Level Curvatures and Conductances: A Numerical Study of the Thouless Relation

TL;DR

This study numerically investigates the Thouless relation in the Anderson model, demonstrating proportionality between conductance and level curvature in diffusive regimes and analyzing deviations in ballistic and localized regimes.

Contribution

It provides a detailed numerical validation of the Thouless conjecture, linking conductance to spectral sensitivity across different transport regimes in disordered systems.

Findings

In diffusive regime, conductance is proportional to mean absolute curvature.

Contact resistance affects conductance near ballistic regime, but curvature measures bulk conductance.

In localized regime, the logarithms of curvature and conductance are proportional.

Abstract

The Thouless conjecture states that the average conductance of a disordered metallic sample in the diffusive regime can be related to the sensitivity of the sample's spectrum to a change in the boundary conditions. Here we present results of a direct numerical study of the conjecture for the Anderson model. They were obtained by calculating the Landauer-B\"uttiker conductance $g_L$ for a sample connected to perfect leads and the distribution of level curvatures for the same sample in an isolated ring geometry, when the ring is pierced by an Aharonov-Bohm flux. In the diffusive regime ($L\gg l_e$) the average conductance $g_L$ is proportional to the mean absolute curvature $< |c| >$: $ g_L = \pi <| c | > / \Delta$, provided the system size $L$ is large enough, so that the contact resistance can be neglected. $l_e$ is the elastic mean free path, $\Delta$ is the mean level spacing. When approaching the ballistic regime, the limitation of the conductance due to the contact resistance becomes essential and expresses itself in a deviation from the above proportionality. However, in both regimes and for all system sizes the same proportionality is recovered when the contact resistance is subtracted from the inverse conductance, showing that the ``curvatures measure the conductance in the bulk''. In the localized regime, the mean logarithm of the absolute curvature and the mean logarithm of the Landauer-B\"uttiker conductance are proportional.