No description available.

Thesis advisor: Fazel Tafti / The search for an ideal quantum spin-liquid (QSL) material which can host a QSL ground state as well as exotic excitations has been one of the leading research topics in condensed matter physics over the past few decades. Out of all the proposals to realize the physics of a QSL, the Kitaev model is the most promising proposal with a QSL ground state. The Kitaev Hamiltonian is exactly solvable via fractionalization of its spin degrees of freedom into Majorana excitations, and it can be engineered in real materials. Among all the proposed Kitaev candidates, α-Li2IrO3, Na2IrO3, Li2RhO3, and α-RuCl3 are the most promising candidates. During my Ph.D. research I explored new physics related to Kitaev materials via modification of the symmetry and structural properties of these known Kitaev candidates. First, I studied how modification of the inter-layer chemistry can alter the thermodynamic properties of Kitaev candidate α-Li2IrO3 via an enhancement of the spin-orbit coupling (SOC) effect. The light, octahedrally-coordinated inter-layer Li atoms are replaced with heavier, linearly-coordinated Ag atoms to synthesize Ag3LiIr2O6. In addition to these structural modifications to the parent compound α-Li2IrO3, having heavier elements between the honeycomb layers in the Ag compound increased the effect of SOC in the honeycomb layers and led to a decrease in the long-range ordering temperature in Ag3LiIr2O6 compared to its parent compound. Second, I studied the effect of local crystal distortion in the presence of a weak SOC effect to explore a new spin-orbital state different from the Jeff=1/2 state. Based on theoretical predictions, the ground states of Kitaev materials can be tuned to other exotic spin-orbital states such as an Ising spin-1/2 state. To provide the proper conditions for a competition between the trigonal crystal distortion and the SOC effect, I modified the crystal environment around the magnetic elements in the parent compound Li2RhO3 via a topo-chemical method and synthesized Ag3LiRh2O6. An increase in the strength of trigonal distortion in Ag3LiRh2O6, in the presence of weak SOC, led to a transition from the Jeff=1/2 ground state (Kitaev limit) in the parent compound to an Ising spin-1/2 ground state (Ising limit) in the product. This change in spin-orbital state resulted in a dramatic change in magnetic behavior. Whereas Li2RhO3 shows a spin-freezing transition at 6 K, Ag3LiRh2O6 reveals a robust long-range antiferromagnetic transition at 94 K. This is the first realization of a change of ground state between the Kitaev and Ising limits in the same structural family. Lastly, I studied how the crystal symmetry can be an important factor in the physics of Kitaev materials. Honeycomb layered materials can be crystallized in space groups C2/m, C2/c, and P6_322. However, the crystal symmetry of most Kitaev candidates is described by the C2/m space group. We successfully synthesized a polymorph of a 3d Kitaev candidate, hexagonal Na2Co2TeO6 (P6_322 space group) in space group C2/m. The change in crystal symmetry of this cobalt tellurate replaced three anti-ferromagnetic (AFM) orders at 27, 15, 7 K in the hexagonal polymorph by a single AFM peak at 9.6 K in the monoclinic Na2Co2TeO6. / Thesis (PhD) — Boston College, 2023. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

This dissertation is devoted to investigation of quantum chaos in the Sachdev-Ye-Kitaev (SYK) and supersymmetric SYK models. First, a unified minimal scheme is developed to classify quantum chaos in the SYK and supersymmetric SYK models and also work out the structure of the energy levels in one periodic table. The SYK with even q-body or supersymmetric SYK with odd q-body interaction, with N even or odd number of sites, are put on an equal footing in the minimal Hilbert space; N (mod 8), q (mod 4) double Bott periodicity, and a reflection relation are identified. Then, exact diagonalizations are performed to study both the bulk energy level statistics and hard-edge behaviors. Excellent agreements between the exact diagonalization results and the symmetry classifications are demonstrated. This compact and systematic method can be transformed to map out more complicated periodic tables of SYK models with more degrees of freedom, tensor models, or symmetry protected topological phases.

