The Fermi-Hubbard model and its limiting cases as a testbed for techniques and phenomena
Loading...
Date
2021
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Trotz mittlerweile etwa sechs Jahrzehnten Forschung am Fermi-Hubbard-Modell (FHM) geht von kaum einem anderen Modell eine vergleichbar hohe Faszination aus. Das Modell ist von konzeptionell einfacher Struktur und enthält dabei doch die wesentliche Grundzutat dessen, was Festkörper ausmacht: Wechselwirkung. Ebendiese Wechselwirkung ist es, die abschließenden Lösungen des FHM im Gleich- sowie Nichtgleichgewicht diametral entgegensteht und hohe Ansprüche an die verwendeten methodischen Zugänge stellt.
In dieser Arbeit wenden wir uns dem FHM sowie einigen hieraus ableitbaren Modellen zu, etwa dem t-J-Modell oder dem Heisenberg-Modell, diskutieren Kernfragen der aktuellen Forschung und nutzen bekannte Ansätze wie die CET oder TPQS sowie neue Techniken wie die iEoM, um zur Klärung einiger zentraler Fragestellungen beizutragen. Wir starten mit Analysen von Äquilibration und Thermalisierung in gequenchten von der Umgebung abgeschlossenen Quantensysteme und bestätigen und erweitern bisherige Annahmen. Ferner widerlegen wir die Vermutung eines dynamischen Phasenübergangs in einer Dimension. In einem zweiten Schritt reduzieren wir das FHM auf ein effektives t-J-Modell, das besonders im Kontext von Hochtemperatursupraleitung Beachtung findet, und betrachten hierin die Ladungsträgerdynamik. Darüber hinaus bestätigen wir Annahmen zur quantitativen Vorhersagbarkeit der Autokorrelation in dichten Spin-Systemen. Im letzten Schritt schlagen wir einen neuartigen theoretischen Ansatz mittels iEoM für die systematische Berechnung von Greenfunktionen in reduzierten Operator-Unterräumen vor und motivieren seine Anwendbarkeit im Rahmen einer exemplarischen Rechnung.
Despite about six decades of research on the Fermi-Hubbard model (FHM), there is hardly any other model that holds a comparable fascination. The model has a conceptually simple structure and yet contains the essential ingredient of a solid: interaction. Precisely this interaction, however, opposes conclusive solutions of the FHM in equilibrium and non-equilibrium and imposes high demands on methodological approaches. In the course of this work, we review the FHM and related models which can be derived from it, e.g. the t-J model or the Heisenberg model, discuss key topics of current research and use both well-known methods like CET and TPQS and new techniques like iEoM to contribute to the elucidation of some central questions. We begin with the analyses of equilibration and thermalisation in quenched closed quantum systems and confirm as well as extend previously made hypotheses. Furthermore, we challenge and falsify the assumption of a dynamical phase transition in one dimension. In a second step, we reduce the FHM to an effective t-J model, which is of particular interest in the context of high-temperature superconductivity, and consider the resulting charge carrier dynamics. Moreover, we confirm ideas on the quantitative predictability of the autocorrelation in dense spin systems. In the last step, we propose a novel theoretical approach based on iEoM for the systematic evaluation of Green's functions in reduced operator subspaces and motivate its applicability by means of an exemplary calculation.
Despite about six decades of research on the Fermi-Hubbard model (FHM), there is hardly any other model that holds a comparable fascination. The model has a conceptually simple structure and yet contains the essential ingredient of a solid: interaction. Precisely this interaction, however, opposes conclusive solutions of the FHM in equilibrium and non-equilibrium and imposes high demands on methodological approaches. In the course of this work, we review the FHM and related models which can be derived from it, e.g. the t-J model or the Heisenberg model, discuss key topics of current research and use both well-known methods like CET and TPQS and new techniques like iEoM to contribute to the elucidation of some central questions. We begin with the analyses of equilibration and thermalisation in quenched closed quantum systems and confirm as well as extend previously made hypotheses. Furthermore, we challenge and falsify the assumption of a dynamical phase transition in one dimension. In a second step, we reduce the FHM to an effective t-J model, which is of particular interest in the context of high-temperature superconductivity, and consider the resulting charge carrier dynamics. Moreover, we confirm ideas on the quantitative predictability of the autocorrelation in dense spin systems. In the last step, we propose a novel theoretical approach based on iEoM for the systematic evaluation of Green's functions in reduced operator subspaces and motivate its applicability by means of an exemplary calculation.
Description
Table of contents
Keywords
Festkörpertheorie, Stark korrelierte Systeme, High-performance computing, Algorithmendesign