Spin glasses 1 are disordered magnetic systems with competing ferromagnetic and antiferromagnetic interactions, which generate frustration. These systems exhibit a freezing transition to a low temperature phase where the spins are aligned in fixed but random directions. Examples of spin glasses are found in metallic alloys, e.g. CuMn, with RKKY interactions between the spins, and in insulators, e.g. Eu x Sr 1-x S, with exchange interactions between first and second neighbours.

Disorder and frustration are the two key features of spin glasses. Concepts and techniques developed in the study of these complex systems have had an impact on a variety of other subjects, such as combinatorial optimization, neural networks, prebiotic evolution and protein folding.

Despite the enormous amount of work dedicated over the past three decades to the study of spin glasses, no consensus has yet been reached on the most fundamental properties of these systems, namely, the nature and complexity of the glassy phase and the existence of a transition in a nonzero magnetic field. Two different pictures have been proposed for the spin glass. One corresponds to the mean field theory for Ising spin glasses, provided by the Parisi solution for the infinite-range, Sherrington-Kirkpatrick model, which predicts a glassy phase described by an infinite number of pure states organized in an ultrametric structure, and a phase transition occurring in a magnetic field. The alternative is the droplet model, which claims that the real, short-range systems behave quite differently, the glassy phase being described by only two pure states, related by a global inversion of the spins, and no phase transition occurring in a magnetic field. A fundamental step toward the clarification of the controversy, and therefore toward the understanding of spin glasses, lies in the investigation of how the fluctuations, associated into the finite-range interactions, modify the mean-field picture.

Our work concentrates on the study of short-range Ising spin glasses in a magnetic field, considering the role of fluctuations. We use the replica method to average over the quenched disorder and describe the system within a field theory in replica space. Renormalization group techniques are used to study the critical behaviour. We derived a general replica symmetric field theory appropriate to study the high temperature phase of a spin glass in a magnetic field, and analysed the spin glass transitions in zero and nonzero fields. 2 ,3 We are now deriving a field theory to study the low temperature phase of a spin glass with replica symmetry breaking, which will allow the calculation of properties within the glassy phase.

This work has been carried out in collaboration with C. De Dominicis (SPhT, CEA Saclay, France ) and T. Temesvari ( Eötvös University , Budapest , Hungary ).

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