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In this thesis we discussed the expansion behaviour of an ultracold bosonic gas from an initial harmonic confinement. We studied the reaction of the non-interacting system to changes of the trap frequency ω and of the strongly interacting system to changes of the number of Mott insulating particles NMI in the initial state and the interaction U/J. The total number of particles is kept constant for the different simulations, which are performed by means of the Bosonic Gutzwiller approach...
Folgend auf den ersten Realisierungen von Bose-Einstein Kondensaten erschienen weitere innovative Experimente, die sich in den optischen Gittern gefangenen Quantengasen widmeten. In diesen zahlreichen, wissenschaftlichen Untersuchungen konnten die Eigenschaften von Bose-Einstein Kondensaten besser verstanden werden. Das Prinzip von Vielteilchensystemen, gefangen in einem periodischen Potential, bot eine Plattform zur Untersuchung weiterer Quantenphasen.
Eine konzeptionell einfache Modifikation von solchen Systemen erhält man durch die Kopplung der Grundzustände der gefangenen Teilchen an hoch angeregten Zuständen mithilfe einer externen Lichtquelle. Im Falle dessen, dass diese Zustände nahe der Ionisationsgrenze des Atoms liegen, spricht man von Rydberg-Zuständen und Atome, welche zu diesen Zuständen angeregt werden, bezeichnet man als Rydberg-Atome. Eines der vielen charakteristischen Eigenschaften von Rydberg-Atomen ist die Fähigkeit über große Entfernungen jenseits der atomaren Längenskalen zu wechselwirken. Im Rahmen von Vielteilchensystemen wurden dementsprechend Kristallstrukturen aus gefangenen Rydberg-Atomen experimentell beobachtet.
Nun stellt sich die Frage, was mit einem gefangenen Bose-Einstein Kondensat passiert, dessen Teilchen an langreichweitig wechselwirkenden Zuständen gekoppelt sind. Gibt es ein Parameterregime, in dem sowohl Kristallstruktur als auch Suprafluidität in solchen Systemen koexistieren können? Dies ist die zentrale Frage dieser Arbeit, die sich mit der Theorie von gefangenen Quantengasen gekoppelt an Rydberg-Zuständen auseinandersetzt.
The study of systems whose properties are governed by electronic correlations is a corner stone of modern solid-state physics. Often, such systems feature unique and distinct properties like Mott metal-insulator transitions, rich phase diagrams, and high sensitivity to subtle changes in the applied conditions. Whereas the standard approach to electronic structure calculations, density functional theory (DFT), is able to address the complexity of real-world materials but is known to have serious limitations in the description of correlations, the dynamical mean-field theory (DMFT) has become an established method for the treatment of correlated fermions, first on the level of minimal models and later in combination with DFT, termed LDA+DMFT.
This thesis presents theoretical calculations on different materials exhibiting correlated physics, where we aim at covering a range in terms of systems --from rather weakly correlated to strongy correlated-- as well as in terms of methods, from DFT calculations to combined LDA+DMFT calculations. We begin with a study on a selection of iron pnictides, a recently discovered family of high-temperature superconductors with varying degree of correlation strength, and show that their magnetic and optical properties can be assessed to some degree within DFT, despite the correlated nature of these systems. Next, extending our analysis to the inclusion of correlations in the framework of LDA+DMFT, we discuss the electronic structure of the iron pnictide LiFeAs which we find to be well described by Fermi liquid theory with regard to many of its properties, yet we see distinct changes in its Fermi surface upon inclusion of correlations. We continue the study of low-energy properties and specifically Fermi surfaces on two more iron pnictides, LaFePO and LiFeP, and predict a topology change of their Fermi surfaces due to the effect of correlations, with possible implications for their superconducting properties. In our last study, we close the circle by presenting LDA+DMFT calculations on an organic molecular crystal on the verge of a Mott metal-insulator transition; there, we find the spectral and optical properties to display signatures of strong electronic correlations beyond Fermi liquid theory.
In this thesis we study strongly correlated electron systems within the Density Functional Theory (DFT) in combination with the Dynamical Mean-Field Theory (DMFT).
First, we give an introduction into the theoretical methods and then apply them to study realistic materials. We present results on the hole-doped 122-family of the iron-based superconductors and the transition-metal oxide SrVO3. Our investigations show that a proper treatment of strong electronic correlations is necessary to describe the experimental observations.
