Random field Ising model: dimensional reduction or spin-glass phase?

TL;DR

This paper investigates the stability of the random field Ising model against spin glass fluctuations, revealing that the spin glass transition is signaled by free-energy fluctuations and that in dimensions less than six, the spin glass phase appears before ferromagnetism, challenging the dimensional reduction hypothesis.

Contribution

The study provides a new analysis of the RFIM's stability against spin glass fluctuations, showing that the spin glass transition always occurs before ferromagnetism in dimensions below six, contradicting dimensional reduction.

Findings

Spin glass transition signals via free-energy fluctuations.

In dimensions less than six, SG transition precedes ferromagnetism.

Dimensional reduction fails in low dimensions due to early SG transition.

Abstract

The stability of the random field Ising model (RFIM) against spin glass (SG) fluctuations, as investigated by M\'ezard 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 {\sl fluctuations\/} only, and not in the free energy itself. Eigenvalues of the inverse SG susceptibility matrix is then approached by the Rayleigh Ritz method which provides an upper bound. Coming from the paramagnetic phase {\sl on the Curie line,\/} one is able to use a virial-like relationship generated by scaling the {\sl 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 {\sl distinct\/} replicas, it follows that, despite the repulsive coupling of the RFIM the effective pair coupling is {\sl attractive\/} (at least for small values of the parameter $ g\bar \Delta , $ $ g $ the coupling and $ \bar \Delta $ the effective random field fluctuation). As a result, \lq\lq bound states\rq\rq\ associated with replica pairs (negative eigenvalues) provide the instability signature. {\sl 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.