Some physical systems exhibit topological properties in the form of topological invariants— features of the system that remain constant unless the system undergoessignificant changes i.e. changes that require closing the energy gap of the Hamiltonian.This work studies one example of a system with topological properties — a Kitaevchain. Here, this model is studied when it is coupled to an environment. We studythe effect of the coupling on the topology of the system and attempt to find signaturesof topological phases in the dynamics of the system. By using the Lindblad equationdefined in the formalism of third quantization, we study the time evolution of thesystem numerically by using the Euler method. We find that the dynamics of theentanglement spectrum of half of the chain is different in the topological and trivialphases: if the system undergoes a quench from trivial to topological phase, the entanglementspectrum exhibits crossings as the system evolves in time. We also studythe topological phases when disorder is added to the system. We test the stabilityof the topological phases of the system against disorder and find that the topologicalphases are not affected by a weak disorder. Moreover, by studying the statistics of theminimum entanglement spectrum gap, we find that, in general, a stronger disordermakes the crossings less likely to appear in the topological phase and more likely toappear in the trivial phase. / Det finns fysiska system som visar topologiska egenskaper i form av topologiska invarianter,som ändras inte så länge systemet genomgår ändringar som inte stängerHamiltonianens energigap. I det här arbetet undersöker vi ett exempel av ett systemmed topologiska egenskaper — en Kitaev kedja. Denna modell är studerat närden är kopplad till en omgivning. Vi undersöker kopplingens påverkan på systemetstopologi och vi försöker hitta tecken på topologiska faser i systemets dynamik. Vianvänder Lindblads ekvation definierat i tredje kvantiserings formalism för att studerasystemets tidsutveckling numeriskt, genom att använda Eulers metod. Vi upptäckeratt det finns skillnader i tidsutveckling av kvantsammanflätningsspektrumav häften av kedjan som beror på systems topologiska fas. Om systemet genomgåren kvantsläckning från den triviala till den topologiska fasen, kommer det finnas korsningari kvantsammanflätningensspektrum som uppstår under dess tidsutveckling.Dessutom studerar vi de topologiska faserna när det finns oordning i systemet. Viundersöker topologiska fasernas stabilitet mot oordning och upptäcker att en svagoordning påverkar inte de topologika faserna. Dessutom, genom att studera den minstakvantsammanflätningsspektrumsgap upptäcker vi att en starkare oordning ledertill kvantsammanflätningsspektrumskorsningar att vara mindre sannolika i den topologiskafasen och mer sannolika i den triviala fasen.

This thesis presents details of x-ray and neutron scattering experiments used to probe quantum magnets with strong spin-orbit interaction. The first of these systems are the three-dimensional iridate compounds, in which the three-fold co-ordination of IrO₆ octahedra has been theoretically hypothesized to stabilize anisotropic exchange between Ir⁴⁺ ions. This novel interaction between these spin-orbital entangled, J_eff=1/2 moments is described by a Hamiltonian first proposed by Kitaev, and would be the first physical realization of this Hamiltonian in a condensed matter system. This thesis details the determination of the structure of a new polytype within these compounds, the 'stripyhoneycomb' &gamma;-Li₂IrO₃. Furthermore, through resonant magnetic x-ray diffraction experiments on single crystals of &beta;-Li₂IrO₃ and &gamma;-Li₂IrO₃, an incommensurate, non-coplanar structure with counter-rotating moments is found. The counter-rotating moment structure is a rather counter-intuitive result, as it is not stabilizied by Heisenberg exchange between magnetic sites, however, the Kitaev exchange naturally accounts for this feature. As such, these experiments reveal, for the first time, systems which exhibit dominant Kitaev interactions. The ordering wavevector of both &beta;- and &gamma;-Li₂IrO₃ polytypes are found to be identical, suggesting that the same magnetic interactions are responsible for stabilizing magnetic order in both materials, despite their different lattice topologies. Following this, the spinel FeSc₂S₄ is considered. Here, despite the presence of strong exchange between Fe<sup>2+,/sup>, and the fact that these ions sit in a Jahn-Teller active environment, the system does not order in the spin or orbital degrees of freedom. A 'spin-orbital singlet' has been theoretically proposed to describe the groundstate of this system, and here inelastic neutron scattering (INS) is used to probe the resulting triplon excitations. This allows determination of microscopic parameters in the single ion and exchange Hamiltonians, and moreover experiments in external magnetic field reveal the true spin-and-orbital nature of these triplon excitations. Finally, Ba₃CoSb₂O₉, a physical realization of the canonical triangular antiferromagnet model is explored with INS and the high energy excitations from the 120 degree magnetic structure are found to display significant differences from those calculated by linear spin wave theory, suggesting the presence of quantum dynamics not captured in the 1/S linear spin wave expansion.

No description available.

This thesis deals with a series of quantum computer implementation issues from the Kane 31P in 28Si architecture to Shor’s integer factoring algorithm and beyond. The discussion begins with simulations of the adiabatic Kane CNOT and readout gates, followed by linear nearest neighbor implementations of 5-qubit quantum error correction with and without fast measurement. A linear nearest neighbor circuit implementing Shor’s algorithm is presented, then modified to remove the need for exponentially small rotation gates. Finally, a method of constructing optimal approximations of arbitrary single-qubit fault-tolerant gates is described and applied to the specific case of the remaining rotation gates required by Shor’s algorithm.

In the search for novel magnetic materials, systems with strong spin-orbit coupling are a focus. 5d Ir-oxides and 4d Ru-halide, in particular, are associated in this context with a flurry of new theoretical concepts, models, and predictions, and more recently to various exotic topological states. In this thesis, we use computational quantum-chemistry methods to determine nearest-neighbor (NN) magnetic interactions in such systems. We also explore different routes to tune NN exchange couplings and provide guidelines for material design. In the first chapter, an introduction to concepts of electron correlations, spin-orbit coupling and magnetic interactions is provided. Many-body quantum chemistry methods used to determine electronic and magnetic properties of the transition metal systems in this work are outlined in the second chapter. In chapter 3, we determine multiplet-structure, magnetic g factors as well as NN magnetic interaction for the edge-shared 4d5 honeycomb lattice-based system, i.e., α-RuCl3. We find that the the magnetic anisotropy shows up in the form of bond-dependent Kitaev couplings, which defines the largest superexchange energy scale in this system. Magnetic couplings obtained by mapping the ab initio data onto an effective spin Hamiltonian are then used in the the subsequent exact diagonalization calculation to retrieve the magnetic phase diagram as a function of second and third NN coupling. Further, in chapter 4, we investigate the effects of uniform pressure and strain on the magnetic interactions in honeycomb and related lattice-based systems. We find that the Heisenberg and Kitaev terms are affected differently: for strain, in particular, the Heisenberg component decreases more rapidly than the Kitaev counterpart. This suggests a scenario where strain can stabilize a spin liquid state in such materials. In chapter 5, we discuss another factor that allows to modify magnetic couplings, i.e., the electrostatics between layered stackings with different metallic species. We examine magnetic interactions between Ir moments in H3LiIr2O6, a recently proposed Kitaev spin liquid candidate, and clarify the effect of interlayer electrostatics on the anisotropic Kitaev exchange . We show that the precise position of H+ cations between magnetically active [LiIr2O6]3− honeycomb-like layers has a strong impact on the magnitude of Kitaev interactions. In the last chapter, we examine Ir-oxides on the pyrochlore lattice. In these corner-sharing systems the NN anisotropic exchange occurs in the form of antisymmetric exchange, also known as Dzyaloshinskii-Moriya (DM) coupling. Our calculations predict that a highly unusual regime can be realized in such systems due to the vanishing NN Heisenberg interaction, making the antisymmetric DM exchange to be the dominant interaction in the oxides where the Ir-O-Ir links show bond-angles less than 125◦. We also confirm the accuracy of the employed quantum-chemistry methods by reproducing experimental data for Sm2Ir2O7.:Table of contents 1 Introduction 1 1.1 Electronic correlations 2 1.2 Crystal fields and d-level splitting 5 1.3 Spin-orbit Coupling 8 1.4 Magnetic interactions 10 1.5 Conclusions 13 2 Quantum Chemistry Methods 15 2.1 Introduction 15 2.2 Motivation for using quantum chemical approach 17 2.3 The Hartree-Fock approach 19 2.4 Multiconfigurational approach 22 2.5 Multireference configuration interaction 26 2.5.1 Recent developments towards performing FCI 27 2.6 Embedded cluster approach 28 2.7 Conclusions 30 3 Anisotropic spin interactions in α-RuCl3 31 3.1 Introduction 31 3.2 Spin-orbit ground state and excitations 33 3.2.1 Structural details .34 3.2.2 Computational details 37 3.2.3 Results and Discussions 40 3.3 Intersite exchange interactions for j=1/2 moments 44 3.3.1 Kitaev-Heisenberg model and symmetric anisotropies 45 3.3.2 Computational details 49 3.3.3 Results and Discussion 53 3.4 Conclusions 61 x Table of contents 4 Strain and pressure tuned magnetic interactions in Kitaev materials 63 4.1 Introduction 64 4.2 Qualitative analysis: Kitaev-Heisenberg model 65 4.3 Quantitative analysis: ab initio results 66 4.3.1 Computational approach 69 4.3.2 Results and discussion 70 4.4 Experimental results for pressurized α-RuCl3 74 4.4.1 Pressure induced dimerization 75 4.4.2 Ab initio calculations 76 4.5 Conclusions 78 5 Impact of inter-layer species on in-plane magnetism in H3LiIr2O6 79 5.1 Introduction 79 5.2 Structural details 81 5.3 Computational approach 82 5.4 Results and discussion 85 5.4.1 Magnetic couplings 85 5.4.2 Phase diagram and longer-range interactions 86 5.4.3 Position of H cations and effect on in-plane interactions 88 5.4.4 Angle dependence, the Kitaev limit 91 5.5 Conclusions 92 6 Anisotropic spin interactions in pyrochlore iridates 95 6.1 Introduction 95 6.2 Structural details 97 6.3 Computational details 98 6.3.1 Embedded cluster and basis sets 98 6.3.2 Quantum chemistry calculations 99 6.3.3 Effective spin model Hamiltonian 99 6.4 Results and Discussion 101 6.4.1 Magnetic couplings 101 6.4.2 Spin Dynamics 103 6.4.3 Magnetic ground state 105 6.5 Conclusions 109 Summary 111

The study of topological phases in condensed matter physics has seen remarkable advancements, primarily focusing on systems with a well-defined bulk and boundary. However, the emergence of topological phenomena on self-similar systems, characterized by the absence of a clear distinction between bulk and boundary, presents a fascinating challenge. This thesis focuses on the topological phases on self-similar systems, shedding light on their unique properties through the lens of adiabatic charge pumping. We observe that the spectral flow in these systems exhibits striking qualitative distinctions from that of translationally invariant non-interacting systems subjected to a perpendicular magnetic field. We show that the instantaneous eigenspectra can be used to understand the quantization of the charge pumped over a cycle, and hence to understand the topological character of the system. Furthermore, we establish a correspondence between the local contributions to the Hall conductivity and the spectral flow of edge-like states. We also find that the edge-like states can be approximated as eigenstates of the discrete angular-momentum operator, with their chiral characteristics stemming from this unique perspective. We also investigate the effect of local structure on the topological phases on self-similar structures embedded in two dimensions. We study a geometry dependent model on two self-similar structures having different coordination numbers, constructed from the Sierpinski gasket. For different non-spatial symmetries present in the system, we numerically study and compare the phases on both structures. We characterize these phases by the localization properties of the single-particle states, their robustness to disorder, and by using a real-space topological index. We find that both structures host topologically nontrivial phases and the phase diagrams are different on the two structures, emphasizing the interplay between non-spatial symmetries and the local structure of the self-similar unit in determining topological phases. Furthermore, we demonstrate the presence of topologically ordered chiral spin liquid on fractals by extending the Kitaev model to the Sierpinski Gasket. We show a way to perform the Jordan-Wigner transformation to make this model exactly solvable on the Sierpinski Gasket. This system exhibits a fractal density of states for Majorana modes and showcases a transition from a gapped to a gapless phase. Notably, the gapped phase features symmetry-protected Majorana corner modes, while the gapless phase harbors robust zero-energy and low-energy self-similar Majorana edge-like modes. We also study the vortex excitations, characterized by remarkable localization properties even in small fractal generations. These localized excitations exhibit anyonic behavior, with preliminary calculations hinting at their fundamental differences from Ising anyons observed in the Kitaev model on a honeycomb lattice.

