Supersymmetry, Replica and Dynamic Treatments of Disordered Systems: a Parallel Presentation

I briefly review the three nonperturbative methods for the treatment of disordered systems — supersymmetry, replicas and dynamics — with a parallel presentation that highlights their connections and differences. Disordered systems need to be treated with a method that allows to perform averages over the sample realisation. There is no universal way to do this that can be applied efficiently to all problems. For Gaussian systems, the method of supersymmetry is as good as one can expect: it involves a minimum of variables, it is elegant and rigorous. Although one can still apply it for some non-Gaussian problems, in many of the interesting cases – as for example spin-glasses – it only gives limited information. The replica trick was introduced to tackle such ‘complex’ problems. It has been extensively used and has yielded some of the most innovative solutions in disordered systems. It has however the problem that it is very far from being controlled, let alone rigorous. This is because the space itself – a vector space with noninteger dimension – does not have a general definition other than the ansatz itself or perturbations around it. The dynamic method consists of solving exactly the evolution of the system in contact with a heat bath. If the system reaches equilibrium one 1 recovers all the thermodynamic information. Surprisingly enough, one can treat this way all the problems one can solve with replicas. A problem arises, however, when equilibrium cannot be achieved: then the long-time out of equilibrium regime may be of interest in itself (as in the case of glasses), or it may be viewed as an obstacle for exploring the deepest levels in phase-space (as for example in optimisation problems). Although the dynamic method was initially proposed as a way to obtain equilibrium results, this tendency has reverted in the last few years, at least in the field of glasses, where replicas are now used mostly to mimic the out of equilibrium dynamics. The aim is of this paper is not to make a complete presentation of either of the three methods — there are very complete reviews of this [1, 2, 3] (including some very recent ones [4]) but rather to put the three methods ‘side by side’ so that the connections can be better appreciated. To the best of my knowledge this has not been done for supersymmetry, replicas and dynamics simultaneously, as the practitioners of each method tend to belong to different communities. The Problem Consider an energy EJ = i 2 ∑ ij (λδij − Jij)sisj ; EJ(h) = EJ − ∑ i hisi (1) where si (i = 1, ..., N) are real variables, and Jij is a random matrix. We take λ with negative imaginary part. This energy can be used to calculate the averaged Green function: G(λ) ≡ Tr[λI − J ]−1 (2) from which one obtains the eigenvalue distribution. (Here and in what follows the overline denotes averages over the disorder J). This is done by defining the partition function ZJ(h) = ∫ ds eJ (h) (3) and computing: G(λ) = −iT ∑