc=2. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range D-dimensional systems is investigated. At the critical threshold value of ρ, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.