Die Arbeit beschäftigt sich mit der Herstellung sowie der strukturellen und magnetischen Charakterisierung von zwei Materialklassen von kupferbasierten zweidimensionalen Quanten-Spin-Systemen: Quadratische Gitter von Dimeren sowie geometrisch frustrierte Kagomé Gitter. In beiden Systemen werden Substitutionen vorgestellt die zu verbesserten Eigenschaften führen.
Topological phases set themselves apart from other phases since they cannot be understood in terms of the usual Landau theory of phase transitions. This fact, which is a consequence of the property that topological phase transitions can occur without breaking symmetries, is reflected in the complicated form of topological order parameters. While the mathematical classification of phases through homotopy theory is known, an intuition for the relation between phase transitions and changes to the physical system is largely inhibited by the general complexity.
In this thesis we aim to get back some of this intuition by studying the properties of the Chern number (a topological order parameter) in two scenarios. First, we investigate the effect of electronic correlations on topological phases in the Green's function formalism. By developing a statistical method that averages over all possible solutions of the manybody problem, we extract general statements about the shape of the phase diagram and investigate the stability of topological phases with respect to interactions. In addition, we find that in many topological models the local approximation, which is part of many standard methods for solving the manybody lattice model, is able to produce qualitatively correct phase transitions at low to intermediate correlations.
We then extend the statistical method to study the effect of the lattice, where we evaluate possible applications of standard machine learning techniques against our information theoretical approach. We define a measure for the information about particular topological phases encoded in individual lattice parameters, which allows us to construct a qualitative phase diagram that gives a more intuitive understanding of the topological phase.
Finally, we discuss possible applications of our method that could facilitate the discovery of new materials with topological properties.
Diese Thesis befasst sich mit dem Problem korrelierter Elektronensysteme in realen Materialien. Ausgangspunkt hierbei ist die quantenmechanische Beschreibung dieser Systeme im Rahmen der sogenannten Kohn-Scham Dichtefunktionaltheorie, welche die Elektronen der Kristallsysteme als effektiv nicht-wechselwirkende Teilchen beschreibt.
Während diese Modellierung im Falle vieler Materialklassen erfolgreich ist, unterscheiden sich die korrelierten Elektronensysteme dadurch, dass der kollektive Charakter der Elektronendynamik nicht zu vernachlässigen ist.
Um diese Korrelationseffekte genauer zu untersuchen, verwenden wir in dieser Arbeit das Hubbard-Modell, welches mit der projektiven Wannierfunktionsmethode aus der Kohn-Scham Dichtefunktionaltheorie konstruiert werden kann.
Das Hubbard-Modell umfasst hierbei nur die lokale Elektron-Elektron-Wechselwirkung auf einem Gitter. Auch wenn das Modell augenscheinlich sehr simpel ist, existieren exakte Lösungen nur in bestimmten Grenzfällen. Dies macht die Entwicklung approximativer Ansätze erforderlich, wobei die Weiterentwicklung der sogenannten Two-Particle Self-Consistent Methode (TPSC) eine zentrale Rolle dieser Arbeit einnimmt.
Bei TPSC handelt es sich um eine Vielteilchenmethode, die in der Sprache funktionaler Ableitungen und sogenannter conserving approximations hergeleitet werden kann.
Der zentrale Gedanke dabei ist, den effektiven Wechselwirkungsvertex als statisch und lokal zu approximieren. Dies wiederum erlaubt die Bewegungsgleichung des Systems
erheblich zu vereinfachen, sodass eine numerische approximative Lösung des Hubbard-Modells möglich wird. Vorsetzung hierbei ist nur, dass sich das System in der normalleitenden Phase befindet und die bei Phasenübergängen entstehenden Fluktuationen nicht zu groß sind.
Während diese Methode ursprünglich von Y. M. Vilk und A.-M. Tremblay für das Ein-Orbital Hubbard-Modell entwickelt wurde, stellen wir in dieser Arbeit eine Erweiterung auf Viel-Orbital-Systeme vor.
Im Falle mehrerer Orbitale treten in der TPSC-Herleitung einzelne Komplikationen auf, die mit weiteren Approximationen behandelt werden müssen. Diese werden anhand eines einfachen Zwei-Orbital Modell-Systems diskutiert und die TPSC-Ergebnisse werden darüber hinaus mit den Ergebnissen der etablierten dynamischen Molekularfeldnährung verglichen.
In diesem Zusammenhang werden auch mögliche zukünftige Erweiterungen bzw. Verbesserungen von TPSC diskutiert.
