Mesoscopic charging effects in the $\sigma$-model for granular metals

TL;DR

This paper develops a $\sigma$-model for granular metals that captures non-perturbative charging effects and Coulomb blockade phenomena across different temperature regimes, revealing topological properties and soliton solutions.

Contribution

It introduces a novel $\sigma$-model action with a new term for studying Coulomb blockade effects in granular metals, valid for any temperature and level spacing relation.

Findings

The model describes Coulomb blockade via solitons at low temperatures.

It shows a transition from delocalized solitons to localized phase instantons with increasing temperature.

Topological properties encode charge discreteness in the $\sigma$-model.

Abstract

We derive the $\sigma$-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 $\sigma$-model describes non-perturbative charging effects in homogeneous disordered metals. The $\sigma$-model action contains a novel term which is crucial for studying Coulomb blockade effects. Topological properties of the $\sigma$-model encoding charge discreteness are studied. For $T$ below the escape rate from the grain, $\Gamma$, Coulomb blockade effects are described by solitons of the $Q$-matrix which are delocalized in real space. At $T> \Gamma $ the solitons transform into phase instantons localized at a single tunneling contact.