Dimensional dependence of the metal-insulator transition

TL;DR

This study investigates how the Anderson transition's properties depend on spatial dimensionality, revealing that the upper critical dimension is infinite and spectral correlations weaken with increasing dimensions.

Contribution

It combines numerical and theoretical analysis to explore the dimensional dependence of the Anderson transition, including the critical behavior and effectiveness of random matrix models.

Findings

Upper critical dimension for localization is infinite.

Spectral correlations weaken as dimension increases.

Eigenvalue correlations decay exponentially in certain ranges.

Abstract

We study the dependence on the spatial dimensionality of different quantities relevant in the description of the Anderson transition by combining numerical calculations in a $3 \leq d \leq 6$ disordered tight binding model with theoretical arguments. Our results indicate that, in agreement with the one parameter scaling theory, the upper critical dimension for localization is infinity. Typical properties of the spectral correlations at the Anderson transition such as level repulsion or a linear number variance are still present in higher dimensions though eigenvalues correlations get weaker as the dimensionality of the space increases. It is argued that such a critical behavior can be traced back to the exponential decay of the two-level correlation function in a certain range of eigenvalue separations. We also discuss to what extent different effective random matrix models proposed in the literature to describe the Anderson transition provide an accurate picture of this phenomenon. Finally, we study the effect of a random flux in our results.