Ein weiterer wichtiger Aspekt ist die Anwendung von TPSC auf reale Materialien.
In diesem Zusammenhang werden in dieser Arbeit die supraleitenden Eigenschaften der organischen K-(ET)2X Systeme untersucht. Hierbei lassen die TPSC-Resultate darauf schließen, dass das populäre Dimer-Modell, welches zur Beschreibung dieser Materialien herangezogen wird, nicht genügt um die experimentell bestimmten kritischen Temperaturen zu erklären und dass das komplexere Molekülmodell weitere exotische supraleitende Lösungen zulässt.
Schließlich untersuchen wir außerdem die elektronischen Eigenschaften des eisenbasierten Supraleiters LiFeAs und diskutieren inwieweit nicht-lokale Korrelationseffekte, welche durch TPSC aufgelöst werden können, die experimentellen Daten reproduzieren.
The term superconductivity describes the phenomenon of vanishing electrical resistivity in a certain material, then called a superconductor, below a critical typically very low temperature. Since the discovery of superconductivity in mercury in 1911 many other superconductors have been found and the critical temperature below which superconductivity occurs could recently be raised to the temperatures encountered in a cold antarctic winter.
Superconductors are promising materials for applications. They can serve as nearly loss-free cables for energy transmission, in coils for the generation of high magnetic fields or in various electronic devices, such as detectors for magnetic fields. Despite their obvious advantages, the cost for using superconductors, however, depends a lot on the cooling effort needed to realize the superconducting state. Therefore, the search for a superconductor with critical temperature above room-temperature, which would avoid the need for any specialized cooling system, is one of the main projects of contemporary research in condensed matter physics.
While a theory of superconductivity in simple metals has already been developed in the 1950s, it has meanwhile been recognized that many superconductors are unconventional in the sense that their behavior does not follow the aforementioned theory. Unconventional superconductors differ from conventional superconductors mainly by the momentum- and real-space symmetry of the order parameter, which is associated with the superconducting state. While conventional superconductors have a uniform order parameter, unconventional superconductors can have an order parameter that bears structure. Of course, alternative theoretical descriptions have been suggested, but the discussion on the right theory for unconventional superconductivity has not yet been settled. Ultimately, this lack of a general theory of superconductivity prevents a targeted search for the room-temperature superconductor. Any new theoretical approach must, however, prove its value by correctly predicting the structure of the superconducting order parameter and further material properties.
In this work we participate in the search for a theory of unconventional superconductivity. We discuss the theory of superconductivity mediated by electron-electron interactions, which has been popular in the last few decades due to its success in explaining various properties of the copper-based superconductors that emerged in the 1980s. We give a detailed derivation of the so-called random phase approximation for the Hubbard model in terms of a diagrammatic many-body theory and apply it in conjunction with low-energy kinetic Hamiltonians, which we construct from first principles calculations in the framework of density functional theory. Density functional theory is an established technique for calculating the electronic and magnetic properties of materials solely based on their crystal structure. Its practical implementations in computer codes, however, do for example not describe complicated many-electron phenomena like the superconducting state that we are interested in here. Nevertheless, it can provide important information about the properties of the normal state of the material, which superconductivity emerges from. In our theory we use these information and approach the superconducting state from the normal state.
Such an interfacing of different calculational techniques requires a lot of implementation work in the form of computer code. Inclusion of the computer code into this work would consume by far too much space, but since some of the decisions on approximations in the calculational formalism are guided by the feasibility of the associated computer calculations, we discuss the numerical implementation in great detail.
We apply the developed methods to quasi-two-dimensional organic charge transfer salts and iron-based superconductors. Finally, we discuss implications of our findings for the interpretation of various experiments.
