@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}
}
@phdthesis{Riegler2022,
  author    = {Riegler, David},
  title     = {Emergent phenomena in strongly correlated electron systems: Auxiliary particle approach to the many-body problem},
  doi       = {10.25972/OPUS-27473},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-274737},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {Emergent phenomena in condensed matter physics like, e.g., magnetism, superconductivity, or non-trivial topology often come along with a surprise and exert great fascination to researchers up to this day. Within this thesis, we are concerned with the analysis of associated types of order that arise due to strong electronic interactions and focus on the high-\(T_c\) cuprates and Kondo systems as two prime candidates. The underlying many-body problem cannot be solved analytically and has given rise to the development of various approximation techniques to tackle the problem. In concrete terms, we apply the auxiliary particle approach to investigate tight-binding Hamiltonians subject to a Hubbard interaction term to account for the screened Coulomb repulsion. Thereby, we adopt the so-called Kotliar-Ruckenstein slave-boson representation that reduces the problem to non-interacting quasiparticles within a mean-field approximation. Part I provides a pedagogical review of the theory and generalizes the established formalism to encompass Gaussian fluctuations around magnetic ground states as a crucial step to obtaining novel results. Part II addresses the two-dimensional one-band Hubbard model, which is known to approximately describe the physics of the high-\(T_c\) cuprates that feature high-temperature superconductivity and various other exotic quantum phases that are not yet fully understood. First, we provide a comprehensive slave-boson analysis of the model, including the discussion of incommensurate magnetic phases, collective modes, and a comparison to other theoretical methods that shows that our results can be massively improved through the newly implemented fluctuation corrections. Afterward, we focus on the underdoped regime and find an intertwining of spin and charge order signaled by divergences of the static charge susceptibility within the antiferromagnetic domain. There is experimental evidence for such inhomogeneous phases in various cuprate materials, which has recently aroused interest because such correlations are believed to impact the formation of Cooper pairs. Our analysis identifies two distinct charge-ordering vectors, one of which can be attributed to a Fermi-surface nesting effect and quantitatively fits experimental data in \(\mathrm{Nd}_{2-\mathrm{x}}\mathrm{Ce}_\mathrm{x}\mathrm{CuO}_4\) (NCCO), an electron-doped cuprate compound. The other resembles the so-called Yamada relation implying the formation of periodic, double-occupied domain walls with a crossover to phase separation for small dopings. Part III investigates Kondo systems by analyzing the periodic Anderson model and its generalizations. First, we consider Kondo metals and detect weakly magnetized ferromagnetic order in qualitative agreement with experimental observations, which hinders the formation of heavy fermions. Nevertheless, we suggest two different parameter regimes that could host a possible Kondo regime in the context of one or two conduction bands. The part is concluded with the study of topological order in Kondo insulators based on a three-dimensional model with centrosymmetric spin-orbit coupling. Thereby, we classify topologically distinct phases through appropriate \(\mathbb{Z}_2\) invariants and consider paramagnetic and antiferromagnetic mean-field ground states. Our model parameters are chosen to specifically describe samarium hexaboride (\(\mbox{SmB}_6\)), which is widely believed to be a topological Kondo insulator, and we identify topologically protected surface states in agreement with experimental evidence in that material. Moreover, our theory predicts the emergence of an antiferromagnetic topological insulator featuring one-dimensional hinge-states as the signature of higher-order topology in the strong coupling regime. While the nature of the true ground state is still under debate, corresponding long-range magnetic order has been observed in pressurized or alloyed \(\mbox{SmB}_6\), and recent experimental findings point towards non-trivial topology under these circumstances. The ability to understand and control topological systems brings forth promising applications in the context of spintronics and quantum computing.},
  subject      = {Elektronenkorrelation},
  language  = {en}
}
@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{Eckl2004,
  author    = {Eckl, Thomas},
  title     = {Phenomenological phase-fluctuation model for the underdoped cuprates},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-12115},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2004},
