Mesoscopic charging effects in the σ model for granular metals

We derive the σ model for granular metals. It is valid for any relation between the temperature T and the grain mean level spacing. The continuum limit of the σ model describes nonperturbative charging effects in homogeneous disordered metals. The σ-model action contains a term which is crucial for studying Coulomb blockade effects. Topological properties of the σ model encoding charge discreteness are studied. For T below the escape rate from the grain, Γ, Coulomb blockade effects are described by solitons of the Q matrix which are delocalized in real space. At T>Γ the solitons transform into phase instantons localized at a single tunneling contact.