The ab-initio molecular dynamics framework has been the cornerstone of computational solid state physics in the last few decades. Although it is already a mature field it is still rapidly developing to accommodate the growth in solid state research as well as to efficiently utilize the increase in computing power. Starting from the first principles, the ab-initio molecular dynamics provides essential information about structural and electronic properties of matter under various external conditions. In this thesis we use the ab-initio molecular dynamics to study the behavior of BaFe2As2 and CaFe2As2 under the application of external pressure. BaFe2As2 and CaFe2As2 belong to the family of iron based superconductors which are a novel and promising superconducting materials. The application of pressure is one of two key methods by which electronic and structural properties of iron based superconductors can be modified, the other one being doping (or chemical pressure). In particular, it has been noted that pressure conditions have an important effect, but their exact role is not fully understood. To better understand the effect of different pressure conditions we have performed a series of ab-initio simulations of pressure application. In order to apply the pressure with arbitrary stress tensor we have developed a method based on the Fast Inertial Relaxation Engine, whereby the unit cell and the atomic positions are evolved according to the metadynamical equations of motion. We have found that the application of hydrostatic and c axis uniaxial pressure induces a phase transition from the magnetically ordered orthorhombic phase to the non-magnetic collapsed tetragonal phase in both BaFe2As2 and CaFe2As2. In the case of BaFe2As2, an intermediate tetragonal non-magnetic tetragonal phase is observed in addition. Application of the uniaxial pressure parallel to the c axis reduces the critical pressure of the phase transition by an order of magnitude, in agreement with the experimental findings. The in-plane pressure application did not result in transition to the non-magnetic tetragonal phase and instead, rotation of the magnetic order direction could be observed. This is discussed in the context of Ginzburg-Landau theory. We have also found that the magnetostructural phase transition is accompanied by a change in the Fermi surface topology, whereby the hole cylinders centered around the Gamma point disappear, restricting the possible Cooper pair scattering channels in the tetragonal phase. Our calculations also permit us to estimate the bulk moduli and the orthorhombic elastic constants of BaFe2As2 and CaFe2As2.
To study the electronic structure in systems with broken translational symmetry, such as doped iron based superconductors, it is necessary to develop a method to unfold the complicated bandstructures arising from the supercell calculations. In this thesis we present the unfolding method based on group theoretical techniques. We achieve the unfolding by employing induced irreducible representations of space groups. The unique feature of our method is that it treats the point group operations on an equal footing with the translations. This permits us to unfold the bandstructures beyond the limit of translation symmetry and also formulate the tight-binding models of reduced dimensionality if certain conditions are met. Inclusion of point group operations in the unfolding formalism allows us to reach important conclusions about the two versus one iron picture in iron based superconductors.
And finally, we present the results of ab-initio structure prediction in the cases of giant volume collapse in MnS2 and alkaline doped picene. In the case of MnS2, a previously unobserved high pressure arsenopyrite structure of MnS2 is predicted and stability regions for the two competing metastable phases under pressure are determined. In the case of alkaline doped picene, crystal structures with different levels of doping were predicted and used to study the role of electronic correlations.
This thesis contains three theoretical works about certain aspects of the interplay of electronic correlations and topology in the Hubbard model.
In the first part of this thesis, the applicability of elementary band representations (EBRs) to diagnose interacting topological phases, that are protected by spatial symmetries and time-reversal-symmetry, in terms of their single-particle Matsubara Green’s functions is investigated. EBRs for the Matsubara Green’s function in the zero-temperature limit can be defined via the topological Hamiltonian. It is found that the Green’s function EBR classification can only change by (i) a gap closing in the spectral function at zero frequency, (ii) the Green’s function becoming singular i.e. having a zero eigenvalue at zero frequency or (iii) the Green’s function breaking a protecting symmetry. As an example, the use of the EBRs for Matsubara Green’s functions is demonstrated on the Su-Schriefer-Heeger model with exact diagonalization.
In the second part the Two-Particle Self-Consistent approach (TPSC) is extended to include spin-orbit coupling (SOC). Time-reversal symmetry, that is preserved in the presence of SOC, is used to derive new TPSC self-consistency equations including SOC. SOC breaks spin rotation symmetry which leads to a coupling of spin and charge channel. The local and constant TPSC vertex then consists of three spin vertices and one charge vertex. As a test case to study the interplay of Hubbard interaction and SOC, the Kane-Mele-Hubbard model is studied. The antiferromagnetic spin fluctuations are the leading instability which confirms that the Kane-Mele-Hubbard model is an XY antiferromagnet at zero temperature. Mixed spin-charge fluctuations are found to be small. Moreover, it is found that the transversal spin vertices are more strongly renormalized than the longitudinal spin vertex, SOC leads to a decrease of antiferromagnetic spin fluctuations and the self-energy shows dispersion and sharp features in momentum space close to the phase transition.
