Complexes containing a transition metal atom with a 3d^4 - 3d^7 electron configuration typically have two low-lying, high spin (HS) and low spin (LS) states. The adiabatic energy difference between these states, known as the spin-crossover energy, is small enough to pose a challenge even for electronic structure methods that are well known for their accuracy and reliability. In this work we analyze the quality of electronic structure approximations for spin-crossover energies of iron complexes with four different ligands by comparing energies from self-consistent and post-self-consistent calculations for methods based on the random phase approximation and the Fermi-L\"{o}wdin self-interaction correction. Considering that Hartree-Fock densities were found by Song et al. J. Chem. Theory Comput. 14,2304 (2018) to eliminate the density error to a large extent, and that the Hartree-Fock method and the Perdew-Zunger-type self-interaction correction share some physics, we compare the densities obtained with these methods to learn about their resemblance. We find that evaluating non-empirical exchange-correlation energy functionals on the corresponding self-interaction-corrected densities can mitigate the strong density errors and improves the accuracy of the adiabatic energy differences between HS and LS states. / Physics

Die Dichtefunktionaltheorie (DFT) einschließlich Dispersionkorrekturen (+D) wird mit der Random-Phase-Approximation (RPA) für die Adsorption von Alkanen auf der Pt(111)-Oberfläche verglichen. RPA wird zuerst im Hinblick auf relevante technische Parameter evaluiert und für die Methanadsorption an der Pt(111)-Oberfläche getestet. Im Vergleich zum Perdew-Burke-Ernzerhof-Funktional (PBE) mit Tkatchenkos Many-Body-Dispersionskorrektur (PBE+MBD) liefert RPA gute Ergebnisse. Auch reproduziert RPA experimentelle Adsorptionsenergien bei verschiedenen, physikalisch sinnvollen Beladungsstufen der Pt(111) Oberfläche mit Alkanmolekülen. Für Platin in der hexagonal dichtesten Kugelpackung sagt RPA richtigerweise die Methanadsorption an der hollow-tripod-Stelle voraus, während mit PBE+MBD die Adsorption an einer anderen Stelle bevorzugt wäre. Dies geht aus Schwingungsspektren hervor. Da periodisches RPA sehr rechenaufwändig ist, wird ein QM:QM Hybridansatz (QM=Quantenmechanik) angewendet, wobei periodisches PBE(+D) mithilfe von RPA Rechnungen an Clustern korrigiert wird (RPA:PBE(+D)). In einem Test verschiedener Dispersionskorrekturen schneiden RPA:PBE und RPA:PBE+MBD am besten ab. Diese Arbeit ist wegbereitend für die Anwendung des QM:QM Hybridansatzes zur Beschreibung der Adsorptionsprozesse an Metalloberflächen ‒ bei hoher Genauigkeit und deutlich verringertem Rechenaufwand. Auch Kresses low-scaling RPA Algorithmus wird getestet. Dieser Algorithmus ermöglicht, große Systeme, wie z.B. die Methan-, Ethan-, Propan- und n-Butanadsorption an Pt(111), zu untersuchen. Der Vergleich mit experimentellen Daten zeigt, dass mit RPA stets die beste Übereinstimmung erreicht wird. Dabei wird eine deutliche Verbesserung gegenüber allen untersuchten Dichte-Funktionalen erzielt. Obwohl Bindungen mit RPA etwas zu schwach vorhergesagt werden, ist es die derzeit beste Methode zur Untersuchung der Adsorption an Metalloberflächen und damit der Benchmark für diese Systeme. / Density Functional Theory (DFT) including dispersion (+D) is compared against the Random Phase Approximation (RPA) for the adsorption of alkanes on the Pt(111) surface. RPA is first benchmarked with respect to technical parameters and tested for methane adsorption on Pt(111). It is found to perform well relative to the Perdew–Burke–Ernzerhof (PBE) functional augmented with the many-body dispersion scheme of Tkatchenko (PBE+MBD). It also compares well relative to experimentally derived adsorption energies at physically relevant coverages. RPA correctly assigns the adsorption of methane to the hcp (hexagonal close packed) hollow tripod site, matching vibrational spectra, whereas PBE+MBD found another site. Given the high cost of periodic RPA, a high-level: low-level QM:QM (QM = quantum mechanics) hybrid approach is applied using RPA (RPA:PBE(+D)), which has also been tested with several dispersion corrections, with RPA:PBE and RPA:PBE+MBD performing best. This extends the QM:QM hybrid approach to the study of adsorption on metal surfaces, resulting in high accuracy at significantly reduced cost. Finally we test the performance of the low-scaling RPA algorithm of Kresse and co-workers. This algorithm enables the study of larger systems and is applied to the first four n-alkanes (C1-C4) on the Pt(111) surface. Comparison against experiment indicates that RPA offers the best agreement, consistently better than any studied density functional. RPA underbinds slightly but is still found to be the best method for studying adsorption on metal surfaces and is the current benchmark for such systems.

