Boundary conditions at the mobility edge

TL;DR

This paper demonstrates that boundary conditions influence the universal spectral statistics at the Anderson transition, but do not affect the multifractal properties of wave functions in the bulk.

Contribution

It reveals the boundary condition dependence of spectral statistics while confirming the invariance of the multifractal exponent D_2 at the transition.

Findings

Spacing distribution depends on boundary conditions

Spectral rigidity varies with boundary conditions at low energies

Multifractal exponent D_2 remains unaffected by boundary conditions

Abstract

It is shown that the universal behavior of the spacing distribution of nearest energy levels at the metal--insulator Anderson transition is indeed dependent on the boundary conditions. The spectral rigidity $\Sigma^2(E)$ also depends on the boundary conditions but this dependence vanishes at high energy $E$. This implies that the multifractal exponent $D_2$ of the participation ratio of wave functions in the bulk is not affected by the boundary conditions.