On the Phase Diagram of the Random Field Ising Model on the Bethe Lattice

    loading  Checking for direct PDF access through Ovid


The ferromagnetic Ising model on the Bethe lattice of degree k is considered in the presence of a dichotomous external random field ξx = ±α and the temperature T ≥ 0. We give a description of a part of the phase diagram of this model in the T − α plane, where we are able to construct limiting Gibbs states and ground states. By comparison with the model with a constant external field we show that for all realizations ξ = {ξx = ±α} of the external random field: (i) the Gibbs state is unique for T > Tc (k ≥ 2 and any α) or for α > 3 (k = 2 and any T); (ii) the ±-phases coexist in the domain {T < Tc, α ≤ HF(T)}, where Tc is the critical temperature and HF(T) is the critical external field in the ferromagnetic Ising model on the Bethe lattice with a constant external field. Then we prove that for almost all ξ: (iii) the ±-phases coexist in a larger domain{T < Tc, α ≤HF(T) + ε(T)}, where ε(T)>0; and (iv) the Gibbs state is unique for 3≥α≥2 at any T. We show that the residual entropy at T = 0 is positive for 3≥α≥2, and we give a constructive description of ground states, by so-called dipole configurations.

Related Topics

    loading  Loading Related Articles