{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:63324"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:63324","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Simulating quantum systems on classical computers with matrix product states","abstract":"In this thesis, the numerical simulation of strongly-interacting many-body quantum-mechanical systems using matrix product states (MPS) is considered. Compared to classical systems, quantum many-body systems possess an exponentially enlarged number of degrees of freedom, significantly complicating a simulation on a classical computer. Matrix-Product-States are a novel representation of arbitrary quantum many-body states. Using quantum information theory, it is possible to show that Matrix-Product-States provide a polynomial-sized representation of one-dimensional quantum systems, thus allowing an efficient simulation of one-dimensional quantum system on classical computers. Matrix-Product-States form the conceptual framework of the density-matrix renormalization group (DMRG). Based upon this connection, deeper understanding of the density matrix renormalization group can be obtained. After a general introduction in the first chapter of this thesis, the second chapter deals with Matrix-Product-States, focusing on the development of fast and stable algorithms. It is possible to extend the Matrix-Product-States approach to be able to represent arbitrary operators, the so-called Matrix-Product-Operators, which allows a fast and flexible calculation of arbitrary expectation values. To obtain algorithms to efficiently calculate groundstates, the density-matrix renormalization group is reformulated using the Matrix-Product-States framework. Further, time-dependent problems are considered. Two different algorithms are presented, one based on a Trotter decomposition of the time-evolution operator, the other one on Krylov subspaces. Finally, the evaluation of dynamical spectral functions is discussed, and a correction vector-based method is presented. In the following chapters, the methods presented in the second chapter, are applied to a number of different physical problems. The third chapter deals with the existence of chiral phases in isotropic one-dimensional quantum spin systems. A preceding analytical study based on a mean-field approach indicated the possible existence of those phases in an isotopic Heisenberg model with a frustrating zig-zag interaction and a magnetic field. In this thesis, the existence of the chiral phases will be shown numerically by using Matrix-Product-States-based algorithms. A key effect of interacting one-dimensional quantum-mechanical many-body systems is the spin-charge separation. However, up to now only signs of the spin-charge separation have been observed in experiments. In the fourth chapter, we propose an experiment using ultracold atomic gases in optical lattices, which allows a well controlled observation of the spin-charge separation (of different hyperfine states of the ultracold atoms) with current state of the art experimental techniques. Ultracold atoms in optical lattices are well described by (Bose)-Hubbard models. In order to support this proposal, we present numerical results for realistic system parameters. Matrix-Product-States are an excellent tool for the simulation of one-dimensional quantum systems, however, they are not well suited for the simulation of higher dimensional systems. For strongly-correlated systems, for instance cuprates-based high-temperature superconductors, quantum fluctuations play an essential role. Classical mean-field theories neglect any kind of fluctuations, thus they are not suitable to describe strongly-correlated systems. The dynamical mean-field theory (DMFT) fully takes local quantum fluctuations into account but neglects any kind of spatial fluctuations. The many-body problem on the lattice is mapped onto an impurity problem, which needs to be solved self-consistently. In the last chapter of this thesis, Matrix-Product-States-based algorithms are used to solve the impurity problem of the dynamical mean-field theory. We present results for a Hubbard model on a one-dimensional lattice and on a Bethe lattice obtained by the dynamical mean-field and compare them with exact results.","abstract_html":"In this thesis, the numerical simulation of strongly-interacting many-body quantum-mechanical systems using matrix product states (MPS) is considered. Compared to classical systems, quantum many-body systems possess an exponentially enlarged number of degrees of freedom, significantly complicating a simulation on a classical computer. Matrix-Product-States are a novel representation of arbitrary quantum many-body states. Using quantum information theory, it is possible to show that Matrix-Product-States provide a polynomial-sized representation of one-dimensional quantum systems, thus allowing an efficient simulation of one-dimensional quantum system on classical computers. Matrix-Product-States form the conceptual framework of the density-matrix renormalization group (DMRG). Based upon this connection, deeper understanding of the density matrix renormalization group can be obtained. After a general introduction in the first chapter of this thesis, the second chapter deals with Matrix-Product-States, focusing on the development of fast and stable algorithms. It is possible to extend the Matrix-Product-States approach to be able to represent arbitrary operators, the so-called Matrix-Product-Operators, which allows a fast and flexible calculation of arbitrary expectation values. To obtain algorithms to efficiently calculate groundstates, the density-matrix renormalization group is reformulated using the Matrix-Product-States framework. Further, time-dependent problems are considered. Two different algorithms are presented, one based on a Trotter decomposition of the time-evolution operator, the other one on Krylov subspaces. Finally, the evaluation of dynamical spectral functions is discussed, and a correction vector-based method is presented. In the following chapters, the methods presented in the second chapter, are applied to a number of different physical problems. The third chapter deals with the existence of chiral phases in isotropic one-dimensional quantum spin systems. A