Effective $\sigma$ Model Formulation for Two Interacting Electrons in a Disordered Metal

TL;DR

This paper develops an analytical effective sigma model for two interacting electrons in a disordered metal, revealing subdiffusive pair propagation and logarithmic growth of pair size, aligning with recent numerical findings.

Contribution

It introduces a novel sigma model formulation for two-electron interactions in disordered systems, linking it to disordered metal theory and analyzing pair dynamics.

Findings

Pair propagation is subdiffusive.

Pair size grows logarithmically with time.

The model aligns with recent numerical results.

Abstract

We derive an analytical theory for two interacting electrons in a $d$--dimensional random potential. Our treatment is based on an effective random matrix Hamiltonian. After mapping the problem on a nonlinear $\sigma$ model, we exploit similarities with the theory of disordered metals to identify a scaling parameter, investigate the level correlation function, and study the transport properties of the system. In agreement with recent numerical work we find that pair propagation is subdiffusive and that the pair size grows logarithmically with time.