Various experiments on Yb₂Ti₂O₇ have shown evidence of strange magnetic behaviour at low temperatures. Specific heat measurements on powder samples of Yb₂Ti₂O₇ show evidence of a sharp peak, indicating the occurence of a first order phase transition. Meanwhile, neutron scattering, Mössbauer absorption, and μSR measurements find no evidence of long range order below the temperature of this phase transition, leaving the nature of the low temperature phase a mystery. Quantifying the magnetic interactions in this material should allow us to better understand the low temperature behaviour of this material. In this study, we fit a symmetry allowed nearest-neighbour bilinear exchange model to quasi-elastic neutron scattering data collected well above the temperature of the experimentally observed phase transition. This neutron scattering data shows evidence of rods of scattering intensity along the ⟨111⟩ crystallographic directions. Neutron scattering probes the correlations between magnetic moments in a material, so fitting an interaction model to the neutron scattering is equivalent to fitting the interactions to the magnetic correlations. These correlations are driven by the interactions between the magnetic moments, so the neutron scattering should give us direct access to the form of these interactions. Using this method we successfully identify an anisotropic nearest-neighbour bilinear exchange model that reproduces the experimentally observed quasi-elastic neutron scattering. With this model we then proceed to compute real space correlation functions, finding that the rods of neutron scattering arise from the presence of strong correlations along nearest-neighbour chains. We also compute the bulk susceptibility and local susceptibility, obtaining very good fits to experiment with no variation of the model determined from the neutron scattering. The success of these calculations provides a further independent confirmation of the success of our interaction model in describing the magnetic interactions in Yb₂Ti₂O₇. Finally, we present a brief summary of ongoing work based on our anisotropic exchange model, including mean field calculations to determine the low temperature ground state of this model and classical Monte Carlo simulations to study the phase transition present in this model. We also discuss potential further studies of this and other models.

Various experiments on Yb₂Ti₂O₇ have shown evidence of strange magnetic behaviour at low temperatures. Specific heat measurements on powder samples of Yb₂Ti₂O₇ show evidence of a sharp peak, indicating the occurence of a first order phase transition. Meanwhile, neutron scattering, Mössbauer absorption, and μSR measurements find no evidence of long range order below the temperature of this phase transition, leaving the nature of the low temperature phase a mystery. Quantifying the magnetic interactions in this material should allow us to better understand the low temperature behaviour of this material. In this study, we fit a symmetry allowed nearest-neighbour bilinear exchange model to quasi-elastic neutron scattering data collected well above the temperature of the experimentally observed phase transition. This neutron scattering data shows evidence of rods of scattering intensity along the ⟨111⟩ crystallographic directions. Neutron scattering probes the correlations between magnetic moments in a material, so fitting an interaction model to the neutron scattering is equivalent to fitting the interactions to the magnetic correlations. These correlations are driven by the interactions between the magnetic moments, so the neutron scattering should give us direct access to the form of these interactions. Using this method we successfully identify an anisotropic nearest-neighbour bilinear exchange model that reproduces the experimentally observed quasi-elastic neutron scattering. With this model we then proceed to compute real space correlation functions, finding that the rods of neutron scattering arise from the presence of strong correlations along nearest-neighbour chains. We also compute the bulk susceptibility and local susceptibility, obtaining very good fits to experiment with no variation of the model determined from the neutron scattering. The success of these calculations provides a further independent confirmation of the success of our interaction model in describing the magnetic interactions in Yb₂Ti₂O₇. Finally, we present a brief summary of ongoing work based on our anisotropic exchange model, including mean field calculations to determine the low temperature ground state of this model and classical Monte Carlo simulations to study the phase transition present in this model. We also discuss potential further studies of this and other models.