The overall objective of this thesis is to perform DFT based microscopic modeling of real cobaltates viewed as quantum spin liquid candidates and estimate their magnetic exchange parameters. Using the FPLO code with its Wannier function module, we estimated the onsite and intersite transfer integrals that are key quantities for a realistic material-specific description of the electron structure. Based on these results, theoretical approaches in the framework of hexagonal edge-sharing octahedra were applied to address the magnetic properties. We studied the three cobaltates Na2BaCo(PO4)2, Li3Co2SbO6 and Na3Co2SbO6 in detail. For each cobaltate that we calculated, we first examined the crystal structure. In the case of Na2 BaCo(PO4)2, we figured out uncertainties in the published crystal structure and investigated a plausible rotation of the O ligand octahedra, which we confirmed. For the two honeycomb cobaltates Li3Co2SbO6 and Na3Co2SbO6, we relaxed the crystal structure and numerically applied uniaxial strain along the c axis with the amount of ±0.05. We confirmed our suggestion that tensile strain brings the compressed octahedra closer to cubic symmetry. In the standard DFT approach with the GGA functional, we obtained wrong metallic behavior for insulating materials. This is a well-known shortcoming of the non-magnetic treatment and the insufficient account of the strong electron correlation. To overcome this issue, magnetic DFT+U calculations were performed. We were able to achieve excellent Wannier function fits on the calculated band structures for the two models: d and dp. The latter includes the O p states, which is reasonable because of the strong hybridization. For the honeycomb cobaltates Li3Co2SbO6 and Na3Co2SbO6 it was necessary to include the Sb 5s state located in the center of the hexagonal void to achieve such excellent Wannier function fits. With the help of the additional dp model, we can distinguish between different direct and indirect hopping processes. These Wannier function analyses give us the onsite properties and intersite processes which are the key quantities of the microscopic model and determine the magnetic behavior. The derived onsite properties of all three cobaltates are used to determine the crystal field parameters and the spin-orbit coupling constants by two methods: diagonalization and matrix comparison. In both methods, we achieved proper values for the cubic splitting, the charge transfer gap and the spin-orbit coupling constant. The sign of the trigonal splitting is at odds with the simple point charge model of the respective distorted octahedra. For Na2 BaCo(PO4)2 , we go a step further and use the crystal field parameters to calculate the multiplet energy levels and g-factors with ELISA. Compared with ESR measurements, we found that the results of the published structure are in a better agreement where the structure seems to be inaccurate. Evaluating the intersite hopping processes for all three cobaltates shows a good agreement with the cubic-symmetry-allowed hoppings. This reveals the closeness of the structures with honeycomb and cubic symmetry of the octahedra, respectively. The leading nearest neighbor process found for the z-bond is t3 between the two d_xy orbitals. The extended dp model allows for a resolution in direct and indirect hopping processes. We found that t3 in Na2BaCo(PO4)2 is dominated by indirect contributions, while in Li3Co2SbO6 and Na3Co2SbO6 the direct contributions dominate. One important outcome of this thesis is the sensitivity of hopping terms regarding structural modifications like optimization, relaxation or applying strain. Especially the honeycomb cobaltates Li3Co2SbO6 and Na3Co2SbO6 show high sensitivity on the crystal structure. Even small modifications by relaxing the crystal structure alter the hierarchy of the leading hoppings. Furthermore, the status quo between direct and indirect contributions can change dramatically. As the important 3rd neighbor hoppings in Li3Co2SbO6 and Na3Co2SbO6 are indirect, the major hopping paths were discovered. In the literature we found three different theoretical approaches to calculating the magnetic exchange parameters. With these and the onsite and intersite properties estimated before, we determined the magnetic exchange parameters in dependence on strain for Li3Co2SbO6 and Na3Co2SbO6. A direct comparison revealed that results for each approach are rather different. Hence, a universal theory model for these systems is still in development. With that we confirm the substantial controversy regarding the strength and the sign of the magnetic exchange terms. Following from the sensitivity of the