In the third part TPSC with SOC is used to calculate the spin Hall conductivity in the Kane-Mele-Hubbard model at finite temperature. The spin Hall conductivity is calculated once using the conductivity bubble and once including vertex corrections. Vertex corrections for the spin Hall conductivity within TPSC corresponds to the analogues of the Maki-Thompson contributions which physically correspond to the excitation and reabsorption of a spin, a charge or a mixed spin-charge excitation by an electron. At all temperatures, the vertex corrections show a large contribution in the vicinity of the phase transition to the XY antiferromagnet where antiferromagnetic spin fluctuations are large. It is found that vertex corrections are crucial to recover the quantized value of −2e^2/h in the zero-temperature limit. Further, at non-zero temperature, increasing the Hubbard interaction leads to a decrease of the spin Hall conductivity. The results indicate that scattering of electrons off antiferromagnetic spin fluctuations renormalize the band gap. Decreasing the gap can be interpreted as an effective increase of temperature leading to a decrease of the spin Hall conductivity.
In this thesis we investigate the thermodynamic and dynamic properties of the D-dimensional quantum Heisenberg ferromagnet within the spin functional renormalization group (FRG); a
formalism describing the evolution of the system’s observables as the magnetic exchange inter-action is artificially deformed. Following an introduction providing a self contained summary of the conceptual and mathematical background, we present the spin FRG as developed by Krieg and Kopietz in references [1] and [2] in chapter two. Thereto, the generating functional of the imaginary time-spin correlation functions and its exact flow equation describing the deformation process of the exchange interaction are introduced. In addition, it is highlighted that - in contrast to conventional field-theoretic FRG approaches - the related Legendre trans-formed functional cannot be defined if the exchange interaction is initially switched off. Next, we show that this limitation can be circumvented within an alternativ hybrid approach, which treats transverse and longitudinal spin fluctuations differently. The relevant functionals are introduced and the relations of the corresponding functional Taylor coefficients with the spin correlation functions are discussed. Lastly, the associated flow equations are derived and the possibility of explicit or spontaneous symmetry breaking is taken into account.
In chapter three, we benchmark the hybrid formalism against a calculation of the thermo-dynamic properties of the one and two-dimensional Heisenberg model at low temperatures T and finite magnetic field H. For this purpose, we devise an anisotropic deformation scheme of the exchange interaction which allows for a controlled truncation of the infinite hierarchy of FRG flow equations. Thereby, contact with mean-field and spin-wave theory is made and the violation of the Mermin-Wagner theorem is discussed. To fulfill the latter, the truncation scheme is then complemented by a Ward identity relating the transverse self-energy and the magnetization. The resulting magnetization M (H, T ) and isothermal susceptibility χ(H, T ) are in quantitative agreement with the literature and the established behavior of the transverse correlation length and the zero-field susceptibility close to the critical point is qualitatively reproduced in the limit H → 0.
Finally, we investigate the longitudinal dynamics at low temperatures. To this end, the hierarchy of flow equations is solved within the same anisotropic deformation scheme complemented by an expansion in the inverse interaction range, and the resulting longitudinal dynamic structure factor is calculated within a low-momentum expansion. In D = 3, the large phase space accessible for the decay into transverse magnons yields only a broad hump centered at zero frequency whose width scales linearly in momentum. In contrast, at low temperatures and in a certain range of magnetic fields, a well-defined quasiparticle peak with linear dispersion emerges in D ≤ 2, which we identify as zero-magnon sound. Sound velocity and damping are discussed as a function of temperature and magnetic field, and the relevant momentum-frequency window is estimated and compared to the hydrodynamic
second-magnon regime.
This thesis has two main parts.
The first part is based on our publication [1], where we use perturbation theory to calculate decay rates of magnons in the Kitaev-Heisenberg-Γ (KHΓ) model. This model describes the magnetic properties of the material α-RuCl 3 , which is a candidate for a Kitaev spin liquid. Our motivation is to validate a previous calculation from Ref. [2]. In this thesis, we map out the classical phase diagram of the KHΓ model. We use the Holstein-Primakoff
transformation and the 1/S expansion to describe the low temperature dynamics of the Kitaev-Heisenberg-Γ model in the experimentally relevant zigzag phase by spin waves. By parametrizing the spin waves in terms of hermitian fields, we find a special parameter region within the KHΓ model where the analytical expressions simplify. This enables us to construct the Bogoliubov transformation analytically. For a representative point in the special parameter region, we use these results to numerically calculate the magnon damping, which is to leading order caused by the decay of single magnons into two. We also calculate the dynamical structure factor of the magnons.