>Magister Scientiae - MSc / Observation of neutrinoless double beta decay (0 ) is currently the only means by which one could establish the Majorana nature of neutrinos. Additionally, such an observation would determine the absolute neutrino mass scale. However, this requires that the matrix element for a given 0 decay process is accurately calculated. The objective of this project is to provide useful nuclear structure information that aim to improve future theoretical calculations for the nuclear matrix element (NME) of 136Xe 0 decay to 136Ba. We studied neutron pairing correlations in 134Ba using the 136Ba(p; t) reaction to stringently test the Bardeen-Cooper-Schrie er (BCS) approximation in the A = 136 mass region. This is because many theoretical calculations of the NME's for 0 decay are performed using the quasiparticle random phase approximation (QRPA), which uses the BCS approximation to describe the ground states of the even-even parent and daughter nuclei. Our results show a signi cant fragmentation of the neutron-pair transfer to excited 0+ states, implying a breakdown of the BCS approximation in this mass region.

In this thesis, we study the unconventional superconductivity in Sr2RuO4 (SRO) and related systems. The superconducting state in SRO remains a puzzle after more than 28 years of study. Early experiments had pointed toward a topological non-trivial time-reversal symmetry breaking (TRSB) chiral p-wave order. This pairing candidate has attracted a large amount of attention, partly in relation to the possibility of topological quantum computation, and has stimulated studies on higher chirality superconducting systems. In the first part of this thesis, we study the spontaneous edge current in chiral d- and f-wave superconductors. We show that these currents, which vanish in the continuum limit at zero temperature, are generally non-vanishing but tiny, compared to the simplest chiral p-wave case. In the presence of strong surface roughness, the direction of the edge current in the chiral d-wave case can be reversed, compared with that of a specular ideal surface with specular scattering. However, it is shown that this current reversal is non-universal beyond the continuum limit. The chiral p-wave scenario in SRO is overturned by recent Knight shift measurements, highlighting the importance of exploring different pairing symmetries for SRO. Recently, $d_{x^2-y^2} \pm ig_{(x^2-y^2)xy}$, $s' \pm id_{xy}$ and mixed helical p-wave pairings have been proposed as order parameter candidates. However, the stability of these states, especially of the $d_{x^2-y^2} \pm ig$ pairing, remains unclear. In the second part of the thesis, we study the leading superconducting instabilities in SRO in the presence of sizable atomic spin-orbit coupling (SOC), non-local SOC, and non-local interactions. We find that it is difficult to stabilize chiral p-wave pairing in SRO models; this is because, among the triplet p-wave states, the atomic SOC favors helical states over the chiral state. The presence of both d- and g-wave pairings, including a $d_{x^2-y^2} \pm ig$ state, is found when the second nearest neighbor (in-plane) repulsions, together with orbital-anisotropy of the non-local interactions and/or the B2g channel non-local SOC are included. We further analyze the properties, such as nodal structures, in-plane field spin-susceptibility, and spontaneous edge current, of the realized $d_{x^2-y^2} \pm ig$ pairing and find that this state is more compatible with existing experimental measurements than the $s' \pm id_{xy}$ and the mixed helical p-wave proposals. / Dissertation / Doctor of Philosophy (PhD)

