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Spin glasses are the paradigm of complex systems. These materials presentreally slow dynamics. However, the nature of the spin glass phase in finitedimensional systems is still controversial. Different theories describing thelow temperature phase have been proposed: droplet, replica symmetry breakingand chaotic pairs. We present analytical studies of critical properties of spinglasses, in particular, critical exponents at and below the phase transition,existence of a phase transition in a magnetic field, computation of the lowercritical dimension (in presence/absence of a magnetic field). We also introducesome rigorous results based on the concept of metastate. Finally, we reportsome numerical results regarding the construction of the Aizenman-Wehrmetastate, scaling of the correlation functions in the spin glass phase andexistence of a phase transition in a field, confronting these results with thepredictions of different theories.

Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses