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

As propriedades do estado fundamental do modelo de Heisenberg antiferroinagnético quântico de spin-1/2 na rede quadrada e na rede cúbica espacialmente anisotrópica são investigadas através de um novo método de Monte Carlo, baseado na estimativa do maior autovalor de uma matriz de elementos não negativos. A energia do estado fundamental e a magnetização \"staggered\" destes sistemas são calculadas em redes relativamente grandes com até 24 x 24 sítios para o caso de redes quadradas e 8 x 8 x 8 sítios para o caso de redes cúbicas. O método desenvolvido também pode ser usado como um novo algoritmo para a determinação direta da entropia de sistemas de spins de Ising através de simulações usuais de Monte Carlo. Usando este método, calculamos a entropia do antiferromagneto de Ising na presença de um campo magnético externo nas redes triangular e cúbica de face centrada. / The ground state properties of the antiferromagnetic quantum Heisenberg model with spin-112 defined on a square lattice and on a cubic lattice with spatial anisotropy are investigated through a new Monte Carlo method, based on the estimation of the largest eigenvalue of a matrix with nonnegative elements. The ground state energy and the staggered magnetization of these systems are calculated in relatively large lattices with up to 24 x 24 sites for the square lattices and 8 x 8 x 8 sites for cubic lattices. The method developped can also be used as a new algorithm for the direct determination of the entropy of Ising spin systems through ordinary Monte Car10 simulations. By using this method we calculate the entropy of the Ising antiferromagnetic in the presence of a magnetic field in the triangular and face centered cubic lattices.

Neste trabalho foram estudadas as fases topológicas não triviais presentes em sistemas formados pela deposição de átomos de grafeno. Encontramos que quando um átomo hibridiza fortemente com o grafeno, apresenta um momento magnético e um forte spin-órbirta é possível a formação de uma rede hexagonal de baricentros que efetivamente gera uma estrutura de bandas característica de um efeito hall quântico anômalo. Especificamente, determinamos que o Ru satisfaz estas características. Quando este metal é depositado em uma configuração triangular no grafeno ocorrem picos na densidade de estados localizados no centro geométrico (baricentro) dos triângulos formados pelos Ru. Estes picos estão distribuídos de forma hexagonal e efetivamente geram uma estrutura de bandas que nas proximidades do nível de Fermi apresenta uma configuração de spin característica do efeito Hall quântico anômalo. Adicionalmente, encontramos que o sistema composto pela absorção de Ba ou Sr no grafeno favorece a formação do efeito Hall quântico de spin. Neste sistema, o acoplamento spin-órbita (SOC) gera um gap mais de 1000 vezes maior ao período no grafeno prístino. Para o estudo destes sistemas, implementamos no código SIESTA a aproximação on-site do acoplamento spin-órbita via o formalismo dos pseudopotenciais relativísticos de norma conservada. Nossa implementação foi testada a partir do estudo de fenômenos já conhecidos: i) o strong spin-splitting gerado no grafeno pela adsorção de Au, ii) o efeito hall quântico de spin no poço quântico de HgTe/CdTe e, iii) a formação de estados topológicos na superfície do Bi2Se3 e as fases magnéticas deste material com átomos de Mn adsorvidos. / In this work, were studied the non-trivial topological phases present in systems formed by deposition of atoms in graphene. We found that when an atom hybridizes strongly with grapheme, has a magnetic moment and a strong spin-orbit it is possible the formation of a hexagonal network of barycentres that effectively generates a structure band characteristic of a quantum anomalous Hall effect. Specifically, we determined that Ru satisfies these characteristics. When this metal is deposited in a triangular configuration in grapheme, peaks occur in the density of localized states in the geometric center (centroid) of the triangles formed by Ru. These peaks are distributed in a hexagonal structure and effectively generates a band structure that near the Fermi level has a spin configuration characteristic of the spin quantum Hall effect anomalous. Additionally, we found that the system composed by the adsorption of Ba or Sr in grapheme, promotes the formation of spin quantum Hall effect. In this system, the spin-orbit coupling (SOC) generates a gap more than 1000 times grater that predicted in pristine praphene. To study these systems, wu implemented in the code SIESTA the on-site approach of the spin-orbit coupling throught the formalism of norm conserved relativistic pseudo potentials. Our implementation was tested from the study of phenomena already known: i) the strong spin-splitting generated in graphene by adsorption of Au, ii) the quantum spin Hall effect in quantum well of HgTe / CdTe and, iii) formation of topological states in the surface of Bi2Se3 and the magnetic of this material with Mn atoms adsorved.