Relativistic effects crucially influence the fundamental properties of many quantum materials. In the accelerated reference frame of an electron, the electric field of the nuclei is transformed into a magnetic field that couples to the electron spin. The resulting interaction between an electron spin and its orbital angular momentum, known as spin-orbit coupling (SOC), is hence fundamental to the physics of many condensed matter phenomena. It is particularly important quantitatively in low-dimensional quantum systems, where its coexistence with inversion symmetry breaking can lead to a splitting of spin degeneracy and spin momentum locking. Using the paradigm of Landau Fermi liquid theory, the physics of SOC can be adequately incorporated in an effective single particle picture. In a weak coupling approach, electronic correlation effects beyond single particle propagator renormalization can trigger Fermi surface instabilities such as itinerant magnetism, electron nematic phases, superconductivity, or other symmetry broken states of matter. In this thesis, we use a weak coupling-based approach to study the effect of SOC on Fermi surface instabilities and, in particular, superconductivity. This encompasses a weak coupling renormalization group formulation of unconventional superconductivity as well as the random phase approximation. We propose a unified formulation for both of these two-particle Green’s function approaches based on the notion of a generalized susceptibility. In the half-Heusler semimetal and superconductor LuPtBi, both SOC and electronic correlation effects are prominent, and thus indispensable for any concise theoretical description. The metallic and weakly dispersive surface states of this material feature spin momentum locked Fermi surfaces, which we propose as a possible domain for the onset of unconventional surface superconductivity. Using our framework for the analysis of Fermi surface instability and combining it with ab-initio density functional theory calculations, we analyse the surface band structure of LuPtBi, and particularly its propensity towards Cooper pair formation. We study how the presence of strong SOC modifies the classification of two-electron wave functions as well as the screening of electron-electron interactions. Assuming an electronic mechanism, we identify a chiral superconducting condensate featuring Majorana edge modes to be energetically favoured over a wide range of model parameters. / Relativistische Effekte bestimmen die Eigenschaften vieler Quantenmaterialien entscheidend. Im beschleunigten Bezugssystem eines Elektrons transformiert sich das elektrische Feld des Kerns in ein Magnetfeld, welches an den Spin des Elektrons koppelt. Die resultierende Wechselwirkung zwischen dem Spin eines Elektrons und seinem Bahndrehimpuls, bekannt als Spin-Bahn-Kopplung (engl. spin-orbit coupling SOC), ist für viele Phänomene der kondensierten Materie von grundlegender Bedeutung. In niedrigdimensionalen Quantensystemen, wo die Koexistenz von SOC und Inversionssymmetriebrechung zu einer Aufspaltung der Spinentartung und Kopplung des Spins an den Impulsfreiheitsgrad führen kann, besonders wichtig. Mit dem Paradigma von Landaus Fermi-Flüssigkeits-Theorie lässt sich die Physik des SOC in einem effektiven Ein-Teilchenbild gut modellieren. Ausgehend von einem schwach gekoppelten Bild können elektronische Korrelationseffekte, die über diese einfache Theorie hinausgehen, eine Instabilität der Fermi-Fläche auslösen, die zu Magnetismus, elektronisch-nematischen Phasen, Supraleitung oder anderen symmetriegebrochenen Materialzuständen führt. In dieser Dissertation verwenden wir einen auf schwacher Kopplung basierenden Ansatz, um die Wirkung von SOC auf Instabilitäten der Fermi-Fläche und insbesondere auf Supraleitung zu untersuchen. Wir betrachten eine störungstheoretische Renormierungsgruppenformulierung für unkonventionellen Supraleitung die Random-Phase-Approximation (RPA). Auf Grundlage der verallgemeinerten Suszeptibilität entwickeln wir eine einheitliche Formulierung für diese beiden Ansätze. Im Halb-Heusler-Halbmetall und Supraleiter LuPtBi sind sowohl SOC- als auch elektronische Korrelationseffekte für jede theoretische Beschreibung von großer Bedeutung. Der metallische und schwach dispersive Oberflächenzustand dieses Materials weist Fermi-Flächen mit gekoppeltem Spin und Impuls auf, die wir als mögliche Domäne für den Beginn unkonventioneller Oberflächensupraleitung vorschlagen. Wir kombinieren ab-initio Dichtefunktionaltheorieberechnungen für die Oberflächenbandstruktur von LuPtBi mit der Renormierungsgruppe und der RPA für eine Analyse der Fermi-Oberflächeninstabilitäten and der Kristalloberfläche. Wir untersuchen, wie die Existenz von starkem SOC die Klassifizierung von Zwei-Elektronen-Wellenfunktionen sowie die Abschirmung von Elektron-Elektronen-Wechselwirkungen modifiziert. Unter der Annahme eines elektronischen Mechanismus identifizieren wir ein chirales supraleitendes Kondensat mit Majorana-Randmoden, das über einen weiten Bereich von Modellparametern energetisch begünstigt ist.

