@phdthesis{Hausoel2022,
  author    = {Hausoel, Andreas},
  title     = {Electronic magnetism in correlated systems: from quantum materials down to Earth's core},
  doi       = {10.25972/OPUS-25444},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-254444},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {In the last decade continuous-time quantum Monte Carlo in the hybridization expansion (CTHYB) was one of the most successful Monte Carlo techniques to describe correlated quantum phenomena in conjunction with dynamical mean field theory (DMFT). The first part of the thesis consists of algorithmical developments regarding CTHYB and DMFT. I provide a complete derivation and an extensive discussion of the expansion formula. We generalized it to treat spin-orbit coupling, and invented the superstate sampling algorithm to make it efficient enough for describing systems with general interactions, crystal fields and spin-orbit coupling at low temperatures. But CTHYB is known to fail in the standard implementation for equal-time correlators, certain higher-order Green's functions and the atomic limit; we discovered that its estimator for the Greens function is also inconsistent for Anderson impurities with finite, discrete baths. I focus then on further improvements of CTHYB that we have conceived and worked on, in particular for f-orbitals and for taking physical symmetries into account in the calculation of the Monte Carlo observables. The second part of the thesis presents selected physical applications of these methods. I show DMFT calculations of highest accuracy for elemental iron and nickel and discover a new mechanism of magnetic ordering in nickel: the ordering of band structure-induced local moments. Then we analyze the stability of this phenomenon under pressure and temperatures, that characterize in the Earth's core. We find, that the mechanism survives these conditions and may give a significant contribution to the generation of the Earth's magnetic field. The next topic is the stability of double Dirac fermions against electronic correlations. We find, that the Coulomb interaction in the corresponding material Bi2 CuO4 are strong enough to destroy the double Dirac cone, and substantial uniform pressure is necessary to restore them. In the last chapter I derive the properties of Higgs and Goldstone bosons from Ginzburg-Landau theory, and identify these excitations in a model of an excitonic magnet.},
  subject      = {Monte-Carlo-Simulation},
  language  = {en}
}
@phdthesis{Schrauth2021,
  author    = {Schrauth, Manuel},
  title     = {Critical Phenomena in Topologically Disordered Systems},
  doi       = {10.25972/OPUS-23499},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-234998},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2021},
  abstract  = {Clearly, in nature, but also in technological applications, complex systems built in an entirely ordered and regular fashion are the exception rather than the rule. In this thesis we explore how critical phenomena are influenced by quenched spatial randomness. Specifically, we consider physical systems undergoing a continuous phase transition in the presence of topological disorder, where the underlying structure, on which the system evolves, is given by a non-regular, discrete lattice. We therefore endeavour to achieve a thorough understanding of the interplay between collective dynamics and quenched randomness. According to the intriguing concept of universality, certain laws emerge from collectively behaving many-body systems at criticality, almost regardless of the precise microscopic realization of interactions in those systems. As a consequence, vastly different phenomena show striking similarities at their respective phase transitions. In this dissertation we pursue the question of whether the universal properties of critical phenomena are preserved when the system is subjected to topological perturbations. For this purpose, we perform numerical simulations of several prototypical systems of statistical physics which show a continuous phase transition. In particular, the equilibrium spin-1/2 Ising model and its generalizations represent -- among other applications -- fairly natural approaches to model magnetism in solids, whereas the non-equilibrium contact process serves as a toy model for percolation in porous media and epidemic spreading. Finally, the Manna sandpile model is strongly related to the concept of self-organized criticality, where a complex dynamic system reaches a critical state without fine-tuning of external variables. Our results reveal that the prevailing understanding of the influence of topological randomness on critical phenomena is insufficient. In particular, by considering very specific and newly developed lattice structures, we are able to show that -- contrary to the popular opinion -- spatial correlations in the number of interacting neighbours are not a key measure for predicting whether disorder ultimately alters the behaviour of a given critical system.},
  subject      = {Ising-Modell},
  language  = {en}
}
@phdthesis{Hofmann2020,
  author    = {Hofmann, Johannes Stephan},
  title     = {On the interplay of topology and interaction: A quantum Monte Carlo study},
  doi       = {10.25972/OPUS-20507},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-205071},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2020},
