Continuous matrix product states with periodic boundary conditions and an application to atomtronics

We introduce a time evolution algorithm for one-dimensional quantum field theories with periodic boundary conditions. This is done by applying the Dirac-Frenkel time-dependent variational principle to the set of translational invariant continuous matrix product stateswith periodic boundary conditions. Moreover, the ansatz is accompanied with additional boundary degrees of freedom to study quantum impurity problems. The algorithm allows for a cutoff in the spectrum of the transfer matrix and thus has an efficient computational scaling. In particular we study the prototypical example of an atomtronic system-an interacting Bose gas rotating in a ring shaped trap in the presence of a localized barrier potential

Bosons in the lowest Landau-level in an anharmonic trap

Der Gegenstand der Diplomarbeit ist ein Vielteilchenmodell zur Beschreibung von kalten Bose-Gasen in einer sehr schnell rotierenden anharmonischen Falle. Das System wird im niedrigsten Landau-Niveau durch ein mean-field Energiefunktional beschrieben. Mittels kohärenter Zustände wird die Konvergenz der Grundzustandsenergie des Vielteilchenmodells zu der mean-field Energie bewiesen.The topic of the thesis is a many-body modell which describes a Bose gas in an fast rotating anharmonic trap. The system will be considered to be in the lowest Landau-level and it will be described by a mean field energy functional. The convergence of the ground state energy of the many-body problem towards the mean field energy shall be shown by the use of coherent states

Particles, holes and solitons: a matrix product state approach

We introduce a variational method for calculating dispersion relations of translation invariant (1+1)-dimensional quantum field theories. The method is based on continuous matrix product states and can be implemented efficiently. We study the critical Lieb-Liniger model as a benchmark and excelent agreement with the exact solution is found. Additionally, we observe solitonic signatures of Lieb's Type II excitation. In addition, a non-integrable model is introduced where a U(1)-symmetry breaking term is added to the Lieb-Liniger Hamiltonian. For this model we find evidence of a non-trivial bound-state excitation in the dispersion relation

Quantum Gross-Pitaevskii Equation

We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system ---including entanglement and correlations--- and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.Comment: 4.{\epsilon} pages + references and 4 pages supplementary material (small revisions + extended discussion of periodic potential example

Transfer matrices and excitations with matrix product states

We use the formalism of tensor network states to investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low-energy excitations. In particular, we show that the matrix product state transfer matrix (MPS-TM)—a central object in the computation of static correlation functions—provides important information about the location and magnitude of the minima of the low-energy dispersion relation(s), and we present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson's numerical renormalization group along the virtual (imaginary time) dimension of the system

On the quantum Gross-Pitaevskii equation

Gegenstand dieser Dissertation ist die Entwicklung neuer variationeller Algorithmen zur Untersuchung von stark korrelierten eindimensionalen Quantenfeldtheorien. Zu diesem Zweck wird das Dirac-Frenkel zeitabhaengige Variationsprinzip auf die Klasse der kontinuierlichen Matrix Produkt Zustaende (cMPS) angewandt. Die Dissertation beinhaltet im wesentlichen drei Hauptresultate: einen Ansatz zur Beschreibung von Anregungszustaenden niedriger Energie, ein Algorithmus fuer die Zeitentwicklung von Systemen mit offenen Randbedingungen sowie ein weiterer fuer jene mit periodischen Randbedingungen. Darueber hinaus verallgemeinern wir die renommierte Theorie von Gross und Pitaevskii, omnipraesent in der theoretischen Beschreibung von ultrakalten Bose Gasen, auf den Fall von stark korrelierten, eindimensionalen Systemen wo Molekularfeldnaeherungen typischerweise nicht anwendbar sind. Diese Verallgemeinerung beinhaltet die Gross-Pitaevskii Gleichung im Molekularfeldlimes aber geht weit darueber hinaus indem Verschraenkung und Quantenkorrelationen mitberuecksichtigt werden. Die Linearisierung dieser sogenannten Quantum Gross-Pitaevskii Gleichungen fuehrt zu einer quantenmechanischen Version der Bogoliubov de-Gennes Gleichungen. Diese Methoden werden dann dazu verwendet die Anregungen und die dynamischen Korrelationsfunktionen des Lieb-Liniger Models und einer nicht integrablen Erweiterung desselben zu untersuchen, wobei im Spektrum des ersteren solitonische Anregungen und im Spektrum des letzteren ein nicht trivialer, gebundener Zustand identifiziert wurde. Des Weiteren wird ein wechselwirkendes Bose Gas studiert, welches in der Anwesenheit eines $U(1)$-Eichpotentials auf eine ringfoermige Geometrie beschraenkt ist. Hierbei liegt der Fokus auf der Berechnung der dissipationsfreien Teilchen Stroeme, wobei zusaetzlich eine lokale Barriere in das System eingefuehrt wurde. Darueber hinaus werden Grundzustandseigenschaften eines zwei-komponentigen Bose Gases studiert.This thesis is concerned with the development of new variational algorithms to study strongly correlated one dimensional quantum field theories. To this end we apply the Dirac-Frenkel time-dependent variational principle to the class of continuous Matrix Product States (cMPS). Three main results are presented in this thesis: an ansatz for low lying excitations, a time-evolution algorithm for systems with open boundary conditions and a time evolution algorithm for systems with periodic boundary conditions. These algorithms can be applied equally well to finite translationally and non-translationally invariant systems, to systems in the thermodynamic limit and to both relativistic and non-relativistic theories. Hence, we provide an almost complete toolbox for the numerical study of strongly correlated one dimensional quantum field theories. Moreover, we thereby generalize the renowned theory of Gross and Pitaevskii, central to the theoretical study of ultracold Bose gases, to the case of strongly correlated one dimensional systems where mean-field descriptions typically fail. Our generalization includes the Gross-Pitaevskii equation in the mean-field limit but goes well beyond this regime by capturing entanglement and quantum correlations. Linearizing these quantum Gross-Pitaevskii equations gives then rise to a quantum version of the Bogoluibov de-Gennes equations which allow the study of non-perturbative linear response or quantum control theory. These methods are then used to study excitations and dynamical correlation functions of the Lieb-Liniger model, as well as a non-integrable extension thereof where we identify clear solitonic signatures in the spectrum of the former one and a non-trivial bound state in the spectrum of the latter one. We then investigate an interacting Bose gas loaded into a ring shaped geometry in the presence of a $U(1)$-gauge potential, and focus on the persistent currents behaviour in the presence of a barrier. In addition, ground state properties of the two-component Bose gas are studied thereby paving the way towards quasi two (or higher) dimensional systems which are currently not accessible with cMPS based methods

Quantum Gross-Pitaevskii Equation

We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system ---including entanglement and correlations--- and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential

Quantum Gross-Pitaevskii Equation

We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system ---including entanglement and correlations--- and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.Copyright J. Haegeman et a