Random Field Ising Model: Dimensional Reduction or Spin-Glass Phase?

(Received 23 December 1994, received in final form and accepted 9 May 1995)

Abstract
The stability of the random field Ising model (RFIM) against spin glass (SG) fluctuations, as investigated by Mézard and Young, is naturally expressed via Legendre transforms, stability being then associated with the non-negativeness of eigenvalues of the inverse of a generalized SG susceptibility matrix. It is found that the signal for the occurrence of the SG transition will manifest itself in free-energy fluctuations only, and in the free energy itself. Eigenvalues of the inverse SG susceptibility matrix are then investigated by the Rayleigh Ritz method which provides an upper bound. Coming from the paramagnetic phase on the Curie line, one is able to use a virial-like relationship generated by the single unit length ( D<6; in higher dimension a new length sets in, the inverse momentum cut off). Instability towards a SG phase being probed on pairs of distinct replicas, it follows that, despite the repulsive coupling of the RFIM the effective pair coupling is attractive (at least for small values of the parameter the coupling and the effective random field fluctuation). As a result, "bound states" associated with replica pairs (negative eigenvalues) provide the instability signature. Away from the Curie line, the attraction is damped out till the SG transition line is reached and paramagnetism restored. In D<6, the SG transition always precedes the ferromagnetic one, thus the domain in dimension where standard dimensional reduction would apply (on the Curie line) shrinks to zero.