  abstract  = {Adding interactions to topological (non-)trivial free fermion systems can in general have four different effects: (i) In symmetry protected topological band insulators, the correlations may lead to the spontaneous breaking of some protecting symmetries by long-range order that gaps the topological boundary modes. (ii) In free fermion (semi-)metal, the interaction could vice versa also generate long-range order that in turn induces a topological mass term and thus generates non-trivial phases dynamically. (iii) Correlation might reduce the topological classification of free fermion systems by allowing adiabatic deformations between states of formerly distinct phases. (iv) Interaction can generate long-range entangled topological order in states such as quantum spin liquids or fractional quantum Hall states that cannot be represented by non-interacting systems. During the course of this thesis, we use numerically exact quantum Monte Carlo algorithms to study various model systems that (potentially) represent one of the four scenarios, respectively. First, we investigate a two-dimensional \$d_{xy}\$-wave, spin-singlet superconductor, which is relevant for high-\$T_c\$ materials such as the cuprates. This model represents nodal topological superconductors and exhibits chiral flat-band edge states that are protected by time-reversal and translational invariance. We introduce the conventional Hubbard interaction along the edge in order to study their stability with respect to correlations and find ferromagnetic order in case of repulsive interaction as well as charge-density-wave order and/or additional \$i\$s-wave pairing for attractive couplings. A mean-field analysis that, for the first time, is formulated in terms of the Majorana edge modes suggests that any order has normal and superconducting contributions. For example, the ferromagnetic order appears in linear superposition with triplet pairing. This finding is well confirmed by the numerically exact quantum Monte Carlo investigation. Second, we consider spinless electrons on a two-dimensional Lieb lattice that are subject to nearest-neighbor Coulomb repulsion. The low energy modes of the free fermion part constitute a spin-\$1\$ Dirac cone that might be gapped by several mass terms. One option breaks time-reversal symmetry and generates a topological Chern insulator, which mainly motivated this study. We employ two flavors of quantum Monte Carlo methods and find instead the formation of charge-density-wave order that breaks particle-hole symmetry. Additionally, due to sublattices of unequal size in Lieb lattices, this induces a finite chemical potential that drives the system away from half-filling. We argue that this mechanism potentially extends the range of solvable models with finite doping by coupling the Lieb lattice to the target system of interest. Third, we construct a system with four layers of a topological insulators and interlayer correlation that respects one independent time-reversal and a unitary \$\mathbb{Z}_2\$ symmetry. Previous studies claim a reduced topological classification from \$\mathbb{Z}\$ to \$\mathbb{Z}_4\$, for example by gapping out degenerate zero modes in topological defects once the correlation term is designed properly. Our interaction is chosen according to this analysis such that there should exist an adiabatic deformation between states whose topological invariant differs by \$\Delta w=\pm4\$ in the free fermion classification. We use a projective quantum Monte Carlo algorithm to determine the ground-state phase diagram and find a symmetry breaking regime, in addition to the non-interacting semi-metal, that separates the free fermion insulators. Frustration reduces the size of the long-range ordered region until it is replaced by a first order phase transition. Within the investigated range of parameters, there is no adiabatic path deforming the formerly distinct free fermion states into each other. We conclude that the prescribed reduction rules, which often use the bulk-boundary correspondence, are necessary but not sufficient and require a more careful investigation. Fourth, we study conduction electron on a honeycomb lattice that form a Dirac semi-metal Kondo coupled to spin-1/2 degrees of freedom on a Kagome lattice. The local moments are described by a variant of the Balents-Fisher-Girvin model that has been shown to host a ferromagnetic phase and a \$\mathbb{Z}_2\$ spin liquid at strong frustration. Here, we report the first numerical exact quantum Monte Carlo simulation of the Kondo-coupled system that does not exhibit the negative-sign problem. When the local moments form a ferromagnet, the Kondo coupling induces an anti-ferromagnetic mass term in the conduction-electron system. At large frustration, the Dirac cone remains massless and the spin system forms a \$\mathbb{Z}_2\$ spin liquid. Owing to the odd number of spins per unit cell, this constitutes a non-Fermi liquid that violates Luttinger's theorem which relates the Fermi volume to the particle density in a Fermi liquid. This phase is a specific realization of the so called 'fractional Fermi liquid` as it has been first introduced in the context of heavy fermion models.},
  subject      = {Monte-Carlo-Simulation},
  language  = {en}
}
@phdthesis{Beyl2020,
  author    = {Beyl, Stefan},
  title     = {Hybrid Quantum Monte Carlo for Condensed Matter Models},
  doi       = {10.25972/OPUS-19122},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-191225},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2020},
