Spin Glasses: a brief introduction

Spin Glasses are dilute magnetic alloys where the interactions between spins are randomly ferromagnetic or anti-ferromagnetic, and are considered as paradigmatic examples of frozen disorder. The presence of disorder (the random interactions) induces frustration and a greater difficulty for the system to find optimal configurations. As a consequence, these systems exhibit non trivial thermodynamic and dynamic properties, different and richer than those observed in their non disordered counterpart.
Spin glasses can be modeled using Ising-like Hamiltonians where the bonds between spins can be positive or negative at random. Due to the heterogeneity of the couplings, there are many triples or loops of spin sequences which are frustrated, that is for which there is no way of choosing the orientations of the spins without frustrating at least one bond. As a consequence, even the best possible arrangement of the spins comprises for a large proportion of frustrated bonds. More importantly, since there are many configurations with similar degree of frustration one may expect the existence of many local minima of the free energy.

In mean field models the effects of frustration are enhanced, and the thermodynamic scenario that emerges is novel and surprising. As compared to ordinary ferromagnets, where two pure states are present below the transition, mean field spin glasses exhibit at low temperature a complex structure of infinitely many equilibrium states, organized in a hierarchical structure. At a formal level, the thermodynamics can be exactly solved using the Replica Theory, a mathematical approach that allows to appropriately deal with the quenched disorder present in the Hamiltonian. The static order parameter is a function describing the structure in phase space of the (many) pure equilibrium states.
At low temperature spin glasses also exhibit a non trivial dynamical behaviour, with off-equilibrium dynamics and aging. Time translation invariance is typically broken and the standard fluctuation-dissipation relations between correlation and response functions do not hold. At mean field level, such a behaviour can be formally described using a Langevin approach, and analytic computations can be performed revealing a new and more general form of the fluctuation-dissipation theorem.