Transfer matrix approach to thermodynamics and dynamics of one-dimensional quantum systems
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Date
2002-12-18
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Universität Dortmund
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Abstract
In dieser Arbeit wird die Thermodynamik eindimensionaler Quantensysteme untersucht, indem das eindimensionale quantenmechanische Modell auf ein zweidimensionales klassisches System mittels einer Trotter-Suzuki Zerlegung abgebildet wird. Neben der häufig angewandten Schachbrettzerlegung wird eine neue Trotter-Suzuki Abbildung vorgestellt, die zu einem klassischen Modell mit alternierenden Reihen führt und die Definition einer einspaltigen Quantentransfermatrix erlaubt. Innerhalb des Transfermatrix-Zugangs kann der thermodynamische Limes exakt durchgeführt werden. Um die Transfermatrix insbesondere für tiefe Temperaturen zu untersuchen, wird eine Variante der Dichtematrix-Renormierungsgruppe, die sog. TMRG, verwendet. Mit dieser Methode wird das t-J Modell untersucht, wobei besonderer Wert auf die Berechnung von Korrelationslängen gelegt wird, um den Übergang vom Hochtemperaturbereich in den quantenkritischen Bereich zu studieren. Zusätzlich wird das reichhaltige Phasendiagramm untersucht. Die numerischen Resultate werden detailliert mit analytischen Ergebnissen aus konformer Feldtheorie verglichen. Ferner wird ein für das Übergangsmetalloxid YVO_3 relevantes Spin-Orbital Modell untersucht. Es wird eine Dimerisierung bei endlicher Temperatur gefunden, die durch den zugehörigen Entropiegewinn getrieben wird. Dieser Mechanismus scheint die bisher unverstandene Stabilität der C-Phase in YVO_3 zu erklären. Im letzten Teil wird eine neue Gitter-Pfadintegraldarstellung vorgestellt, die es erlaubt, dynamische Korrelationsfunktionen zu reellen Zeiten ohne analytische Fortsetzung zu berechnen. Erste Resultate für die Spin-Autokorrelationsfunktion des XXZ-Modells bei unendlicher Temperatur werden vorgestellt.
In this thesis thermodynamics of one-dimensional quantum systems is investigated. The one-dimensional quantum model is mapped onto a two-dimensional classical system by a Trotter-Suzuki decomposition. Apart from the usual checkerboard decomposition, a novel Trotter-Suzuki mapping is introduced which leads to a classical model with alternating rows and makes the definition of a one-column transfer matrix possible. Within the transfer matrix approach the thermodynamic limit can be performed exactly. To investigate the transfer matrix especially at low temperatures, a variant of the density-matrix-renormalization group, the so-called TMRG, is used. By applying this method, the t-J model is investigated with special emphasize put on the calculation of correlation lengths to study the crossover from the high-T lattice into the quantum critical regime. Additionally the rich phase diagram is investigated. The numerical results are compared in detail with analytical results from conformal field theory. Furthermore, a spin-orbital model relevant for the transition metal oxide YVO_3 is investigated. A dimerization at finite temperature is found which is driven by the large entropy content of the dimer state. This mechanism seems to explain the stability of the C-phase in YVO_3 which has not been understood before. In the last part a new lattice-pathintegral representation is introduced which makes the calculation of real-time dynamical correlation functions possible without performing an analytical continuation. First results for the spin autocorrelation function of the XXZ-model at infinite temperature are shown.
In this thesis thermodynamics of one-dimensional quantum systems is investigated. The one-dimensional quantum model is mapped onto a two-dimensional classical system by a Trotter-Suzuki decomposition. Apart from the usual checkerboard decomposition, a novel Trotter-Suzuki mapping is introduced which leads to a classical model with alternating rows and makes the definition of a one-column transfer matrix possible. Within the transfer matrix approach the thermodynamic limit can be performed exactly. To investigate the transfer matrix especially at low temperatures, a variant of the density-matrix-renormalization group, the so-called TMRG, is used. By applying this method, the t-J model is investigated with special emphasize put on the calculation of correlation lengths to study the crossover from the high-T lattice into the quantum critical regime. Additionally the rich phase diagram is investigated. The numerical results are compared in detail with analytical results from conformal field theory. Furthermore, a spin-orbital model relevant for the transition metal oxide YVO_3 is investigated. A dimerization at finite temperature is found which is driven by the large entropy content of the dimer state. This mechanism seems to explain the stability of the C-phase in YVO_3 which has not been understood before. In the last part a new lattice-pathintegral representation is introduced which makes the calculation of real-time dynamical correlation functions possible without performing an analytical continuation. First results for the spin autocorrelation function of the XXZ-model at infinite temperature are shown.
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Thermodynamik, Stark korrelierte Elektronensysteme, Eindimensionale Quantensysteme, Trotter-Suzuki Abbildung, Transfermatrix, Renormierungsgruppe, TMRG, DMRG, t-J Modell, Tomonaga-Luttinger Flüssigkeit, Luther-Emery Phase, Phasenseparation, Korrelationslängen, Korrelationsfunktionen, Konforme Feldtheorie, Spin-Orbital Modell, Übergangsmetalloxide, Dynamik, Autokorrelationen, thermodynamics, strongly correlated electron systems, one-dimensional quantum systems, Trotter-Suzuki decomposition, transfer matrix, renormalization group, TMRG, DMRG, t-J model, Tomonaga-Luttinger liquid, Luther-Emery phase, phase separation, correlation lengths, correlation functions, conformal field theory, spin-orbital model, transition metal oxides, dynamics, autocorrelations