Dependence of critical level statistics on the sample shape

TL;DR

This study investigates how the shape of 3D samples affects the critical level-spacing distribution at the metal-insulator transition, revealing shape dependence in the critical statistics while the critical disorder remains constant.

Contribution

It demonstrates that the critical level statistics depend on sample shape and tests the applicability of a known small-$s$ behaviour expression at the critical point.

Findings

Critical level statistics vary with sample aspect ratio.

Critical disorder $W_c$ remains unchanged at 16.4.

Critical conductance $g_c$ for cubic samples matches previous calculations.

Abstract

The level-spacing distribution of consecutive energy eigenvalues is calculated numerically at the metal insulator transition for 3d systems with different cuboid shapes. It is found that the scale independent critical $P_c(s)$ changes as a function of the aspect ratio of the samples while the critical disorder $W_c/V=16.4$ remains the same. We use our data to test whether an expression for the small-$s$ behaviour of the level statistics proposed by Kravtsov and Mirlin for the metallic regime is applicable also at the critical point. For this reason, a shape dependent dimensionless critical conductance $g_c$ has been extracted from the small-$s$ behaviour of the critical level statistics. Our result for a cubic sample, $g_c=0.112\pm 0.005$, is in good agreement with a value obtained previously from calculations using the Kubo-formula.