Universal conductance fluctuations in non-integer dimensions

TL;DR

This paper introduces a universal formula for conductance fluctuations across continuous dimensions from 0 to 4, validated against known cases and supported by numerical simulations, revealing the breakdown of universality in 4D.

Contribution

It presents a new Ansatz for universal conductance fluctuations applicable in non-integer dimensions and confirms its validity through comparison with known results and numerical analysis.

Findings

The Ansatz matches known formulas for 1D, 2D, and 3D.

Numerical simulations in 4D show a diverging logarithmic plateau.

Universality of conductance fluctuations breaks down in 4D.

Abstract

We propose an Ansatz for Universal conductance fluctuations in continuous dimensions from 0 up to 4. The Ansatz agrees with known formulas for integer dimensions 1, 2 and 3, both for hard wall and periodic boundary conditions. The method is based solely on the knowledge of energy spectrum and standard assumptions. We also study numerically the conductance fluctuations in 4D Anderson model, depending on system size L and disorder W. We find a small plateau with a value diverging logarithmically with increasing L. Universality gets lost just in 4D.