The second part of this thesis is based on our publication [3], where we use the functional renormalization group to analyze a discontinuous quantum phase transition towards a non-Fermi liquid phase in the Sachdev-Ye-Kitaev (SYK) model. In this thesis, we perform a disorder average over the random interactions in the SYK model. We argue that in the thermodynamic limit, the average renormalization group (RG) flow of the SYK model is identical to the RG flow of an effective disorder averaged model. Using the functional RG, we find a fixed point describing the discontinuous phase transition to the non-Fermi liquid phase at zero temperature. Surprisingly, we find a finite anomalous dimension of the fermions, which indicates critical fluctuations and is unusual for a discontinuous transition. We also determine the RG flow at zero temperature, and relate it to the phase diagram known from the literature.
The present thesis is primarily concerned with the application of the functional renormalization group (FRG) to spin systems. In the first part, we study the critical regime close to the Berezinskii-Kosterlitz-Thouless (BKT) transition in several systems. Our starting point is the dual-vortex representation of the two-dimensional XY model, which is obtained by applying a dual transformation to the Villain model. In order to deal with the integer-valued field corresponding to the dual vortices, we apply the lattice FRG formalism developed by Machado and Dupuis [Phys. Rev. E 82, 041128 (2010)]. Using a Litim regulator in momentum space with the initial condition of isolated lattice sites, we then recover the Kosterlitz-Thouless renormalization group equations for the rescaled vortex fugacity and the dimensionless temperature. In addition to our previously published approach based on the vertex expansion [Phys. Rev. E 96, 042107 (2017)], we also present an alternative derivation within the derivative expansion. We then generalize our approach to the O(2) model and to the strongly anisotropic XXZ model, which enables us to show that weak amplitude fluctuations as well as weak out-of-plane fluctuations do not change the universal properties of the BKT transition.
In the second part of this thesis, we develop a new FRG approach to quantum spin systems. In contrast to previous works, our spin functional renormalization group (SFRG) does not rely on a mapping to bosonic or fermionic fields, but instead deals directly with the spin operators. Most importantly, we show that the generating functional of the irreducible vertices obeys an exact renormalization group equation, which resembles the Wetterich equation of a bosonic system. As a consequence, the non-trivial structure of the su(2) algebra is fully taken into account by the initial condition of the renormalization group flow. Our method is motivated by the spin-diagrammatic approach to quantum spin system that was developed more than half a century ago in a seminal work by Vaks, Larkin, and Pikin (VLP) [Sov. Phys. JETP 26, 188 (1968)]. By embedding their ideas in the language of the modern renormalization group, we avoid the complicated diagrammatic rules while at the same time allowing for novel approximation schemes. As a demonstration, we explicitly show how VLP's results for the leading corrections to the free energy and to the longitudinal polarization function of a ferromagnetic Heisenberg model can be recovered within the SFRG. Furthermore, we apply our method to the spin-S Ising model as well as to the spin-S quantum Heisenberg model, which allows us to calculate the critical temperature for both a ferromagnetic and an antiferromagnetic exchange interaction. Finally, we present a new hybrid formulation of the SFRG, which combines features of both the pure and the Hubbard-Stratonovich SFRG that were published recently [Phys. Rev. B 99, 060403(R) (2019)].
The objective of this work is twofold. First, we explore the performance of the density functional theory (DFT) when it is applied to solids with strong electronic correlations, such as transition metal compounds. Along this direction, particular effort is put into the refinement and development of parameterization techniques for deriving effective models on a basis of DFT calculations. Second, within the framework of the DFT, we address a number of questions related to the physics of Mott insulators, such as magnetic frustration and electron-phonon coupling (Cs2CuCl4 and Cs2CuBr4), high-temperature superconductivity (BSCCO) and doping of Mott insulators (TiOCl). In the frustrated antiferromagnets Cs2CuCl4 and Cs2CuBr4, we investigate the interplay between strong electronic correlations and magnetism on one hand and electron-lattice coupling on the other as well as the effect of this interplay on the microscopic model parameters. Another object of our investigations is the oxygen-doped cuprate superconductor BSCCO, where nano-scale electronic inhomogeneities have been observed in scanning tunneling spectroscopy experiments. By means of DFT and many-body calculations, we analyze the connection between the structural and electronic inhomogeneities and the superconducting properties of BSCCO. We use the DFT and molecular dynamic simulations to explain the microscopic origin of the persisting under doping Mott insulating state in the layered compound TiOCl.
