@phdthesis{Balzer2008,
  author    = {Balzer, Matthias},
  title     = {F{\"u}llungs- und wechselwirkungsabh{\"a}ngiger Mott-{\"U}bergang: Quanten-Cluster-Rechnungen im Rahmen der Selbstenergiefunktional-Theorie},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-35266},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2008},
  abstract  = {Die Untersuchung stark korrelierter Elektronensysteme anhand des zweidimensionalen Hubbard-Modells bildet das zentrale Thema dieser Arbeit. Wir analysieren das Schicksal des Mott-Isolators bei Dotierung als auch bei Reduzierung der Wechselwirkungsst{\"a}rke. Die numerische Auswertung erfolgt mit Hilfe von Quanten-Cluster-Approximationen, die eine thermodynamisch konsistente Beschreibung der Grundzustandseigenschaften garantieren. Der hier verwendete Rahmen der Selbstenergiefunktional-Theorie bietet eine große Flexibilit{\"a}t bei der Konstruktion von Cluster-N{\"a}herungen. Eine detaillierte Analyse gibt Aufschluss {\"u}ber die Qualit{\"a}t und das Konvergenzverhalten unterschiedlicher Cluster-N{\"a}herungen innerhalb der Selbstenergiefunktional-Theorie. Wir verwenden f{\"u}r diese Untersuchungen das eindimensionale Hubbard-Modell und vergleichen unsere Resultate mit der exakten L{\"o}sung. In zwei Dimensionen finden wir als Grundzustand des Teilchen-Loch-symmetrischen Modells bei Halbf{\"u}llung einen antiferromagnetischen Isolator unabh{\"a}ngig von der Wechselwirkungsst{\"a}rke. Die Ber{\"u}cksichtigung kurzreichweitiger r{\"a}umlicher Korrelationen durch unsere Cluster-N{\"a}herung f{\"u}hrt, im Vergleich mit der dynamischen Mean-Field-Theorie, zu einer deutlichen Verbesserung des antiferromagnetischen Ordnungsparameters. Dar{\"u}berhinaus beobachten wir in der paramagnetischen Phase einen Metall-Isolator-{\"U}bergang als Funktion der Wechselwirkungsst{\"a}rke, der sich qualitativ vom reinen Mean-Field-Szenario unterscheidet. Ausgehend vom antiferromagnetischen Mott-Isolator zeigt sich ein f{\"u}llungsgetriebener Metall-Isolator-{\"U}bergang in eine paramagnetische metallische Phase. Abh{\"a}ngig von der verwendeten Cluster-Approximation tritt dabei zun{\"a}chst eine antiferromagnetische metallische Phase auf. Neben langreichweitiger antiferromagnetischer Ordnung haben wir in unseren Rechnungen auch Supraleitung ber{\"u}cksichtigt. Das Verhalten des supraleitenden Ordnungsparameters als Funktion der Dotierung ist dabei in guter {\"U}bereinstimmung sowohl mit anderen numerischen Verfahren als auch mit experimentellen Ergebnissen.},
  subject      = {Festk{\"o}rpertheorie},
  language  = {de}
}
@phdthesis{Hochkeppel2008,
  author    = {Hochkeppel, Stephan},
  title     = {One- and Two-Particle Correlation Functions in the Dynamical Quantum Cluster Approach},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-28705},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2008},
  abstract  = {This thesis is dedicated to a theoretical study of the 1-band Hubbard model in the strong coupling limit. The investigation is based on the Dynamical Cluster Approximation (DCA) which systematically restores non-local corrections to the Dynamical Mean Field approximation (DMFA). The DCA is formulated in momentum space and is characterised by a patching of the Brillouin zone where momentum conservation is only recovered between two patches. The approximation works well if k-space correlation functions show a weak momentum dependence. In order to study the temperature and doping dependence of the spin- and charge excitation spectra, we explicitly extend the Dynamical Cluster Approximation to two-particle response functions. The full irreducible two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin and/or charge excitations. The effective vertex is calculated by using the Quantum Monte Carlo method on the finite cluster whereas the analytical continuation of dynamical quantities is performed by a stochastic version of the maximum entropy method. A comparison with high temperature auxiliary field quantum Monte Carlo data serves as a benchmark for our approach to two-particle correlation functions. Our method can reproduce basic characteristics of the spin- and charge excitation spectrum. Near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin and single-particle excitations: a collective spin mode emerges at optimal doping and sufficiently low temperatures in the spin response spectrum and exhibits the energy scale of the magnetic exchange interaction J. Simultaneously, the low energy single-particle excitations are characterised by a coherent quasiparticle with bandwidth J. The origin of the quasiparticle can be quite well understood in a picture of a more or less antiferromagnetic ordered background in which holes are dressed by spin-excitations to allow for a coherent motion. By increasing doping, all features which are linked to the spin-polaron vanish in the single-particle as well as two-particle spin response spectrum. In the second part of the thesis an analysis of superconductivity in the Hubbard model is presented. The superconducting instability is implemented within the Dynamical Cluster Approximation by essentially allowing U(1) symmetry breaking baths in the QMC calculations for the cluster. The superconducting transition temperature T_c is derived from the d-wave order parameter which is directly estimated on the Monte Carlo cluster. The critical temperature T_c is in astonishing agreement with the temperature scale estimated by the divergence of the pair-field susceptibility in the paramagnetic phase. A detailed study of the pseudo and superconducting gap is continued by the investigation of the local and angle-resolved spectral function.},
  subject      = {Festk{\"o}rpertheorie},
  language  = {en}
}
@phdthesis{Lang2010,
  author    = {Lang, Thomas C.},
  title     = {Quantum Monte Carlo methods and strongly correlated electrons on honeycomb structures},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-53506},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2010},
  abstract  = {In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.},
  subject      = {Monte-Carlo-Simulation},
  language  = {en}
}
@phdthesis{Luitz2012,
  author    = {Luitz, David J.},
