Multifractal analysis of the metal-insulator transition in anisotropic systems

TL;DR

This study investigates the multifractal properties of eigenstates at the metal-insulator transition in anisotropic three-dimensional systems, revealing differences from isotropic systems and analyzing conductivity behavior.

Contribution

It provides the first detailed analysis of multifractal spectra in anisotropic 3D Anderson models and compares them with isotropic cases, highlighting quantitative differences.

Findings

Multifractal behavior persists at the transition despite strong anisotropy.

Critical disorder strength aligns with previous transfer matrix studies.

Conductivity decreases rapidly in directions with weaker hopping.

Abstract

We study the Anderson model of localization with anisotropic hopping in three dimensions for weakly coupled chains and weakly coupled planes. The eigenstates of the Hamiltonian, as computed by Lanczos diagonalization for systems of sizes up to $48^3$, show multifractal behavior at the metal-insulator transition even for strong anisotropy. The critical disorder strength $W_c$ determined from the system size dependence of the singularity spectra is in a reasonable agreement with a recent study using transfer matrix methods. But the respective spectrum at $W_c$ deviates from the ``characteristic spectrum'' determined for the isotropic system. This indicates a quantitative difference of the multifractal properties of states of the anisotropic as compared to the isotropic system. Further, we calculate the Kubo conductivity for given anisotropies by exact diagonalization. Already for small system sizes of only $12^3$ sites we observe a rapidly decreasing conductivity in the directions with reduced hopping if the coupling becomes weaker.