  abstract  = {In this thesis, a phenomenological phase-fluctuation model for the pseudogap regime of the underdoped cuprates was discussed. The key idea of the phase-fluctuation scenario in the high-T_c superconductors is the notion that the pseudogap observed in a wide variety of experiments arises from phase fluctuations of the superconducting gap. In this scenario, below a mean-field temperature scale T_c^{MF}, a d_{x^2-y^2}-wave gap amplitude is assumed to develop. However, the superconducting transition is suppressed to a considerably lower transition temperature T_c by phase fluctuations. In the intermediate temperature regime between T_c^{MF} and T_c, phase fluctuations of the superconducting order parameter give rise to the pseudogap phenomena. The phenomenological phase-fluctuation model discussed in this thesis consists of a two-dimensional BCS-like Hamiltonian where the phase of the pairing-amplitude is free to fluctuate. The fluctuations of the phase were treated by a Monte Carlo simulation of a classical XY model. First, the density of states was calculated. The quasiparticle tunneling conductance (dI/dV) obtained from our phenomenological phase fluctuation model was able to reproduce characteristic and salient features of recent scanning-tunneling studies of Bi2212 and Bi2201 suggesting that the pseudogap behavior observed in these experiments arises from phase fluctuations of the d_{x^2-y^2}-wave pairing gap. In calculating the single-particle spectral weight, we were further able to show how phase fluctuations influence the experimentally observed quasiparticle spectra in detail. In particular the disappearance of the BCS-Bogoliubov quasiparticle band at T_c and the change from a more V-like superconducting gap to a rather U-like pseudogap above T_c can be explained in a consistent way by assuming that the low-energy pseudogap in the underdoped cuprates is due to phase fluctuations of a local d_{x^2-y^2}-wave pairing gap with fixed magnitude. Furthermore, phase fluctuations can explain why the pseudogap starts closing from the nodal points, whereas it rather fills in along the anti-nodal directions and they can also account for the characteristic temperature dependence of the superconducting (pi,0)-photoemission-peak. Next, we have shown that the "violation" of the low-frequency optical sum rule recently observed in the SC state of underdoped Bi2212, which is associated with a reduction of kinetic energy, can be related to the role of phase fluctuations. The decrease in kinetic energy is due to the sharpening of the quasiparticle peaks close to the superconducting transition at T_c == T_{KT}, where the phase correlation length xi diverges. A detailed analysis of the temperature and frequency dependence of the optical conductivity sigma(omega)=sigma_1(omega)+i sigma_2(omega) revealed a superconducting scaling of sigma_2(omega), which starts already above T_c, exactly as observed in high-frequency microwave conductivity experiments on Bi2212. On the other hand, our model was only able to account for the characteristic peak, which is observed in sigma_1(omega) close to the superconducting transition, after the inclusion of an additional marginal-Fermi-liquid scattering-rate in the optical conductivity formula. Finally, we calculated the static uniform diamagnetic susceptibility. It turned out that the precursor effects of the fluctuating diamagnetism above T_c are very small and limited to temperatures close to T_c in a phase-fluctuation scenario of the pseudogap. Instead, the temperature dependence of the uniform static magnetic susceptibility is dominated by the Pauli spin susceptibility, which displayed a very characteristic temperature dependence, independent of the details of the gap function used in our model. This temperature dependence is qualitatively very similar to the experimentally observed change of the Knight-shift as a function of temperature in underdoped Bi2212.},
  subject      = {Hochtemperatursupraleiter},
  language  = {en}
}
@phdthesis{Dahnken2004,
  author    = {Dahnken, Christopher},
  title     = {Spectral properties of strongly correlated electron systems},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-12238},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2004},
  abstract  = {We investigate the single particle static and dynamic properties at zero temperature within the Hubbard an three-band-Hubbard model for the superconducting copper oxides. Based on the recently proposed self-energy functional approach (SFA) [M.Potthoff, Eur. Phys. J. B 32 429 (2003)], we present an extension of the cluster-perturbation theory (CPT) to systems with spontaneous broken symmetry. Our method accounts for both short-range correlations and long-range order. Short-range correlations are accurately taken into account via the exact diagonalization of finite clusters. Long-range order is described by variational optimization of a ficticious symmetry-breaking field. In comparison with related cluster methods, our approach is more flexible and, for a given cluster size, less demanding numerically, especially at zero temperature. An application of the method to the antiferromagnetic phase of the Hubbard model at half-filling shows good agreement with results from quantum Monte-Carlo calculations. We demonstrate that the variational extension of the cluster-perturbation theory is crucial to reproduce salient features of the single-particle spectrum of the insulating