Discovery of parallelism between the semiclassical dynamics (Heisenberg-Usadell equations) and super-matrix field theory sends several exciting hints. First of all, we could take Δ ≠0 in the Usadell equation and find out the action corresponding to these dynamics. Such field theory describes a dirty superconductor. If <Δ> = 0 but δΔ ≠ 0, we arrive at a gap-less superconductor. We could study the level statistics in a small grain of such a material. When energy ε is larger than the level spacing then the level statistics in a superconductor does not differ from that in a normal metal. The satiation changers if e is comparable with the level spacing. In this case, each level sees its mirror image at -ε and repels from it. Even more interesting situation occurs when one begins to apply the instanton (rare fluctuations) technique to dirty superconductors. For instance, any concentration of paramagnetic impurities destroys the BCS relation between the gap in the spectrum of BCS quasi-particles and the critical temperature. It is well known from the mean field analysis that the large enough concentration of paramagnetic impurities leads to the gap-less situation. The instanton analysis shows that at any concentration of paramagnetic impurities, there is a finite density of states at all energies. Therefore, any superconductor with paramagnetic impurities is gap-less. The difference is that at the high concentration of impurities the states inside the BCS gap are conducting, while, if the concentration of impurities is low, then these states arise from the tail of the density of states and, therefore, they are localized.
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