preceding analytical study based on a mean-field approach indicated the possible existence of those phases in an isotopic Heisenberg model with a frustrating zig-zag interaction and a magnetic field. In this thesis, the existence of the chiral phases will be shown numerically by using Matrix-Product-States-based algorithms. A key effect of interacting one-dimensional quantum-mechanical many-body systems is the spin-charge separation. However, up to now only signs of the spin-charge separation have been observed in experiments. In the fourth chapter, we propose an experiment using ultracold atomic gases in optical lattices, which allows a well controlled observation of the spin-charge separation (of different hyperfine states of the ultracold atoms) with current state of the art experimental techniques. Ultracold atoms in optical lattices are well described by (Bose)-Hubbard models. In order to support this proposal, we present numerical results for realistic system parameters. Matrix-Product-States are an excellent tool for the simulation of one-dimensional quantum systems, however, they are not well suited for the simulation of higher dimensional systems. For strongly-correlated systems, for instance cuprates-based high-temperature superconductors, quantum fluctuations play an essential role. Classical mean-field theories neglect any kind of fluctuations, thus they are not suitable to describe strongly-correlated systems. The dynamical mean-field theory (DMFT) fully takes local quantum fluctuations into account but neglects any kind of spatial fluctuations. The many-body problem on the lattice is mapped onto an impurity problem, which needs to be solved self-consistently. In the last chapter of this thesis, Matrix-Product-States-based algorithms are used to solve the impurity problem of the dynamical mean-field theory. We present results for a Hubbard model on a one-dimensional lattice and on a Bethe lattice obtained by the dynamical mean-field and compare them with exact results.","abstract_has_math":false,"creators":["Kleine, Adrian"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Schollwöck, Ulrich"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-30T19:43:35Z","subjects":["info:eu-repo/classification/ddc/530","Festkörpertheorie","Vielteilchentheorie","Computersimulation","Entropie <Informationstheorie>","Maximum-Entropie-Methode","Quantenmechanisches System","Starke Kopplung","Bose-Einstein-Kondensation","Physik","DMRG","DMFT"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-124759%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-124759%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-124759%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/63324","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Schollwöck, Ulrich"]},{"key":"dc:creator","label":"Author","values":["Kleine, Adrian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2010"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-34590"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/530","Festkörpertheorie","Vielteilchentheorie","Computersimulation","Entropie <Informationstheorie>","Maximum-Entropie-Methode","Quantenmechanisches System","Starke Kopplung","Bose-Einstein-Kondensation","Physik","DMRG","DMFT"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/63324","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-124759%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, the numerical simulation of strongly-interacting many-body quantum-mechanical systems using matrix product states (MPS) is considered. Compared to classical systems, quantum many-body systems possess an exponentially enlarged number of degrees of freedom, significantly complicating a simulation on a classical computer. Matrix-Product-States are a novel representation of arbitrary quantum many-body states. Using quantum information theory, it is possible to show that Matrix-Product-States provide a polynomial-sized representation of one-dimensional quantum systems, thus allowing an efficient simulation of one-dimensional quantum system on classical computers. Matrix-Product-States form the conceptual framework of the density-matrix renormalization group (DMRG). Based upon this connection, deeper understanding of the density matrix renormalization group can be obtained. After a general introduction in the first chapter of this thesis, the second chapter deals with Matrix-Product-States, focusing on the development of fast and stable algorithms. It is possible to extend the Matrix-Product-States approach to be able to represent arbitrary operators, the so-called Matrix-Product-Operators, which allows a fast and flexible calculation of arbitrary expectation values. To obtain algorithms to efficiently calculate groundstates, the density-matrix renormalization group is reformulated using the Matrix-Product-States framework. Further, time-dependent problems are considered. Two different algorithms are presented, one based on a Trotter decomposition of the time-evolution operator, the other one on Krylov subspaces. Finally, the evaluation of dynamical spectral functions is discussed, and a correction vector-based method is presented. In the following chapters, the methods presented in the second chapter, are applied to a number of different physical problems. The third chapter deals with the existence of chiral phases in isotropic one-dimensional quantum spin systems. A preceding analytical study based on a mean-field approach indicated the possible existence of those phases in an isotopic Heisenberg model with a frustrating zig-zag interaction and a magnetic field. In this thesis, the existence of the chiral phases will be shown numerically by using Matrix-Product-States-based algorithms. A key effect of interacting one-dimensional quantum-mechanical many-body systems is the spin-charge separation. However, up to now only signs of the spin-charge separation have been observed in experiments. In the fourth chapter, we propose an experiment using ultracold atomic gases in optical lattices, which allows a well controlled observation of the spin-charge separation (of different hyperfine states of the ultracold atoms) with current state of the art experimental techniques. Ultracold atoms in optical lattices are well described by (Bose)-Hubbard