As propriedades do estado fundamental do modelo de Heisenberg antiferroinagnético quântico de spin-1/2 na rede quadrada e na rede cúbica espacialmente anisotrópica são investigadas através de um novo método de Monte Carlo, baseado na estimativa do maior autovalor de uma matriz de elementos não negativos. A energia do estado fundamental e a magnetização \"staggered\" destes sistemas são calculadas em redes relativamente grandes com até 24 x 24 sítios para o caso de redes quadradas e 8 x 8 x 8 sítios para o caso de redes cúbicas. O método desenvolvido também pode ser usado como um novo algoritmo para a determinação direta da entropia de sistemas de spins de Ising através de simulações usuais de Monte Carlo. Usando este método, calculamos a entropia do antiferromagneto de Ising na presença de um campo magnético externo nas redes triangular e cúbica de face centrada. / The ground state properties of the antiferromagnetic quantum Heisenberg model with spin-112 defined on a square lattice and on a cubic lattice with spatial anisotropy are investigated through a new Monte Carlo method, based on the estimation of the largest eigenvalue of a matrix with nonnegative elements. The ground state energy and the staggered magnetization of these systems are calculated in relatively large lattices with up to 24 x 24 sites for the square lattices and 8 x 8 x 8 sites for cubic lattices. The method developped can also be used as a new algorithm for the direct determination of the entropy of Ising spin systems through ordinary Monte Car10 simulations. By using this method we calculate the entropy of the Ising antiferromagnetic in the presence of a magnetic field in the triangular and face centered cubic lattices.

Ce travail de thèse porte sur l'étude par la résonance paramagnétique électronique (RPE) des sels à transfert de charge quasi-unidimensionnels (TMTTF)$ {2}$X (X=AsF$ {6}$, PF$ {6}$, SbF$ {6}$), matériaux modèles de chaînes de spins quantiques. Tout d'abord, nous avons examiné en onde continue et sur une large gamme de température et de fréquence, la phase d'ordre de charge déjà observée dans ces matériaux en dessous de la température T$ {CO}$. Nous avons mis en évidence deux nouveaux phénomènes à T < T$ {CO}$: la rotation des axes principaux du facteur g et une modification structurale liée à un dédoublement de la maille cristallographique. Un calcul de chimie quantique a été réalisé à l'aide de la méthode DFT confirmant nos résultats expérimentaux. Dans la seconde partie de ces travaux de thèse, nous avons présenté les résultats obtenus par RPE en onde continue et en onde pulsée sur l'étude des défauts corrélés dans les systèmes à chaînes de spins. En onde continue, nous avons détecté pour la première fois une raie RPE fine à basse température, suggérant la présence de défauts corrélés ayant les caractéristiques de solitons. Les mesures par RPE pulsée nous ont permis d'observer les premières oscillations de Rabi de solitons piégés et de déterminer leur caractère robuste. Ces derniers résultats offrent une approche alternative aux qubits à base de spins pour le traitement de l’information quantique. / This thesis focuses on the study by Electron Paramagnetic Resonance (EPR) of the quasi-one-dimensional charge transfer salts (TMTTF)$ {2}$X (X=AsF$ {6}$, PF$ {6}$, SbF$ {6}$), model materials of quantum spin chains. First, we have examined in continuous wave and on a wide range of temperature and frequency, the charge-ordered phase already observed in these materials below the temperature T$ {CO}$. We have identified two new phenomena at T <T$ {CO}$: the rotation of the principal axes of the g factor and a structural change related to a doubling of the unit cell parameter. A quantum chemical calculation was carried out using DFT confirming our experimental results. In the second part of the thesis, we have presented the results obtained by EPR in continuous wave and pulsed wave on the correlated defects study in spin chain systems. In continuous wave, we have detected for the first time a narrow EPR line at low temperature, suggesting the presence of correlated defects having the characteristics of solitons. The pulsed EPR measurements allowed us to observe the first Rabi oscillations of trapped solitons and to determine their robust character. These latter results offer an alternative approach for spin qubits in quantum information processing.

