# Al'tshuler-Aronov correction to the conductivity of a large metallic square network

## TL;DR

This paper analyzes the electron-electron interaction correction to conductivity in a large metallic network, revealing a behavior that interpolates between 1D and 2D regimes and closely resembles 2D logarithmic behavior.

## Contribution

It provides a novel analysis of the Al'tshuler-Aronov correction in large metallic networks, connecting spectral determinants to conductivity corrections across dimensions.

## Key findings

- Correction varies with thermal length similarly to weak localization.
- Results interpolate between 1D and 2D behaviors.
- Interaction parameter exhibits 1D characteristics.

## Abstract

We consider the correction $\Delta\sigma_\mathrm{ee}$ due to electron-electron interaction to the conductivity of a weakly disordered metal (Al'tshuler-Aronov correction). The correction is related to the spectral determinant of the Laplace operator. The case of a large square metallic network is considered. The variation of $\Delta\sigma_\mathrm{ee}(L_T)$ as a function of the thermal length $L_T$ is found very similar to the variation of the weak localization $\Delta\sigma_\mathrm{WL}(L_\phi)$ as a function of the phase coherence length. Our result for $\Delta\sigma_\mathrm{ee}$ interpolates between the known 1d and 2d results, but the interaction parameter entering the expression of $\Delta\sigma_\mathrm{ee}$ keeps a 1d behaviour. Quite surprisingly, the result is very close to the 2d logarithmic behaviour already for $L_T\sim{a}/2$, where $a$ is the lattice parameter.