Ce travail de doctorat s'inscrit dans le cadre des techniques adaptées à la résolution du problème à N corps nucléaire. Il a été motivé par la perspective d'utiliser des méthodes allant au-delà de l'approximation de champ moyen pour améliorer la description des spectres d'excitation des noyaux stables et exotiques, notamment les états de basse énergie et les résonances géantes. À cette fin, l'approche choisie est le développement de modèles basés sur la second random-phase approximation (SRPA) utilisée avec une procédure de soustraction. Ces développements ont pour but d'étendre le champ d'applicabilité du modèle initial et d'inclure des corrélations dans l'état fondamental.Une première partie consiste en l'application de la SRPA avec une méthode de soustraction à l'étude de la réponse dipolaire (comprenant la polarisabilité électrique dipolaire) et quadrupolaire de noyaux de masse moyenne à lourds. Nous vérifions que la SRPA avec soustraction corrige les problèmes observés avec la SRPA standard et améliore la description des spectres d'excitation, comparativement à la random-phase approximation (RPA). Nous étudions également les effets au-delà du champ moyen dûs à la SSRPA avec soustraction, en exploitant la relation entre les modes de respiration axiaux des noyaux et la masse effective de la matière nucléaire.Une seconde partie est dédiée à des extensions.Premièrement, nous étendons les outils numérique initiaux en utilisant l'approximation equal-filling (EFA) afin de permettre les applications aux noyaux ayant une orbitale partiellement occupée. Nous proposons ensuite une méthode d'estimation partielle des effets d'appariement en utilisant des nombres d'occupation corrélés.Une étude des moyens de renormaliser la SRPA avec soustraction est menée en employant un modèle allant au-delà de l'approximation de quasiboson. Cette extension est également basée sur l'utilisation de nombres d'occupation comme moyen d'inclure des corrélations dans l'état fondamental. Nous montrons que les corrélations obtenues par le calcul itératif en RPA des nombres d'occupation ne sont pas suffisantes pour corriger les problèmes de la SRPA standard. / This Ph.D. work falls within the scope of theoretical techniques tailored to the solution of the nuclear many-body problem. It was motivated by the perspective of using beyond-mean-field methods to improve the description of excitation spectra of stable and exotic nuclei, especially the low-energy states and the giant resonances. The chosen path in this direction is the development of models based on the second random-phase approximation (SRPA) used with a subtraction procedure. These developments aim to extend the range of applicability of the initial model and to include correlations in the ground state.A first part consists in applying the SRPA used with a subtraction method to the study of the dipole and quadrupole response in medium to heavy-mass nuclei, including the electric dipole polarizability. We verify that the subtracted SRPA corrects the problems observed with the standard SRPA model and improves the description of excitation spectra compared to the random-phase approximation (RPA). We also study beyond-mean-field effects that arise in the subtracted SRPA by exploiting the relation between the axial breathing modes in nuclei and the effective mass in nuclear matter.A second part is dedicated to extensions.As a first step, we extend the initial numerical tools by employing the equal-filling approximation (EFA), to enable the applications to nuclei with partially-occupied orbitals. We next propose a method to estimate part of the pairing effects using correlated occupation numbers.A study of possible ways to renormalize the subtracted SRPA is carried out by employing a model which goes beyond the quasiboson approximation. This extension also relies on the use of occupation numbers as a means to include ground state correlations. We show that correlations obtained from the computation of occupation numbers in iterative RPA calculations are not sufficient to address the standard SRPA drawbacks.

This thesis explores the theoretical representation of localised electrons in magnetic systems, using Majorana fermions. A motivation is provided for the Majorana fermion representation, which is then developed and applied as a mean-field theory and in the path-integral formalism to the Ising model in transversal-field (TFIM) in one, two and three dimensions, on an orthonormal lattice. In one dimension the development of domain walls precludes long-range order in discrete systems; this is as free energy savings due to entropy outweigh the energetic cost of a domain wall. An argument due to Peierls exists in 2D which allows the formation of domains of ordered spins amidst a disordered background, however, which may be extended to 3D. The forms of the couplings to the bosons used in the Random Phase Analysis (RPA) are considered and an explanation for the non-existence of the phases calculated in this thesis is discussed, in terms of spare degrees of freedom in the Majorana representation. This thesis contains the first known application of Majorana fermions at the RPA level.

Density functional theory is a preferred microscopic method for calculation of nuclear properties over the whole nuclear chart. Besides ground-state properties, which are calculated by Hartree-Fock theory, nuclear excitations can be described by means of Random Phase Approximation (RPA). The main objective of the present work is to give the RPA formalism for spherically symmetric nuclei, using the techniques of angular-momentum coupling. Various auxiliary topics, such as Hartree-Fock theory, Coulomb integral, center-of-mass corrections and pairing, are treated as well. RPA method is derived also for axially deformed nuclei. The derived formulae are then implemented in the computer code and utilized for calculation of some physical results. After thorough investigation of the precision aspects of the calculation, the following topics are treated as examples: toroidal nature of the low-energy (pygmy) part of the E1 resonance, giant resonances of various multipolarities in deformed nucleus 154Sm, and magnetic dipole (M1) transitions in deformed 50Cr. Powered by TCPDF (www.tcpdf.org)