Spectral analysis and quantum chaos in two-dimensional nanostructures

Abstract

This thesis describes a study into the eigenvalues and eigenstates of twodimensional (2D) quantum systems. The research is summarized in four scientific publications by the author. The underlying motivation for this work is the grand question of quantum chaos: how does chaos, as known in classical mechanics, manifest in quantum mechanics? The search and analysis of these quantum fingerprints of chaos requires efficient numerical tools and methods, the development of which is given a special emphasis in this thesis. The first publication in this thesis concerns the eigenspectrum analysis of a nanoscale device. It is shown that a measured addition energy spectrum can be explained by a simple confinement of interacting electrons in a potential well. This result is derived using density-functional theory (DFT) and numerical optimization, guided by the asymmetric geometry of the device. The calculations also show that an observed decrease in the conductance can be explained by a change in the shape of the quantum wave function. The study of quantum chaos by statistical properties of eigenvalues requires a way to solve the eigenvalue spectrum of a quantum system up to highly excited states. The second publication describes a numerical program, itp2d, that uses modern advances in the imaginary time propagation (ITP) method to solve the eigenvalue spectrum of generic 2D systems up to thousands of eigenstates. The program provides means to sophisticated eigenvalue analysis involving long-range correlations, and a unique view to highly excited eigenstates of complicated 2D systems, such as those involving magnetic fields and strong disorder. After the spectrum is solved, the next step in the eigenvalue analysis is to remove the trivial part of the spectrum in a process known as unfolding. Unless the system belongs to a special class for which the trivial part is known, unfolding is an ambiguous process that can cause substantial artifacts to the eigenvalue statistics. Recently it has been proposed that these artifacts can be mitigated by employing the empirical mode decomposition (EMD) algorithm in the unfolding. The third publication describes an efficient implementation of this algorithm, which is also highly useful in other kinds of data analysis. Quantum scarring refers to the condensation of quantum eigenstate probability density around unstable classical periodic orbits in chaotic systems. It represents a useful and visually striking quantum suppression of chaos. The final publication describes the discovery of a new kind of quantum scarring in symmetric 2D systems perturbed by local disorder. These unusually strong quantum scars are not explained by ordinary scar theory. Instead, they are caused by classical resonances and resulting quantum near-degeneracy in the unperturbed system. Wave-packet analysis shows that the scars greatly influence the transport properties of these systems, even to the extent that wave packets launched along the scar path travel with higher fidelity than in the corresponding unperturbed system. This discovery raises interesting possibilities of selectively enhancing the conductance of quantum systems by adding local perturbations.