  title     = {Numerical methods and applications in many fermion systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-75927},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2012},
  abstract  = {This thesis presents results covering several topics in correlated many fermion systems. A Monte Carlo technique (CT-INT) that has been implemented, used and extended by the author is discussed in great detail in chapter 3. The following chapter discusses how CT-INT can be used to calculate the two particle Green's function and explains how exact frequency summations can be obtained. A benchmark against exact diagonalization is presented. The link to the dynamical cluster approximation is made in the end of chapter 4, where these techniques are of immense importance. In chapter 5 an extensive CT-INT study of a strongly correlated Josephson junction is shown. In particular, the signature of the first order quantum phase transition between a Kondo and a local moment regime in the Josephson current is discussed. The connection to an experimental system is made with great care by developing a parameter extraction strategy. As a final result, we show that it is possible to reproduce experimental data from a numerically exact CT-INT model-calculation. The last topic is a study of graphene edge magnetism. We introduce a general effective model for the edge states, incorporating a complicated interaction Hamiltonian and perform an exact diagonalization study for different parameter regimes. This yields a strong argument for the importance of forbidden umklapp processes and of the strongly momentum dependent interaction vertex for the formation of edge magnetism. Additional fragments concerning the use of a Legendre polynomial basis for the representation of the two particle Green's function, the analytic continuation of the self energy for the Anderson Kane Mele Model, as well as the generation of test data with a given covariance matrix are documented in the appendix. A final appendix provides some very important matrix identities that are used for the discussion of technical details of CT-INT.},
  subject      = {Fermionensystem},
  language  = {en}
}
@phdthesis{Klett2021,
  author    = {Klett, Michael},
  title     = {Auxiliary particle approach for strongly correlated electrons : How interaction shapes order},
  doi       = {10.25972/OPUS-24812},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-248121},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2021},
  abstract  = {Since the genesis of condensed matter physics, strongly correlated fermionic systems have shown a variety of fascinating properties and remain a vital topic in the field. Such systems arise through electronic interaction, and despite decades of intensive research, no holistic approach to solving this problem has been found. During that time, physicists have compiled a wealth of individual experimental and theoretical results, which together give an invaluable insight into these materials, and, in some instances, can explain correlated phenomena. However, there are several systems that stubbornly refuse to fall completely in line with current theoretical descriptions, among them the high-\( T_c{}\) cuprates and heavy fermion compounds. Although the two material classes have been around for the better part of the last 50 years, large portions of their respective phase diagram are still under intensive debate. Recent experiments in several electron-doped cuprates compounds, e.g. neodymium cerium copper oxide (Nd\(_{2x}\)Ce\(_x\)CuO\(_4\)), reveal a charge ordering about an antiferromagnetic ground state. So far, it has not been conclusively clarified how this intertwining of charge and spin polarization comes about and how it can be reconciled with a rigorous theoretical description. The heavy-fermion semimetals, on the other hand, have enjoyed renewed scientific interest with the discovery of topological Kondo insulators, a new material class offering a unique interface of topology, symmetry breaking, and correlated phenomena. In this context, samarium hexaboride (SmB\(_6\)) has emerged as a prototypical system, which may feature a topological ground state. In this thesis, we present a spin rotational invariant auxiliary particle approach to investigate the propensities of interacting electrons towards forming new states of order. In particular, we study the onset of spin and charge order in high-\( T_c{}\) cuprate systems and Kondo lattices, as well as the interplay of magnetism and topology. To that end, we use a sophisticated mean-field approximation of bosonic auxiliary particles augmented by a stability analysis of the saddle point via Gaussian fluctuations. The latter enables the derivation of dynamic susceptibilities, which describe the response of the system under external fields and offer a direct comparison to experiments. Both the mean-field and fluctuation formalisms require a numerical tool that is capable of extremizing the saddle point equations, on the one hand, and reliably solving a loop integral of the susceptibility-type, on the other. A full, from scratch derivation of the formalism tailored towards a software implementation, is provided and pedagogically reviewed. The auxiliary particle method allows for a rigorous description of incommensurate magnetic order and compares well to other established numerical and analytical techniques. Within our analysis, we employ the two-dimensional one-band Hubbard as well as the periodic Anderson model as minimal Hamiltonians for the high-\( T_c{}\) cuprates and Kondo systems, respectively. For the former, we observe a regime of intertwined charge- and spin-order in the electron-doped regime, which matches recent experimental observations in the cuprate material Nd\(_{2x}\)Ce\(_x\)CuO\(_4\). Furthermore, we localize the emergence of a Kondo regime in the periodic Anderson model and establish the magnetic phase diagram of the two-band model for topological Kondo insulators. The emerging antiferromagnetic ground state can be characterized by its topological properties and shows, for a non-trivial phase, topologically protected hinge modes.},
  subject      = {Festk{\"o}rpertheorie},
  language  = {en}
}