The challenging intricacies of strongly correlated electronic systems necessitate the use of a variety of complementary theoretical approaches. In this thesis, we analyze two distinct aspects of strong correlations and develop further or adapt suitable techniques. First, we discuss magnetization transport in insulating one-dimensional spin rings described by a Heisenberg model in an inhomogeneous magnetic field. Due to quantum mechanical interference of magnon wave functions, persistent magnetization currents are shown to exist in such a geometry in analogy to persistent charge currents in mesoscopic normal metal rings. The second, longer part is dedicated to a new aspect of the functional renormalization group technique for fermions. By decoupling the interaction via a Hubbard-Stratonovich transformation, we introduce collective bosonic variables from the beginning and analyze the hierarchy of flow equations for the coupled field theory. The possibility of a cutoff in the momentum transfer of the interaction leads to a new flow scheme, which we will refer to as the interaction cutoff scheme. Within this approach, Ward identities for forward scattering problems are conserved at every instant of the flow leading to an exact solution of a whole hierarchy of flow equations. This way the known exact result for the single-particle Green's function of the Tomonaga-Luttinger model is recovered.
In this thesis, we presented the theoretical description of the magnetic properties of various frustrated spin systems. Especially in search of exotic states, such as quantum spin liquids, magnetically frustrated systems have been subject of intense research within the last four decades. Relating experimental observations in real materials with theoretical models that capture those exotic magnetic phenomena has been one of the great challenges within the field of magnetism in condensed matter.
In order to build such a bridge between experimental observations and theoretical models, we followed two complementary strategies in this thesis. One strategy was based on first principles methods that enable the theoretical prediction of electronic properties of real materials without further experimental input than the crystal structure. Based on these predictions, low-energy models that describe magnetic interactions can be extracted and, through further theoretical modelling, can be compared to experimental observations. The second strategy was to establish low-energy models through comparison of data from experiments, such as inelastic neutron scattering intensities, with calculated predictions based on a variety of plausible magnetic models guided by microscopic insights. Both approaches allow to relate theoretical magnetic models with real materials and may provide guidance for the design of new frustrated materials or the investigation of promising models related to exotic magnetic states.
Great interest has emerged recently in the search for Kitaev spin liquid states in real materials. Such states rely on strongly anisotropic magnetic interactions, which have been suggested to exist in a number of candidate materials based on Ir and Ru. This thesis concentrates on two priority purposes. The first is the investigation of electronic and magnetic properties of candidate materials Na2IrO3, α-Li2IrO3, α-RuCl3, γ-Li2IrO3, and Ba3YIr2O9 for Kitaev physics where both spin-orbit coupling and correlation effects are important. The second is the method development for the microscopic description of correlated materials combining many-body methods and density functional theory (DFT). ...
The phenomenon of magnetism has been known to humankind for at least over 2500 years and many useful applications of magnetism have been developed since then, starting from the compass to modern information storage and processing devices. While technological applications are an important part of the continuing interest in magnetic materials, their fundamental properties are still being studied, leading to new physical insights at the forefront of physics. The magnetism of magnetic materials is a pure quantum effect due to the electrons that carry an intrinsic spin of 1/2. The physics of interacting quantum spins in magnetic insulators is the main subject of this thesis.We focus here on a theoretical description of the antiferromagnetic insulator Cs2CuCl4. This material is highly interesting because it is a nearly ideal realization of the two-dimensional antiferromagnetic spin-1/2 Heisenberg model on an anisotropic triangular lattice, where the Cu(2+) ions carry a spin of 1/2 and the spins interact via exchange couplings. Due to the geometric frustration of the triangular lattice, there exists a spin-liquid phase with fractional excitations (spinons) at finite temperatures in Cs2CuCl4. This spin-liquid phase is characterized by strong short-range spin correlations without long-range order. From an experimental point of view, Cs2CuCl4 is also very interesting because the exchange couplings are relatively weak leading to a saturation field of only B_c=8.5 T. All relevant parts of the phase diagram are therefore experimentally accessible. A recurring theme in this thesis will be the use of bosonic or fermionic representations of the spin operators which each offer in different situations suitable starting points for an approximate treatment of the spin interactions. The methods which we develop in this thesis are not restricted to Cs2CuCl4 but can also be applied to other materials that can be described by the spin-1/2 Heisenberg model on a triangular lattice; one important example is the material class Cs2Cu(Cl{4-x}Br{x}) where chlorine is partially substituted by bromine which changes the strength of the exchange couplings and the degree of frustration.