Since the last few decades frustrated spin systems have attracted much interest. These studies are motivated by the rich variety of their unusual magnetic properties and potential applications. In this thesis, excitation spectra of the weakly coupled dimer system Ba3Cr2O8, the spin-1/2 chain material with distorted diamond structure Cu3(CO3)2(OH)2 (natural mineral azurite), and the quasi-twodimensional antiferromagnet with triangle spin structure Cs2CuBr4 have been studied by means of high-field electron spin resonance. Two pairs of gapped modes corresponding to transitions from a spin-singlet ground state to the first excited triplet state with zero-field energy gaps, of 19.1 and 27 K were observed in Ba3Cr2O8. The observation of ground-state excitations clearly indicates the presence of a non-secular term allowing these transitions. Our findings are of crucial importance for the interpretation of the field-induced transitions in this material (with critical fields Hc1 = 12.5 T and Hc2 = 23.6 T) in terms of the magnon Bose-Einstein condensation. The natural mineral azurite, Cu3(CO3)2(OH)2, has been studied in magnetic fields up to 50 T, revealing several modes not observed previously. Based on the obtained data, all three critical fields were identified. A substantial zero-field energy gap, Δ = 9.6 K, has been observed in Cs2CuBr4 above the ordering temperature. It is argued that contrary to the case for the isostructural Cs2CuCl4, the size of the gap can not be explained solely by the uniform Dzyaloshinskii-Moriya interaction, but it is rather the result of the geometrical frustration stabilizing the spin-disordered state in Cs2CuBr4 in the close vicinity of the quantum phase transition between a spiral magnetically ordered state and a 2D quantum spin liquid.

Neste trabalho foram estudadas as fases topológicas não triviais presentes em sistemas formados pela deposição de átomos de grafeno. Encontramos que quando um átomo hibridiza fortemente com o grafeno, apresenta um momento magnético e um forte spin-órbirta é possível a formação de uma rede hexagonal de baricentros que efetivamente gera uma estrutura de bandas característica de um efeito hall quântico anômalo. Especificamente, determinamos que o Ru satisfaz estas características. Quando este metal é depositado em uma configuração triangular no grafeno ocorrem picos na densidade de estados localizados no centro geométrico (baricentro) dos triângulos formados pelos Ru. Estes picos estão distribuídos de forma hexagonal e efetivamente geram uma estrutura de bandas que nas proximidades do nível de Fermi apresenta uma configuração de spin característica do efeito Hall quântico anômalo. Adicionalmente, encontramos que o sistema composto pela absorção de Ba ou Sr no grafeno favorece a formação do efeito Hall quântico de spin. Neste sistema, o acoplamento spin-órbita (SOC) gera um gap mais de 1000 vezes maior ao período no grafeno prístino. Para o estudo destes sistemas, implementamos no código SIESTA a aproximação on-site do acoplamento spin-órbita via o formalismo dos pseudopotenciais relativísticos de norma conservada. Nossa implementação foi testada a partir do estudo de fenômenos já conhecidos: i) o strong spin-splitting gerado no grafeno pela adsorção de Au, ii) o efeito hall quântico de spin no poço quântico de HgTe/CdTe e, iii) a formação de estados topológicos na superfície do Bi2Se3 e as fases magnéticas deste material com átomos de Mn adsorvidos. / In this work, were studied the non-trivial topological phases present in systems formed by deposition of atoms in graphene. We found that when an atom hybridizes strongly with grapheme, has a magnetic moment and a strong spin-orbit it is possible the formation of a hexagonal network of barycentres that effectively generates a structure band characteristic of a quantum anomalous Hall effect. Specifically, we determined that Ru satisfies these characteristics. When this metal is deposited in a triangular configuration in grapheme, peaks occur in the density of localized states in the geometric center (centroid) of the triangles formed by Ru. These peaks are distributed in a hexagonal structure and effectively generates a band structure that near the Fermi level has a spin configuration characteristic of the spin quantum Hall effect anomalous. Additionally, we found that the system composed by the adsorption of Ba or Sr in grapheme, promotes the formation of spin quantum Hall effect. In this system, the spin-orbit coupling (SOC) generates a gap more than 1000 times grater that predicted in pristine praphene. To study these systems, wu implemented in the code SIESTA the on-site approach of the spin-orbit coupling throught the formalism of norm conserved relativistic pseudo potentials. Our implementation was tested from the study of phenomena already known: i) the strong spin-splitting generated in graphene by adsorption of Au, ii) the quantum spin Hall effect in quantum well of HgTe / CdTe and, iii) formation of topological states in the surface of Bi2Se3 and the magnetic of this material with Mn atoms adsorved.