cuprates. Comparison of the dispersion of the low-energy excitations with recent experimental results of angular resolved photoemission spectroscopy (ARPES) allows us to fix a consistent parameter set for the one-band Hubbard model with an additional hopping parameter t' along the lattice diagonal. The doping dependence of the single-particle excitations is studied within the t-t-U Hubbard model with special emphasis on the electron doped compounds. We show, that the ARPES results on the band structure and the Fermi surface of Nd{2-x}Ce_xCuOCl_{4-\delta} are naturally obtained within the t-t-U Hubbard model without further need for readjustment or fitting of parameters, as proposed in recent theoretical considerations. We present a theory for the photon energy and polarization dependence of ARPES intensities from the CuO2 plane in the framework of strong correlation models. The importance of surface states for the observed experimental facts is considered. We show that for electric field vector in the CuO_2 plane the 'radiation characteristics' of the O 2p_{\sigma} and Cu 3d_{x^2-y^2} orbitals are strongly peaked along the CuO_2 plane, i.e. most photoelectrons are emitted at grazing angles. This suggests that surface states play an important role in the observed ARPES spectra, consistent with recent data from Sr_2CuCl_2O_2. We show that a combination of surface state dispersion and Fano resonance between surface state and the continuum of LEED-states may produce a precipitous drop in the observed photoelectron current as a function of in-plane momentum, which may well mimic a Fermi-surface crossing. This effect may explain the simultaneous 'observation' of a hole-like and an electron-like Fermi surfaces in Bi_2Sr_2CaCu_2O_{8+\delta} at different photon energies.},
  subject      = {Hochtemperatursupraleiter},
  language  = {en}
}
@phdthesis{Brehm2009,
  author    = {Brehm, Sascha},
  title     = {Two-Particle Excitations in the Hubbard Model for High-Temperature Superconductors: A Quantum Cluster Study},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-38719},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2009},
  abstract  = {Two-particle excitations, such as spin and charge excitations, play a key role in high-Tc cuprate superconductors (HTSC). Due to the antiferromagnetism of the parent compound the magnetic excitations are supposed to be directly related to the mechanism of superconductivity. In particular, the so-called resonance mode is a promising candidate for the pairing glue, a bosonic excitation mediating the electronic pairing. In addition, its interactions with itinerant electrons may be responsible for some of the observed properties of HTSC. Hence, getting to the bottom of the resonance mode is crucial for a deeper understanding of the cuprate materials . To analyze the corresponding two-particle correlation functions we develop in the present thesis a new, non-perturbative and parameter-free technique for T=0 which is based on the Variational Cluster Approach (VCA, an embedded cluster method for one-particle Green's functions). Guided by the spirit of the VCA we extract an effective electron-hole vertex from an isolated cluster and use a fully renormalized bubble susceptibility chi0 including the VCA one-particle propagators.Within our new approach, the magnetic excitations of HTSC are shown to be reproduced for the Hubbard model within the relevant strong-coupling regime. Exceptionally, the famous resonance mode occurring in the underdoped regime within the superconductivity-induced gap of spin-flip electron-hole excitations is obtained. Its intensity and hourglass dispersion are in good overall agreement with experiments. Furthermore, characteristic features such as the position in energy of the resonance mode and the difference of the imaginary part of the susceptibility in the superconducting and the normal states are in accord with Inelastic Neutron Scattering (INS) experiments. For the first time, a strongly-correlated parameter-free calculation revealed these salient magnetic properties supporting the S=1 magnetic exciton scenario for the resonance mode. Besides the INS data on magnetic properties further important new insights were gained recently via ARPES (Angle-Resolved Photoemission-Spectroscopy) and Raman experiments which disclosed a quite different doping dependence of the antinodal compared to the near-nodal gap. This thesis provides an approach to the Raman response similar to the magnetic case for inspecting this gap dichotomy. In agreement with experiments and one-particle data obtained in the VCA, we recover the antinodal gap decreasing and the near-nodal gap increasing as a function of doping. Hence, our results prove the Hubbard model to account for these salient gap features. In summary, we develop a two-particle cluster approach which is appropriate for the strongly-correlated regime and contains no free parameter. Our results obtained with this new approach combined with the phase diagram and the one-particle excitations obtained in the VCA strongly constitute a Hubbard model description of HTSC cuprate materials.},
  subject      = {Hochtemperatursupraleiter},
  language  = {en}
}