DMRG method for the linear DC-conductance of one-dimensional correlated systems

Publikation: Qualifikations-/StudienabschlussarbeitDissertation

Autoren

  • Jan-Moritz Bischoff

Organisationseinheiten

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Details

OriginalspracheEnglisch
QualifikationDoctor rerum naturalium
Gradverleihende Hochschule
Betreut von
Förderer
  • NTH Nano School for Contacts in Nanosystems
  • Deutsche Forschungsgemeinschaft (DFG)
Datum der Verleihung des Grades16 Dez. 2019
ErscheinungsortHannover
PublikationsstatusVeröffentlicht - 2019

Abstract

Im Bereich der Festkörperphysik und Nanotechnologie haben sich in jüngster Zeit interessante Beobachtungen in elektronischen Systemen machen lassen, die räumlich auf wenige Dimensionen eingeschränkt sind. Entgegen der klassischen Erwartung kommen hier Quantenphänomene der betrachteten Fermionen zum Vorschein. Im eindimensionalen Fall werden diese Systeme daher auch Quanten- oder Nanodrähte genannt. Die Leitfähigkeit in solchen Quantendrähten wird durch quantenmecha- nische Elektronenkorrelationen dominiert und ist quantisiert. In dieser Arbeit wird eine schnelle und effiziente numerische Methode entwickelt, die den Anspruch hat eine Transportmessung an einem Quantendraht möglichst re- alitätsnah zu beschreiben. Diese Methode ist inspieriert von der Arbeit Dan Bohrs et al., der bereits gezeigt hat, dass mithilfe der Dichte-Matrix Renormierungsgrup- penalgorithmus(DMRG) die Kuboformel für die lineare Antwort auf ein Potential berechnet werden kann. Ähnlich dazu wird auch hier die Kuboformel mithilfe der dy- namischen DMRG für eine AC Source-Drain Spannung für endliche Systeme berech- net. Ein spezielles Finite Size Scaling erlaubt es, die Ergebnisse für diese endlichen Systeme in den thermodynamischen Grenzfall zu extrapolieren und schließlich die lineare DC-Leitfähigkeit zu erhalten. Zunächst wird das theoretische Grundgerüst für die Methode erläutert. Im da- rauf folgenden Teil wird die Methode an verschiedensten Modellsystemen getestet. Als erstes Modell wird eine Tight-Binding Kette untersucht, in der die Fermio- nen weder Spin besitzen, noch in irgendeiner Form wechselwirken. Das Modell wird dann langsam erweitert, indem erst homogene Wechselwirkung im Luttinger Flüssigkeitsbereich und dann inhomogene Wechselwirkung in unterschiedlichen Teil- abschnitten der Kette untersucht werden. Des Weiteren wird der Einfluss von Störstellen in diesen Systemen untersucht. Zuletzt werden schließlich die beiden Fälle einer Hubbard Kette mit Spin- 12 Elektronen und des Holstein Modells in einer Dimension mit Einsteinphononen, die mit spinlosen Fermionen wechselwirken, be- trachtet. Die Ergebnisse der Methode sind konsistent mit der Luttinger Flüssigkeits- und mit der Landauer Streutheorie. Außerdem lässt sich ein Einfluss von Phononen auf den Ladungsstrukturfaktor im Luttinger Regime beobachten. Eine Abhängigkeit der Leitfähigkeit durch eine Änderung der Länge des Quantendrahtes und etwaigen Dissipationseffekten ließ sich in allen Fällen nicht beobachten.

Zitieren

DMRG method for the linear DC-conductance of one-dimensional correlated systems. / Bischoff, Jan-Moritz.
Hannover, 2019. 92 S.

