Title: Ginzburg - Landau expansion in strongly disordered attractive Anderson - Hubbard model,
Abstract: We have studied disordering effects on the coefficients of Ginzburg - Landau
expansion in powers of superconducting order - parameter in attractive Anderson
- Hubbard model within the generalized $DMFT+\Sigma$ approximation. We consider
the wide region of attractive potentials $U$ from the weak coupling region,
where superconductivity is described by BCS model, to the strong coupling
region, where superconducting transition is related with Bose - Einstein
condensation (BEC) of compact Cooper pairs formed at temperatures essentially
larger than the temperature of superconducting transition, and the wide range
of disorder - from weak to strong, where the system is in the vicinity of
Anderson transition. In case of semi - elliptic bare density of states disorder
influence upon the coefficients $A$ and $B$ before the square and the fourth
power of the order - parameter is universal for any value of electron
correlation and is related only to the general disorder widening of the bare
band (generalized Anderson theorem). Such universality is absent for the
gradient term expansion coefficient $C$. In the usual theory of "dirty"
superconductors the $C$ coefficient drops with the growth of disorder. In the
limit of strong disorder in BCS limit the coefficient $C$ is very sensitive to
the effects of Anderson localization, which lead to its further drop with
disorder growth up to the region of Anderson insulator. In the region of BCS -
BEC crossover and in BEC limit the coefficient $C$ and all related physical
properties are weakly dependent on disorder. In particular, this leads to
relatively weak disorder dependence of both penetration depth and coherence
lengths, as well as of related slope of the upper critical magnetic field at
superconducting transition, in the region of very strong coupling.