hopping terms, the magnetic exchanges are also quite sensitive to modifications of the crystal structure. We located the cobaltates Li3Co2SbO6 and Na3Co2SbO6 in the phase diagram and found that they mainly occupy the zigzag-y phase also confirmed by experiments. Li3Co2SbO6 shows a promising behavior for increasing tensile strain: crossing the quantum spin liquid phase into the ferromagnetic phase. So the magnetic properties of Li3Co2SbO6 are more sensitive compared to Na3Co2SbO6 . The possibility of reaching the quantum spin liquid phase is theoretically given for Li3Co2SbO6 . With Li3Co2SbO6 covering the quantum spin liquid phase, the octahedra have almost the higher threefold rotational symmetry when checking the angles in the octahedra. In our investigations, we showed that real material simulations give valuable insights into the magnetic properties. Thereby Wannier function analyses are a powerful tool in the estimation of onsite and intersite terms. This makes several desired characteristics and parameters accessible and helps to link theory with experiment. In addition to that, we offered interesting insights into the microscopic behavior and trends when applying numerical strain on the crystals. This provides valuable hints for further research in this field.:List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Material and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1. Magnetism of cobaltates . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1. One-electron Hamiltonian for l = 2 . . . . . . . . . . . . . . 4 2.1.2. Multiplets of the 3d shell . . . . . . . . . . . . . . . . . . . . . 10 2.2. Quantum spin liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1. Kitaev honeycomb lattice model . . . . . . . . . . . . . . . 14 2.3. Slater-Koster terms in a honeycomb lattice . . . . . . . . . . 15 2.4. Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1. Quantum mechanical many-body systems . . . . . . . 21 2.4.2. Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . 22 2.4.3. Exchange-correlation functional and Kohn-Sham equation . . 23 2.4.4. Wannier function projections . . . . . . . . . . . . . . . . . . 24 2.4.5. DFT+U for correlated insulators . . . . . . . . . . . . . . . . 25 2.5. Technical details of the calculations . . . . . . . . . . . . . . . . 26 2.5.1. Choice of the k-mesh . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.2. Cutoff parameter for WFs . . . . . . . . . . . . . . . . . . . . . 28 2.5.3. Relaxation routine . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3. The triangular-lattice cobaltate Na2BaCo(PO4 )2 . . . . . . . 30 3.1. Crystal structure and band structure . . . . . . . . . . . . . . . . 31 3.2. Structural optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3. Wannier function analysis . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1. Onsite properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.2. Intersite processes . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.3. Magnetic exchange parameters . . . . . . . . . . . . . . . . 45 3.4. DFT+U calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5. Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . 48 4. The honeycomb cobaltates Li3 Co2 SbO6 and Na3 Co2 SbO6 . . 50 4.1. Crystal structure and band structure . . . . . . . . . . . . . . . . 51 4.2. Structural optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3. Application of uniaxial strain along the c-axis . . . . . . . . . . 54 4.4. Wannier function analysis . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4.1. Onsite properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.2. Intersite processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.5. Magnetic exchange parameters . . . . . . . . . . . . . . . . . . . . 71 4.6. Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . .75 5. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A.1. Details for the material and methods (Chapter 2) . . . . . . . 81 A.1.1. Cubic representation of the CF Hamiltonian . . . . . . . . . . 81 A.1.2. Derivation of the spin-orbit coupling Hamiltonian . . . . . . .82 A.2. Details for the triangular-lattice cobaltate Na2 BaCo(PO4)2 (Chapter 3) . . 86 A.2.1. DOS of DFT+U calculations . . . . . . . . . . . . . . . . . . . . . . . 86 A.2.2. Nearest neighbor hopping matrices for the y-bond . . . . . 86 A.2.3. Number of exchanges and systems of equations for the different magnetic configurations calculated with DFT+U . . . . . . . 87 A.3. Details for the honeycomb cobaltates (Chapter 4) . . . . . . . 90 A.3.1. Nearest neighbor hopping matrices for the y-bond . . . . . 90 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105