Wave function correlations and the AC conductivity of disordered wires beyond the Mott-Berezinskii law

TL;DR

This paper refines the understanding of wave function correlations and AC conductivity in disordered 1D wires by calculating corrections beyond the Mott-Berezinskii law using an instanton approach and functional determinants.

Contribution

It introduces an exact calculation of wave function correlators beyond leading order, extending the Mott-Berezinskii theory to negative energies and small frequencies.

Findings

Derived corrections to the Mott-Berezinskii law for AC conductivity.

Calculated wave function correlators at different energies beyond leading order.

Validated the approach by comparing results with positive energy cases.

Abstract

In one-dimensional disordered wires electronic states are localized at any energy. Correlations of the states at close positive energies and the AC conductivity $\sigma(\omega)$ in the limit of small frequency are described by the Mott-Berezinskii theory. We revisit the instanton approach to the statistics of wave functions and AC transport valid in the tails of the spectrum (large negative energies). Applying our recent results on functional determinants, we calculate exactly the integral over gaussian fluctuations around the exact two-instanton saddle point. We derive correlators of wave functions at different energies beyond the leading order in the energy difference. This allows us to calculate corrections to the Mott-Berezinskii law (the leading small frequency asymptotic behavior of $\sigma(\omega)$) which approximate the exact result in a broad range of $\omega$. We compare our results with the ones obtained for positive energies.