Publikation: Qualifikations-/StudienabschlussarbeitDissertation

Bischoff, J-M 2019, 'DMRG method for the linear DC-conductance of one-dimensional correlated systems', Doctor rerum naturalium, Gottfried Wilhelm Leibniz Universität Hannover, Hannover. https://doi.org/10.15488/9181
Bischoff, J.-M. (2019). DMRG method for the linear DC-conductance of one-dimensional correlated systems. [Dissertation, Gottfried Wilhelm Leibniz Universität Hannover]. https://doi.org/10.15488/9181
Bischoff JM. DMRG method for the linear DC-conductance of one-dimensional correlated systems. Hannover, 2019. 92 S. doi: 10.15488/9181
Download
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abstract = "Interesting discoveries have recently been made in the area of solid state physics and nano technology regarding electronic systems that are confined to reduced spatial dimensions. Under these circumstances the quantum nature of the electrons begins to surface. In the one-dimensional case this class of systems is therefore called quantum- or nano wires. The conductance in such quantum wires becomes quantized and is dominated by electronic correlations. The goal of this thesis was to develop a fast and efficient procedure to describe a realistic transport measurement for such systems. The method presented here is in- spired by the work of Dan Bohr et al. who used the Density Matrix Renormalization Group algorithm(DMRG) to evaluate the Kubo formula for the linear response to an applied potential. In a similar way the procedure of this thesis also computes the Kubo formula using the dynamical DMRG in order to calculate the low frequency response to an AC source-drain voltage of finite systems. Making use of a special finite size scaling allows to extrapolate the results into the thermodynamic limit and to finally obtain the linear DC-conductance. First, the theoretical framework for the new method was developed. This was fol- lowed by testing the new approach in different model systems. The first model com- prised of a tight-binding chain that was filled with spinless, noninteracting fermions. In subsequent steps the model was expanded by introducing a homogeneous in- teraction in the Luttinger liquid regime followed by inhomogeneous interaction for different chain sections. The potential impact of impurities was assessed in the above models. Lastly, the two cases of the Hubbard chain of spin- 2 1 electrons and of the Holstein model in one dimension, where Einstein phonons interact with spinless fermions, were studied. The results of the proposed method are consistent with Luttinger liquid and Lan- dauer scattering theory. In the Luttinger liquid regime the influence of phonons to the charge structure factor was confirmed. A dependency on the length of the quantum wire in the form of dissipation effects for the conductance was not observed.",
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N2 - Interesting discoveries have recently been made in the area of solid state physics and nano technology regarding electronic systems that are confined to reduced spatial dimensions. Under these circumstances the quantum nature of the electrons begins to surface. In the one-dimensional case this class of systems is therefore called quantum- or nano wires. The conductance in such quantum wires becomes quantized and is dominated by electronic correlations. The goal of this thesis was to develop a fast and efficient procedure to describe a realistic transport measurement for such systems. The method presented here is in- spired by the work of Dan Bohr et al. who used the Density Matrix Renormalization Group algorithm(DMRG) to evaluate the Kubo formula for the linear response to an applied potential. In a similar way the procedure of this thesis also computes the Kubo formula using the dynamical DMRG in order to calculate the low frequency response to an AC source-drain voltage of finite systems. Making use of a special finite size scaling allows to extrapolate the results into the thermodynamic limit and to finally obtain the linear DC-conductance. First, the theoretical framework for the new method was developed. This was fol- lowed by testing the new approach in different model systems. The first model com- prised of a tight-binding chain that was filled with spinless, noninteracting fermions. In subsequent steps the model was expanded by introducing a homogeneous in- teraction in the Luttinger liquid regime followed by inhomogeneous interaction for different chain sections. The potential impact of impurities was assessed in the above models. Lastly, the two cases of the Hubbard chain of spin- 2 1 electrons and of the Holstein model in one dimension, where Einstein phonons interact with spinless fermions, were studied. The results of the proposed method are consistent with Luttinger liquid and Lan- dauer scattering theory. In the Luttinger liquid regime the influence of phonons to the charge structure factor was confirmed. A dependency on the length of the quantum wire in the form of dissipation effects for the conductance was not observed.

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