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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.
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.
Magnetism is a phenomenon ubiquitously found in everyday life. Yet, together with superconductivity and superfluidity, it is among the few macroscopically realized quantum states. Although well-understood on a quasi-classical level, its microscopic description is still far from being solved. The interplay of strong interactions present in magnetic condensed-matter systems and the non-trivial commutator structure governing the underlying spin algebra prevents most conventional approaches in solid-state theory to be applied.
On the other hand, the quantum limit of magnetic systems is fertile land for the development of exotic phases of matter called spin-liquids. In these states, quantum fluctuations inhibit the formation of magnetic long-range order down to the lowest temperatures. From a theoretical point of view, spin-liquids open up the possibility to study their exotic properties, such as fractionalized excitations and emergent gauge fields. However, despite huge theoretical and experimental efforts, no material realizing spin-liquid properties has been unambiguously identified with a three-dimensional crystal structure. The search for such a realization is hindered by the inherent difficulty even for model calculations. As most numerical techniques are not applicable due to the interaction structure and dimensionality of these systems, a methodological gap has to be filled.
In this thesis, to fill this void, we employ the pseudo-fermion functional renormalization group (PFFRG), which provides a scheme to investigate ground state properties of quantum magnetic systems even in three spatial dimensions.
We report the status quo of this established method and extend it by alleviating some of its inherent approximations. To this end, we develop a multi-loop formulation of PFFRG, including hitherto neglected terms in the underlying flow equations consistently, rendering the outcome equivalent to a parquet approximation. As a necessary prerequisite, we also significantly improve the numerical accuracy of our implementation of the method by switching to a formulation respecting the asymptotic behavior of the vertex functions as well as employing state-of-the-art numerical algorithms tailored towards PFFRG. The resulting codebase was made publicly accessible in the open-source code PFFRGSolver.jl.
We subsequently apply the technique to both model systems and real materials. Augmented by a classical analysis of the respective models, we scan the phase diagram of the three-dimensional body-centered cubic lattice up to third-nearest neighbor coupling and the Pyrochlore lattice up to second-nearest neighbor. In both systems, we uncover in addition to the classically ordered phases, an extended parameter regime, where a quantum paramagnetic phase appears, giving rise to the possibility of a quantum spin liquid.
Additionally, we also use the nearest-neighbor antiferromagnet on the Pyrochlore lattice as well as the simple cubic lattice with first- and third-nearest neighbor couplings as a testbed for multi-loop PFFRG, demonstrating, that the inclusion of higher loop orders has quantitative effects in paramagnetic regimes and that the onset of order can be signaled by a lack of loop convergence.
Turning towards material realizations, we investigate the diamond lattice compound MnSc\(_2\)S\(_4\), explaining on grounds of ab initio couplings the emergence of a spiral spin liquid at low temperatures, but above the ordering transition.
In the Pyrochlore compound Lu\(_2\)Mo\(_2\)O\(_5\)N\(_2\), which is known to not magnetically order down to lowest temperatures, we predict a spin liquid state displaying a characteristic gearwheel pattern in the spin structure factor.