In the study of strongly interacting condensed-matter systems controlled microscopic theories hold a key position. Spin-wave theory, large-N expansion, and $epsilon$-expansion are some of the few successful cornerstones. In this doctoral thesis work, we have developed a novel large-$d$ expansion method, $d$ being the spatial dimension, to study model Hamiltonians hosting a quantum phase transition between a paramagnet and a magnetically ordered phase. A highlight of this technique is that it can consistently describe the entire phase diagram of the above mentioned models, including the quantum critical point. Note that most analytical techniques either efficiently describe only one of the phases or suffer from divergences near the critical point. The idea of large-$d$ formalism is that in this limit, non-local fluctuations become unimportant and that a suitable product state delivers exact expectation values for local observables, with corrections being suppressed in powers of $1/d$. It turns out that, due to momentum summation properties of the interaction structure factor, all diagrams are suppressed in powers of $1/d$ leading to an analytic expansion. We have demonstrated this method in two important systems namely, the coupled-dimer magnets and the transverse-field Ising model. Coupled-dimer magnets are Heisenberg spin systems with two spins, coupled by intra-dimer antiferromagnetic interaction, per crystallographic unit cell (dimer). In turn, spins from neighboring dimers interact via some inter-dimer interaction. A quantum paramagnet is realized for a dominant intra-dimer interaction, while a magnetically ordered phase exists for a dominant (or of the same order as intra-dimer interaction) inter-dimer interaction. These two phases are connected by a quantum phase transition, which is in the Heisenberg O(3) universality class. Microscopic analytical theories to study such systems have been restricted to either only one of the phases or involve uncontrolled approximations. Using a non-linear bond-operator theory for spins with S=$1/2$, we have calculated the $1/d$ expansion of static and dynamic observables for coupled dimers on a hypercubic lattice at zero temperature. Analyticity of the $1/d$ expansion, even at the critical point, is ensured by correctly identifying suitable observables using the mean-field critical exponents. This method yields gapless excitation modes in the continuous symmetry broken phase, as required by Goldstone\'s theorem. In appropriate limits, our results match with perturbation expansion in small ratio of inter-dimer and intra-dimer coupling, performed using continuous unitary transformations, as well as the spin-wave theory for spin-$1/2$ in arbitrary dimensions. We also discuss the Brueckner approach, which relies on small quasiparticle density, and derive the same $1/d$ expansion for the dispersion relation in the disordered phase. Another success of our work is in describing the amplitude (Higgs) mode in coupled-dimer magnets. Our novel method establishes the popular bond-operator theory as a controlled approach. In $d=2$, the results from our calculations are in qualitative agreement with the quantum Monte Carlo study of the square-lattice bilayer Heisenberg AF spin-$1/2$ model. In particular, our results are useful to identify the amplitude (Higgs) mode in the QMC data. The ideas of large-$d$ are also successfully applied to the transverse-field Ising model on a hypercubic lattice. Similar to bond operators, we have introduced auxiliary Bosonsic operators to set up our method in this case. We have also discussed briefly the bilayer Kitaev model, constructed by antiferromagnetically coupling two layers of the Kitaev model on a honeycomb lattice. In this case, we investigate the dimer quantum paramagnetic phase, realized in the strong inter-layer coupling limit. Using bond-operator theory, we calculate the mode dispersion in this phase, within the harmonic approximation. We also conjecture a zero-temperature phase diagram for this model.

