Constrained annealing for systems with quenched disorder

We review a method for the study of systems with quenched disorder, such as Ising models with random field or coupling, by means of annealed averages where appropriate constraints are imposed. It allows one to understand the role of the intensive variables of the disorder (mean magnetic field, impurities concentration, frustration...) and improves the annealed approximation of the quenched free energy. The constrained annealing can be obtained by introducing generalized Gibbs-like potential depending on Lagrangian multipliers associated to the intensive variables of the disorder. The minimum of the potential gives a lower bound of the quenched free energy which can be a very accurate estimate for a proper choice of the constraints. The method is first applied to one-dimensional Ising models with random magnetic fields. In this case the frustration is properly taken into account and the quenched free energy density is estimated with a precision higher than the numerical results. As a first step to higher dimensional model we then introduce an Ising model with two competing interactions: nearest neighbour random couplings and a positive infinite range coupling. At low temperature the model exhibits a new type of non-trivial 'ferrimagnetic' order in a region of low temperatures and intermediate disorder strength. The qualitative features of the model (in particular the phase transition line between ferromagnetic and ferrimagnetic phases) is reproduced by the constrained annealing. Finally we apply our method to d-dimensional Ising models with random nearest neighbour coupling. In this case, we also introduce an alternative new way to obtain constrained annealed averages without recurring to the Lagrange multipliers. It requires to perform quenched averages on small volumes, in a analytic or numerical way. We thus give a sequence of converging lower bounds for the quenched free energy. In particular, in 2d the known analytic estimates are considerably improved.