models. In order to support this proposal, we present numerical results for realistic system parameters. Matrix-Product-States are an excellent tool for the simulation of one-dimensional quantum systems, however, they are not well suited for the simulation of higher dimensional systems. For strongly-correlated systems, for instance cuprates-based high-temperature superconductors, quantum fluctuations play an essential role. Classical mean-field theories neglect any kind of fluctuations, thus they are not suitable to describe strongly-correlated systems. The dynamical mean-field theory (DMFT) fully takes local quantum fluctuations into account but neglects any kind of spatial fluctuations. The many-body problem on the lattice is mapped onto an impurity problem, which needs to be solved self-consistently. In the last chapter of this thesis, Matrix-Product-States-based algorithms are used to solve the impurity problem of the dynamical mean-field theory. We present results for a Hubbard model on a one-dimensional lattice and on a Bethe lattice obtained by the dynamical mean-field and compare them with exact results."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 151 S. : Ill., graph. Darst. (2010). = Aachen, Techn. Hochsch., Diss., 2010"]},{"key":"dc:title","label":"Title","values":["Simulating quantum systems on classical computers with matrix product states"]}]}],"canonical_facts":{"dc:contributor":["Schollwöck, Ulrich"],"dc:coverage":["DE"],"dc:creator":["Kleine, Adrian"],"dc:date":["2010"],"dc:description":["In this thesis, the numerical simulation of strongly-interacting many-body quantum-mechanical systems using matrix product states (MPS) is considered. Compared to classical systems, quantum many-body systems possess an exponentially enlarged number of degrees of freedom, significantly complicating a simulation on a classical computer. Matrix-Product-States are a novel representation of arbitrary quantum many-body states. Using quantum information theory, it is possible to show that Matrix-Product-States provide a polynomial-sized representation of one-dimensional quantum systems, thus allowing an efficient simulation of one-dimensional quantum system on classical computers. Matrix-Product-States form the conceptual framework of the density-matrix renormalization group (DMRG). Based upon this connection, deeper understanding of the density matrix renormalization group can be obtained. After a general introduction in the first chapter of this thesis, the second chapter deals with Matrix-Product-States, focusing on the development of fast and stable algorithms. It is possible to extend the Matrix-Product-States approach to be able to represent arbitrary operators, the so-called Matrix-Product-Operators, which allows a fast and flexible calculation of arbitrary expectation values. To obtain algorithms to efficiently calculate groundstates, the density-matrix renormalization group is reformulated using the Matrix-Product-States framework. Further, time-dependent problems are considered. Two different algorithms are presented, one based on a Trotter decomposition of the time-evolution operator, the other one on Krylov subspaces. Finally, the evaluation of dynamical spectral functions is discussed, and a correction vector-based method is presented. In the following chapters, the methods presented in the second chapter, are applied to a number of different physical problems. The third chapter deals with the existence of chiral phases in isotropic one-dimensional quantum spin systems. A preceding analytical study based on a mean-field approach indicated the possible existence of those phases in an isotopic Heisenberg model with a frustrating zig-zag interaction and a magnetic field. In this thesis, the existence of the chiral phases will be shown numerically by using Matrix-Product-States-based algorithms. A key effect of interacting one-dimensional quantum-mechanical many-body systems is the spin-charge separation. However, up to now only signs of the spin-charge separation have been observed in experiments. In the fourth chapter, we propose an experiment using ultracold atomic gases in optical lattices, which allows a well controlled observation of the spin-charge separation (of different hyperfine states of the ultracold atoms) with current state of the art experimental techniques. Ultracold atoms in optical lattices are well described by (Bose)-Hubbard models. In order to support this proposal, we present numerical results for realistic system parameters. Matrix-Product-States are an excellent tool for the simulation of one-dimensional quantum systems, however, they are not well suited for the simulation of higher dimensional systems. For strongly-correlated systems, for instance cuprates-based high-temperature superconductors, quantum fluctuations play an essential role. Classical mean-field theories neglect any kind of fluctuations, thus they are not suitable to describe strongly-correlated systems. The dynamical mean-field theory (DMFT) fully takes local quantum fluctuations into account but neglects any kind of spatial fluctuations. The many-body problem on the lattice is mapped onto an impurity problem, which needs to be solved self-consistently. In the last chapter of this thesis, Matrix-Product-States-based algorithms are used to solve the impurity problem of the dynamical mean-field theory. We present results for a Hubbard model on a one-dimensional lattice and on a Bethe lattice obtained by the dynamical mean-field and compare them with exact results."],"dc:identifier":["https://publications.rwth-aachen.de/record/63324","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-124759%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-34590"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 151 S. : Ill., graph. Darst. (2010). = Aachen, Techn. Hochsch., Diss., 2010"],"dc:subject":["info:eu-repo/classification/ddc/530","Festkörpertheorie","Vielteilchentheorie","Computersimulation","Entropie <Informationstheorie>","Maximum-Entropie-Methode","Quantenmechanisches System","Starke Kopplung","Bose-Einstein-Kondensation","Physik","DMRG","DMFT"],"dc:title":["Simulating quantum systems on classical computers with matrix product states"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:43:35Z"}