Critical behaviour and ultrametricity of Ising spin-glass with long-range interactions

Ising spin-glass systems with long-range interactions (J(r) ∼ r) are considered. A numerical study of the critical behaviour is presented in the non-mean-field region together with an analysis of the probability distribution of the overlaps and of the ultrametric structure of the space of pure states in the frozen phase. Also in presence of diverging thermodynamical fluctuations at the critical point the behaviour of the model is shown to be of the RSB type and there is evidence of a non-trivial ultrametric structure. The parallel tempering algorithm has been used to simulate the dynamical approach to equilibrium of such systems. The Long-Range Spin-Glass Model The greatest incentive to study spin-glasses with long-range interactions is that they are conceptually half-way between the SK model, exactly solvable in mean-field theory, and the more realistic short-range models, with nearestneighbour interactions. Long-range spin-glass models are particularly interesting because already in one dimension they show a phase transition between the paramagnetic and the spin-glass phase. So it is possible to study this transition, also out of the range of validity of mean-field approximation, in a relative easier way in comparison with theories with short-range interactions