Replica symmetry breaking in short-range spin glasses: Theoretical foundations and numerical evidences

We discuss replica symmetry breaking (RSB) in spin glasses. We update worth in this area, From both the analytical and numerical points of view. We gi ve particular attention to the difficulties stressed by Newman and Stein co ncerning the problem of constructing pure states in spin lass systems. We m ainly discuss what happens in finite-dimensional, realistic spin glasses. T ogether with a detailed review of some of the most important features, fact s, data, and phenomena, we present some new theoretical ideas and numerical results. We discuss among others the basic idea of the RSB theory, correla tion functions, interfaces, overlaps, pure states. random Field. and the dy namical approach. We present new numerical results For the behaviors of cou pled replicas and about the numerical verification of sum rules, and we rev iew some of the available numerical results that we consider of larger impo rtanes for example, the determination of the phase transition point. the co rrelation functions, the window overlaps, and the dynamical behavior of the system.