Our first topic is the finite-temperature spin-liquid phase in Cs2CuCl4. We study this regime by using a Majorana fermion representation of the spin-1/2 operators motivated by theoretical and experimental evidence for fermionic excitations in this spin-liquid phase. Within a mean-field theory for the Majorana fermions, we determine the magnetic field dependence of the critical temperature for the crossover from spin-liquid to paramagnetic behavior and we calculate the specific heat and magnetic susceptibility in zero magnetic field. We find that the Majorana fermions can only propagate in one dimension along the direction of the strongest exchange coupling; this reduction of the effective dimensionality of excitations is known as dimensional reduction.
The second topic is the behavior of ultrasound propagation and attenuation in the spin-liquid phase of Cs2CuCl4, where we consider longitudinal sound waves along the direction of the strongest exchange coupling. Due to the dimensional reduction of the excitations in the spin-liquid phase, we expect that we can describe the ultrasound physics by a one-dimensional Heisenberg model coupled to the lattice degrees of freedom via the exchange-striction mechanism. For this one-dimensional problem we use the Jordan-Wigner transformation to map the spin-1/2 operators to spinless fermions. We treat the fermions within the self-consistent Hartree-Fock approximation and we calculate the change of the sound velocity and attenuation as a function of magnetic field using a perturbative expansion in the spin-phonon couplings. We compare our theoretical results with experimental data from ultrasound experiments, where we find good agreement between theory and experiment.
Our final topic is the behavior of Cs2CuCl4 in high magnetic fields larger than the saturation field B_c=8.5 T. At zero temperature, Cs2CuCl4 is then fully magnetized and the ground state is therefore a ferromagnet where the excitations have an energy gap. The elementary excitations of this ferromagnetic state are spin-flips (magnons) which behave as hard-core bosons. At finite temperatures there will be thermally excited magnons that interact via the hard-core interaction and via additional exchange interactions. We describe the thermodynamic properties of Cs2CuCl4 at finite temperatures and calculate experimentally observable quantities, e.g., magnetic susceptibility and specific heat. Our approach is based on a mapping of the spin-1/2 operators to hard-core bosons, where we treat the hard-core interaction by the self-consistent ladder approximation and the exchange interactions by the self-consistent Hartree-Fock approximation. We find that our theoretical results for the specific heat are in good agreement with the available experimental data.
In this thesis, various aspects on the theoretical description of ultracold bosonic atoms in optical lattices are investigated. After giving a brief introduction to the fundamental concepts of BECs, atomic physics, interatomic interactions and experimental procedures in chapter (1), we derive the Bose-Hubbard model from first principles in chapter (2). In this chapter, we also introduce and discuss a technique to efficiently determine Wannier states, which, in contrast to current techniques, can also be extended to inhomogeneous systems. This technique is later extended to higher dimensional, non-separable lattices in chapter (5). The many-body physics and phases of the Bose-Hubbard is shortly presented in chapter (3) in conjunction with Gutzwiller mean-field theory, and the recently devised projection operator approach. We then return to the derivation of an improved microscopic many-body Hamiltonian, which contains higher band contributions in the presence of interactions in chapter (4). We then move on to many-particle theory. To demonstrate the conceptual relations required in the following chapter, we derive Bogoliubov theory in chapter (5.3.4) in three different ways and discuss the connections. Furthermore, this derivation goes beyond the usual version discussed in most textbooks and papers, as it accounts for the fact, that the quasi-particle Hamiltonian is not diagonalizable in the condensate and the eigenvectors have to be completed by additional vectors to form a basis. This leads to a qualitatively different quasi-particle Hamiltonian and more intricate transformation relations as a result. In the following two chapters (7, 8), we derive an extended quasi-particle theory, which goes beyond Bogoliubov theory and is not restricted to weak interactions or a large condensate fraction. This quasi-particle theory naturally contains additional modes, such as the amplitude mode in the strongly interacting condensate. Bragg spectroscopy, a momentum-resolved spectroscopic technique, is introduced and used for the first experimental detection of the amplitude mode at finite quasi-momentum in chapter (9). The closely related lattice modulation spectroscopy is discussed in chapter (10). The results of a time-dependent simulation agree with experimental data, suggesting that also the amplitude mode, and not the sound mode, was probed in these experiments. In chapter (11) the dynamics of strongly interacting bosons far from equilibrium in inhomogeneous potentials is explored. We introduce a procedure that, in conjunction with the collapse and revival of the condensate, can be used to create exotic condensates, while particularly focusing on the case of a quadratic trapping potential. Finally, in chapter (12), we turn towards the physics of disordered systems derive and discuss in detail the stochastic mean-field theory for the disordered Bose-Hubbard model.