Lattice gauge theories are crucial for our understanding of many physical phenomena ranging from fundamental particle interactions in high-energy physics to frustration and topological order in condensed matter. Hence, many equilibrium aspects of these theories have been studied intensively over the past decades. Recent developments, however, have shown that the study of nonequilibrium dynamics in lattice gauge theories also provides a very fertile ground for interesting phenomena. This thesis is devoted to the study of two particular dynamical processes in lattice gauge theories and related quantum spin models. First, we show that an interacting two-dimensional lattice gauge theory can exhibit disorder-free localization: a mechanism for ergodicity breaking due to local constraints imposed by gauge invariance. This result is particularly remarkable as the stability in two dimensions of the more conventional (disorder-induced) many-body localization is still debated. Concretely, we show this type of nonergodic behavior in the quantum link model. Our central result is based on a bound on the localization-delocalization transition, which is established through a concomitant classical percolation problem. Further, we develop a numerical method dubbed “variational classical networks”, to study the quantum dynamics in this system. This technique provides an efficient and perturbatively controlled representation of the wave function in terms of networks of classical spins akin to artificial neural networks. This allows us to identify distinguishing transport properties in the localized and ergodic phases, respectively. In the second problem, we study the dynamics of string breaking, a key process in confining gauge theories, where a string connecting two charges decays due to the creation of new particle-antiparticle pairs. Our main result here is that string breaking can also be observed in quantum Ising chains, in which domain walls get confined either by a symmetry-breaking field or by long-range interactions. We identify, in general, two distinct stages in this process. While at the beginning the initial charges remain stable, the string can exhibit complex dynamics with strong quantum correlations. We provide an effective description of this string motion, and find that it can be highly constrained. In the second stage, the string finally breaks at a timescale that depends sensitively on the initial separation of domain walls. We observe that the second stage can be significantly delayed as a consequence of the dynamical constraints appearing in the first stage. Finally, we discuss the generalization of our results to low-dimensional confining gauge theories. As a general aspect of this work, we discuss how the phenomena studied here could be realized experimentally with current and future technologies in quantum simulation. Furthermore, the methods developed in this thesis can also be applied to other lattice gauge theories and constrained quantum many-body models, not only to address purely theoretical questions but also to provide a theoretical description of experiments in quantum simulators. / Gittereichtheorien sind ein wichtiger Bestandteil im Verständnis vieler physikalischer Phänomene und Grundlage verschiedener Theorien, welche sich von der elementaren Wechselwirkungen in der Hochenergiephysik, Frustration in Spinmodellen bis hin zu topologischer Ordnung in der Festkörperphysik erstrecken. Die Eigenschaften von Eichtheorien im Gleichgewicht waren in den letzten Jahrzehnten ein zentraler Punkt der Forschung. Obwohl sich Untersuchungen der Dynamik jenseits des Gleichgewichs als eine große Herausfordung dargestellt haben, haben kürzliche Erkenntnisse gezeigt, dass die Dynamik in Gittereichtheorien überraschende und interessante Entdeckungen bereithält. Diese Dissertation behandelt zwei zentrale dynamische Prozesse in Gittereichtheorien und verwandten Spinmodellen. Einerseits soll die Dynamik von zweidimensionalen und wechselwirkenden Gittereichtheorien untersucht werden im Falle des sogenan- nten Quanten-Link-Modells untersucht werden. Entgegen der Ergodenhypothese zeigt das System Lokalisierung ohne Unordnung aufgrund lokaler Zwangsbedingungen durch Eininvarianz. Dieses Ergebnis ist insofern bemerkenswert, als die gewöhnliche, durch Unordnung induzierte, Vielteilchenlokalisierung in zwei Dimensionen umstritten ist. Als ein Hauptergebnis finden wir einen Übergang zwischen einer lokalisierten und ergodischen Phase, dessen Existenz durch ein zugehöriges klassisches Perkolationsproblem gezeigt werden konnte. Die quantenmechanischen Transporteigenschaften, elementar verschieden in der lokalisierten und ergodischen Phase, werden charakterisiert und untersucht. Die Lösung der quantenmechanischen Zeitentwicklung wird durch eine methodische Weiterentwicklung der sogenannten „variationellen klassischen Netzwerke“ erreicht Diese Methode stellt eine perturbative, aber kontrollierte Repräsentation von zeitentwickelten quantenmechanischen Wellenfunktionen dar in Form von Netzwerken klassischer Spins, ähnlich wie bei einem künstlichen neuronalen Netz. Im zweiten Teil untersuchen wir die Dynamik eines Schlüsselprozesses in Eichtheorien mit Confinement, welcher als „String-Breaking“ bezeichnet wird In diesem Prozess zerfällt der der Strang, der zwei elementare Ladungen verbindet, durch die Bildung neuer Teilchen-Antiteilchen-Paare. Ein Hauptresultat dieser Arbeit ist die Beobachtung dieses dynamischen Phänomens in Quantum-Ising-Ketten und damit in Systemen ohne Eichinvarianz. Das Confinement entsteht dabei zwischen Domänenwänden entweder durch eine langreichweitige Wechselwirkung zwischen den beteiligten Spins oder durch symmetriebrechende Magnetfelder. Es wird gezeigt, dass während des „String-breaking“ Prozesses das Modell zwei Phasen durchläuft: Während zu Beginn die Anfangsladungen stabil bleiben, weist der Strang eine komplexe Dynamik mit starken Quantenkorrelationen auf. Für diese erste Phase wird eine effektive Beschreibung eingeführt, um die verschiedenen Aspekte zu analysieren und zu verstehen. Die Zeitskalen zur Destabilisierung des Strangs innerhalb einer zweiten Phase zeigen eine starke Abhängigkeit von der anfänglichen Trennung der Domänenwände. Es wird gezeigt, dass die zweite Phase als Konsequenz der dynamischen Beschränkungen der ersten Phase signifikant verzögert werden kann. Diese Resultate können in niedrigdimensionalen Eichtheorien verallgemeinert werden. Weiterführend sollen die Ergebnisse als Grundlage einer experimentellen Realisierung durch Quantensimulationen dienen. Die entwickelten Methoden können auf andere Eichtheorien und verwandten Vielteilchenmodellen angewendet werden und bieten eine Plattform für weitere Ansätze.

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