NONLINEAR SIGMA-MODEL FOR LOCALIZATION IN SUPERCONDUCTORS - ROLE OF ORDER-PARAMETER PHASE FLUCTUATIONS

The phenomenon of Anderson localization of normal excitations in dirty superconductors is considered using the functional representation in terms of the nonlinear sigma-model. The derivation of the effective functional allows for the slow spatial variations of the superconducting order-parameter phase. The thermal fluctuations of the order-parameter phase are shown to violate the equivalence between particle-particle and particle-hole diffusion propagators due to the broken time-reversal symmetry in the presence of random local supercurrents. As a result, a new characteristic length l(f) varies as T-1/2 appears in a system. If the correlation length l(c) varies as \n-n(c)\-nu-connected with the Anderson transition is larger than l(f), the transport properties of normal excitations are the same as in normal metals in a weak magnetic field. At l(c) << xi (xi is a superconducting coherence length) they are the same as in normal metals without magnetic field. The behavior of diffusion propagators in a crossover region-xi << l(c) << l(f) is discussed, and a crossover exponent is calculated.