Glassy transition in the three-dimensional random-field Ising model.

The high temperature phase of the three dimensional random field Ising model is studied using replica symmetry breaking framework. It is found that, above the ferromagnetic transition temperature Tf , there appears a glassy phase at intermediate temperatures Tf < T < Tb while the usual paramagnetic phase exists for T > Tb only. Correlation length at Tb is computed and found to be compatible with previous numerical results. Although a great deal of work has been devoted to the understanding of the random field Ising model (RFIM) [1], some aspects still need to be cleared up. It is now well-known that, in dimension D = 3, long range order is present at sufficiently low temperature and weak random fields with non trivial critical exponents [2], the upper critical dimension of the RFIM being D = 6. Nevertheless, perturbation theory leads to dimensional reduction (critical exponents are incorrectly predicted to be equal to those of the corresponding pure model in dimension D− 2) [3] and therefore does not succeed in describing the critical behaviour of the RFIM. The reason of this failure presumably stems from the very complicated energy landscape due to the quenched disorder, and more precisely, from the existence of a huge number of local minima of the free energy in the space of local magnetisations that usual perturbative expansions do not take into account [4]. Numerical simulations and resolutions of the mean-field equations Unité propre de recherche du CNRS, associée à l’Université de Paris-Sud et à l’ENS. corroborate this picture [5, 6]. Above the ferromagnetic transition temperature Tf , there seems to appear an intermediate “glassy” regime for Tf < T < Tb where many solutions of the local magnetisations mean-field equations coexist, while only one of them subsists in the paramagnetic phase T > Tb. From the theoretical point of view, it was suggested that the techniques of replica symmetry breaking (RSB), which proved to be successful in the mean-field theory of spin glasses [7] where such complicated free energy landscapes arise, could also be applied to the RFIM [8]. Experiments made on diluted anti-ferromagnets also found an irreversibility line above the critical temperature where the anti-ferromagnetic order appears [9]. Recently, Mézard and Young, referred to in the following as M−Y, proposed a variational approach of the RFIM [10] based on a self-consistent expansion in 1/N (where N is the number of spin components) due to Bray [11]. They found that replica symmetry, which gives back dimensional reduction, must be broken at the ferromagnetic transition T = Tf and that RSB solution leads to sensible results for the critical exponents in agreement with already known results [1, 10]. In this paper, using M−Y’s framework, we concentrate upon the non ferromagnetic regime (i.e. T > Tf ). We find that there exist indeed two different phases : a paramagnetic phase at high temperatures T > Tb and a glassy phase at intermediate temperatures Tf < T < Tb. The value of the correlation length at T = Tb where the RSB transition occurs is computed and compared to predictions obtained from numerical resolution of mean-field equations [6]. The model we consider is a N -component version of the RFIM on a three-dimensional lattice including L3 spins Φi = (Φ 1 i , ...,ΦN i ), where i = (i1, i2, i3) and 0 ≤ i1, i2, i3 ≤ L− 1, H(Φ,h) = 1 2 ∑