Magnetization of a Diffusive Ring: Beyond the Perturbation Theory

Average persistent current over a set of diffusive metallic rings with fixed number of electrons is considered. We study the the case where the phase breaking time is much greater than an inverse average inter-level distance. In such a situation, many return events for an electron have to be taken into account. As a result, one arrives at a non-perturbative problem for a fixed by an external magnetic field Cooperon mode. This multi-Cooperon problem has been considered previously by Altland et al., Europhys. Lett., 2, 155 (1992) and in several following papers within the framework of supersymmetric approach. Such an approach involves very tedious calculations which were performed using computer algebra package. Here we solve the problem in question with the help of replica trick. It is demonstrated that the replica trick in combination with a proper analytical continuation in the replica space allows one to obtain the results in much more explicit way.