Spektralstatisitk abseits der Standard Universalitätsklassen

Random matrix theory (RMT) and semi-classical methods are both used to study the spectra of chaotic quantum systems. The former is usually applied to small distances within the spectrum residing on scales of the mean level spacing where the phenomenon of universality emerges. Through the quantum-classical correspondence provided by semi-classics this regime is, conversely, described by long time dynamical properties of the classical system. In the first half of this thesis we explore such a connection for the study of spectral properties of quantum graphs with dynamically broken time reversal invariance. In physical systems this invariance is often not fully broken and to emulate this we include a rank-1 perturbation on the quantum level resulting in a RMT model outside of the usual universality classes. As we further show, the outcome can depend on graph specific properties and the rank of the perturbation. For the second half of the thesis we consider an opposite limit: instead of long times we study short times, on which universality can not be expected, in a chain-like spin system. In these systems the number of spins takes on a similar role as time, therefore the short time behavior in long chains has remarkable similarities to long time dynamics in few-body system. For instance, the formulation of an "evolution" operator in spatial direction is possible. Exploring this spatial-time duality we address long range spectral statistic in many-body systems and, for the first time, resolve periodic orbits in a genuine many-body system from the traces of its quantum evolution.Zufallsmatrixtheorie (RMT) und Semiklassik stellen zwei Methoden zum Studium chaotischer Quantenspektren dar. Erstere beschreibt für gewöhnlich die Spektralstatistik auf Skalen des mittleren Niveauabstandes, für welche man universelle Eigenschaften findet. Ausgehend vom semiklassischen Ansatz, dass eine Korrespondenz zwischen dem quantenmechanischen Spektrum und der klassischen Dynamik eines Systems besteht, wird dieser Bereich gleichermaßen durch klassische Langzeiteigenschaften beschrieben. Im ersten Teil dieser Arbeit nutzen wir diesen Zusammenhang für Studien an einem Quantengraphen mit gebrochener Zeitumkehrinvarianz. In physikalsichen Systemen ist diese Invarianz oft jedoch nicht vollständig gebrochen, was wir durch eine Rang-1 Störung im Quantensystem abbilden. Dies führt zu einem RMT Model abseits der standard Universalitätsklassen. Weiterhin zeigt sich, dass das Ergebnis sowohl von spezifischen Eigenschaften des Graphen als auch vom Rang der Störung abhängt. Der zweite Teil der Arbeit widmet sich einem diametralen Grenzfall: anstelle langer Zeiten untersuchen wir, innerhalb einer Spinkette, kurze Zeitskalen, auf denen Univeralität nicht zu erwarten ist. In diesen Systemen spielt die Teilchenzahl eine ähnliche Rolle wie die Zeit, entsprechend weist das Kurzzeitverhalten bemerkenswerte Übereinstimmungen zum Langzeitverhalten von Systemen mit wenigen Freiheitsgeraden auf. Zum Beispiel ist es möglich einen "Zeitentwicklungsoperator" in räumlicher Richtung aufzustellen. Dieser Zugang ermöglicht es uns Spektralstatistik auf langreichweitigen Energieskalen zu studieren und, zum ersten Mal, periodische Bahnen im Quantenspektrum eines Vielteilchensystems zu identifizieren

Semiclassical Prediction of Large Spectral Fluctuations in Interacting Kicked Spin Chains

While plenty of results have been obtained for single-particle quantum systems with chaotic dynamics through a semiclassical theory, much less is known about quantum chaos in the many-body setting. We contribute to recent efforts to make a semiclassical analysis of many-body systems feasible. This is nontrivial due to both the enormous density of states and the exponential proliferation of periodic orbits with the number of particles. As a model system we study kicked interacting spin chains employing semiclassical methods supplemented by a newly developed duality approach. We show that for this model the line between integrability and chaos becomes blurred. Due to the interaction structure the system features (non-isolated) manifolds of periodic orbits possessing highly correlated, collective dynamics. As with the invariant tori in integrable systems, their presence lead to significantly enhanced spectral fluctuations, which by order of magnitude lie in-between integrable and chaotic cases.Comment: 42 pages, 19 figure

Collectivity and Periodic Orbits in a Chain of Interacting, Kicked Spins

The field of quantum chaos originated in the study of spectral statistics for interacting many-body systems, but this heritage was almost forgotten when single-particle systems moved into the focus. In recent years new interest emerged in many-body aspects of quantum chaos. We study a chain of interacting, kicked spins and carry out a semiclassical analysis that is capable of identifying all kinds of genuin many-body periodic orbits. We show that the collective many-body periodic orbits can fully dominate the spectra in certain cases.Comment: 6 pages, 6 figures, accepted for publication in Acta Physica Polonica A. arXiv admin note: substantial text overlap with arXiv:1611.0574