Statistical Analysis of Magnetic Field Spectra

TL;DR

This paper investigates the statistical properties of magnetic field spectra in quantum dots with chaotic shapes, revealing Gaussian Unitary Ensemble behavior at low values and Poisson-like behavior at higher values, consistent with random matrix theory.

Contribution

It introduces a new analysis of the magnetic-field spectrum in quantum dots, connecting spectral statistics to random matrix theory and eigenvalue problems.

Findings

Lower part of B-spectrum follows Gaussian Unitary Ensemble distribution.

Higher part of B-spectrum exhibits Poisson-like behavior.

Fluctuations are consistent with random matrix theory predictions.

Abstract

We have calculated and statistically analyzed the magnetic-field spectrum (the ``B-spectrum'') at fixed electron Fermi energy for two quantum dot systems with classically chaotic shape. This is a new problem which arises naturally in transport measurements where the incoming electron has a fixed energy while one tunes the magnetic field to obtain resonance conductance patterns. The ``B-spectrum'', defined as the collection of values ${B_i}$ at which conductance $g(B_i)$ takes extremal values, is determined by a quadratic eigenvalue equation, in distinct difference to the usual linear eigenvalue problem satisfied by the energy levels. We found that the lower part of the ``B-spectrum'' satisfies the distribution belonging to Gaussian Unitary Ensemble, while the higher part obeys a Poisson-like behavior. We also found that the ``B-spectrum'' fluctuations of the chaotic system are consistent with the results we obtained from random matrices.