Imry-Ma criterion for long-range random field Ising model: short-/long-range equivalence in a field

The 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 a la Imry-Ma we obtain a threshold value for the power $\rho$ of the long-range interaction, beyond which no critical behavior occurs. The critical threshold value is $\rho_c=3/2$, at a difference with the zero field model in which $\rho_c=2$. This prediction is confirmed by numerical computation of the ground states below, at, and above this threshold value. Some possible implications for the critical behavior of spin-glasses in a field are conjectured.