  abstract  = {In this thesis we consider the hybrid quantum Monte Carlo method for simulations of the Hubbard and Su-Schrieffer-Heeger model. In the first instance, we discuss the hybrid quantum Monte Carlo method for the Hubbard model on a square lattice. We point out potential ergodicity issues and provide a way to circumvent them by a complexification of the method. Furthermore, we compare the efficiency of the hybrid quantum Monte Carlo method with a well established determinantal quantum Monte Carlo method for simulations of the half-filled Hubbard model on square lattices. One reason why the hybrid quantum Monte Carlo method loses the comparison is that we do not observe the desired sub-quadratic scaling of the numerical effort. Afterwards we present a formulation of the hybrid quantum Monte Carlo method for the Su-Schrieffer-Heeger model in two dimensions. Electron-phonon models like this are in general very hard to simulate using other Monte Carlo methods in more than one dimensions. It turns out that the hybrid quantum Monte Carlo method is much better suited for this model . We achieve favorable scaling properties and provide a proof of concept. Subsequently, we use the hybrid quantum Monte Carlo method to investigate the Su-Schrieffer-Heeger model in detail at half-filling in two dimensions. We present numerical data for staggered valence bond order at small phonon frequencies and an antiferromagnetic order at high frequencies. Due to an O(4) symmetry the antiferromagnetic order is connected to a superconducting charge density wave. Considering the Su-Schrieffer-Heeger model without tight-binding hopping reveals an additional unconstrained Z_2 gauge theory. In this case, we find indications for π-fluxes and a possible Z_2 Dirac deconfined phase as well as for a columnar valence bond ordered state at low phonon energies. In our investigations of the several phase transitions we discuss the different possibilities for the underlying mechanisms and reveal first insights into a rich phase diagram.},
  subject      = {Monte-Carlo-Simulation},
  language  = {en}
}
@phdthesis{Weber2019,
  author    = {Weber, Manuel},
  title     = {Action-based quantum Monte Carlo approach to fermion-boson models},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-157643},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2019},
  abstract  = {This work deals with the development and application of novel quantum Monte Carlo methods to simulate fermion-boson models. Our developments are based on the path-integral formalism, where the bosonic degrees of freedom are integrated out exactly to obtain a retarded fermionic interaction. We give an overview of three methods that can be used to simulate retarded interactions. In particular, we develop a novel quantum Monte Carlo method with global directed-loop updates that solves the autocorrelation problem of previous approaches and scales linearly with system size. We demonstrate its efficiency for the Peierls transition in the Holstein model and discuss extensions to other fermion-boson models as well as spin-boson models. Furthermore, we show how with the help of generating functionals bosonic observables can be recovered directly from the Monte Carlo configurations. This includes estimators for the boson propagator, the fidelity susceptibility, and the specific heat of the Holstein model. The algorithmic developments of this work allow us to study the specific heat of the spinless Holstein model covering its entire parameter range. Its key features are explained from the single-particle spectral functions of electrons and phonons. In the adiabatic limit, the spectral properties are calculated exactly as a function of temperature using a classical Monte Carlo method and compared to results for the Su-Schrieffer-Heeger model.},
  subject      = {Monte-Carlo-Simulation},
  language  = {en}
}
@phdthesis{Goth2015,
  author    = {Goth, Florian},
  title     = {Continuous time quantum Monte Carlo Studies of Quenches and Correlated Systems with Broken Inversion Symmetry},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-118836},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2015},
  abstract  = {This thesis deals with quantum Monte Carlo simulations of correlated low dimensional electron systems. The correlation that we have in mind is always given by the Hubbard type electron electron interaction in various settings. To facilitate this task, we develop the necessary methods in the first part. We develop the continuous time interaction expansion quantum algorithm in a manner suitable for the treatment of effective and non-equilibrium problems. In the second part of this thesis we consider various applications of the algorithms. First we examine a correlated one-dimensional chain of electrons that is subject to some form of quench dynamics where we suddenly switch off the Hubbard interaction. We find the light-cone-like Lieb-Robinson bounds and forms of restricted equilibration subject to the conserved quantities. Then we consider a Hubbard chain subject to Rashba spin-orbit coupling in thermal equilibrium. This system could very well be realized on a surface with the help of metallic adatoms. We find that we can analytically connect the given model to a model without spin-orbit coupling. This link enabled us to interpret various results for the standard Hubbard model, such as the single-particle spectra, now in the context of the Hubbard model with Rashba spin-orbit interaction. And finally we have considered a magnetic impurity in a host consisting of a topological insulator. We find that the impurity still exhibits the same features as known from the single impurity Anderson model. Additionally we study the effects of the impurity in the bath and we find that in the parameter regime where the Kondo singlet is formed the edge state of the topological insulator is rerouted around the impurity.},
  subject      = {Elektronenkorrelation},
  language  = {en}
}
@phdthesis{Bercx2014,
  author    = {Bercx, Martin Helmut},
  title     = {Numerical studies of heavy-fermion systems and correlated topological insulators},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-116138},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2014},
  abstract  = {In this thesis, we investigate aspects of the physics of heavy-fermion systems and correlated topological insulators. We numerically solve the interacting Hamiltonians that model the physical systems using quantum Monte Carlo algorithms to access both ground-state and finite-temperature observables. Initially, we focus on the metamagnetic transition in the Kondo lattice model for heavy fermions. On the basis of the dynamical mean-field theory and the dynamical cluster approximation, our calculations point towards a continuous transition, where the signatures of metamagnetism are linked to a Lifshitz transition of heavy-fermion bands. In the second part of the thesis, we study various aspects of magnetic pi fluxes in the Kane-Mele-Hubbard model of a correlated topological insulator. We describe a numerical measurement of the topological index, based on the localized mid-gap states that are provided by pi flux insertions. Furthermore, we take advantage of the intrinsic spin degree of freedom of a pi flux to devise instances of interacting quantum spin systems. In the third part of the thesis, we introduce and characterize the Kane-Mele-Hubbard model on the pi flux honeycomb lattice. We place particular emphasis on the correlations effects along the one-dimensional boundary of the lattice and compare results from a bosonization study with finite-size quantum Monte Carlo simulations.},
  subject      = {Schwere-Fermionen-System},
  language  = {en}
}
@phdthesis{Parragh2013,
  author    = {Parragh, Nicolaus},
  title     = {Strongly Correlated Multi-Orbital Systems : A Continuous-Time Quantum Monte Carlo Analysis},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-85253},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2013},
  abstract  = {In this thesis I present results concerning realistic calculations of correlated fermionic many-body systems. One of the main objectives of this work was the implementation of a hybridization expansion continuous-time quantum Monte Carlo (CT-HYB) algorithm and of a flexible self-consistency loop based on the dynamical mean-field theory (DMFT). DMFT enables us to treat strongly correlated electron systems numerically. After the implementation and extensive testing of the program we investigated different problems to answer open questions concerning correlated systems and their numerical treatment.},
  subject      = {Monte-Carlo-Simulation},
  language  = {en}
}
@article{ChipperfieldDythamHovestadt2011,
  author    = {Chipperfield, Joseph D. and Dytham, Calvin and Hovestadt, Thomas},
  title     = {An Updated Algorithm for the Generation of Neutral Landscapes by Spectral Synthesis},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-68938},
  year      = {2011},
  abstract  = {Background: Patterns that arise from an ecological process can be driven as much from the landscape over which the process is run as it is by some intrinsic properties of the process itself. The disentanglement of these effects is aided if it possible to run models of the process over artificial landscapes with controllable spatial properties. A number of different methods for the generation of so-called 'neutral landscapes' have been developed to provide just such a tool. Of these methods, a particular class that simulate fractional Brownian motion have shown particular promise. The existing methods of simulating fractional Brownian motion suffer from a number of problems however: they are often not easily generalisable to an arbitrary number of dimensions and produce outputs that can exhibit some undesirable artefacts. Methodology: We describe here an updated algorithm for the generation of neutral landscapes by fractional Brownian motion that do not display such undesirable properties. Using Monte Carlo simulation we assess the anisotropic properties of landscapes generated using the new algorithm described in this paper and compare it against a popular benchmark algorithm. Conclusion/Significance: The results show that the existing algorithm creates landscapes with values strongly correlated in the diagonal direction and that the new algorithm presented here corrects this artefact. A number of extensions of the algorithm described here are also highlighted: we describe how the algorithm can be employed to generate landscapes that display different properties in different dimensions and how they can be combined with an environmental gradient to produce landscapes that combine environmental variation at the local and macro scales.},
  subject      = {Landschaft},
  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}
}