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Hofstadter-Hubbard physics
(2020)

The Hofstadter model, besides the Haldane and Kane-Mele models, is the most common tight-binding model which hosts topologically nontrivial states of matter. In its time-reversal-symmetric formulation the model can even describe topological insulators. Experimentally, the Hofstadter model was realized with ultracold quantum gases in optical lattices which is a wellcontrolled way to engineer quantum states of tight-binding Hamiltonians. Another established control parameter in ultracold quantum gases are twoparticle, on-site interactions, also known as Hubbard interactions. This work aims at introducing the reader to the concepts of topological states of matter, a collection of corresponding tight-binding models, and the methodology to treat interacting topological states with dynamical mean-field theory.We present recent results for inhomogeneous, interacting systems, spinimbalanced magnetic systems, propose experimental detection methods, and extensions to three-dimensional topological states.

In the course of this thesis we discuss a certain kind of supersolid, the lattice-supersolid, which can be realized using quantum gases in an optical lattice trap. The lattice-supersolid, which simultaneously possesses off-diagonal and diagonal long-range order in its density matrix and also breaks the discrete translational symmetry of an underlying lattice, is induced by self-ordering of the gas due to strong long-range van der Waals interactions. In the considered scenario, the interactions are facilitated by the excitation of atomic Rydberg states, which exhibit enhanced van der Waals forces.
In the first part of this thesis (chapters 1-3), we review the relevant basics of quantum gases, Rydberg physics and introduce the extended Bose-Hubbard model. We start with the relevant methods and devices of the vast toolbox available in common quantum gas experiments, as well as consider the main concepts behind superfluidity and supersolidity. This is followed by an introduction of some basic concepts of Rydberg atoms in quantum many-body systems, with a focus on the facilitation of long-range interactions and the implementation in a theoretical model. Thereafter a brief introduction is given, on the realization of the Bose-Hubbard model in optical lattice systems and its extension to include Rydberg states, which concludes the introductory part of this thesis.
In the following part (chapters 4-6), we introduce the theoretical tools used to derive the results presented in the final part. First, an introduction to a real-space extension of bosonic dynamical mean-field theory (RB-DMFT) for bosonic systems with long-range interactions in the Hartree approximation is given. This method is based on the non-perturbative self-consistent evaluation of the lattice Green’s function, which also incorporates the effect of nearest neighbor correlations due to the non-condensed particles. Then we focus on a quasiparticle expansion of the Bose-Hubbard model, which has its foundation in linearized fluctuations of a static mean-field ground-state, allowing for the prediction of a vast range of experimentally relevant observables. Lastly, we introduce an efficient truncation scheme for the local bosonic Fock-basis, which allows for the simulation of phases with high condensate density at a vastly reduced computational effort.
In the final part (chapters 7 and 8), we discuss the application of both methods to itinerant bosonic gases in two-dimensional optical lattices, in order to predict the equilibrium ground-state phases, as well as the signatures of supersolidity and its formation in spectral functions and the dynamic and static structure factor. Specifically, we focus on two limiting cases. Firstly, we consider a two-component gas, as realized by two hyperfine ground states, for example, of rubidium-87, where one component is off-resonantly excited to a Rydberg state, which generates a soft-core shaped interaction potential. Secondly, we discuss the opposing limit, using near-resonant excitations
of Rydberg states, where the interacting component now directly corresponds to the Rydberg state, which interacts via a van der Waals potential. In both cases we discuss the rich variety of supersolid phases, which are found for a wide range of parameters. We also discuss how some of these phases can be realized in experiment.
In the subsequent appendices (A to D) we discuss some methodological details. Most notably, we consider the possible Fock-extension of the Hartree approximation (appendix A), introduced in the RB-DMFT treatment of the extended Bose-Hubbard model.

Magnetism is a beautiful example of a macroscopic quantum phenomenon. While known at least since the ancient Greeks, a microscopic theoretical explanation of magnetism could only be achieved with the advent of quantum mechanics at the beginning of the 20th century. Then it was understood that in a certain class of solids the famous Pauli exclusion principle leads to an effective interaction between the microscopic magnetic moments, i.e., the spins, which favors an ordered, and hence macroscopically magnetic, state. Nowadays, magnetic phenomena are used in a host of applications, and are especially relevant for information storage and processing technologies.
Despite the long history of the field, magnetic phenomena are still an active research topic. In particular, in the last decade the fields of spintronics and spin-caloritronics emerged, which manipulate the microscopic spins via charge and heat currents respectively. This opens new avenues to potential applications; including the possibility to use the magnetic spin degrees of freedom instead of charges as carriers of information, which could provide a number of advantages such as reduced losses and further miniaturization.
In this thesis we do not delve any further into the realm of possible applications. Instead we use sophisticated theories to explore the microscopic spin dynamics which is the basis of all such applications. We also focus on a particular compound: Yttrium-iron garnet (YIG), which is a ferrimagnetic insulator. This material has been widely used in experiments on magnetism over the last decades, and is a popular candidate for spintronic devices. Microscopically, the low-energy magnetic properties of YIG can be described by a ferromagnetic Heisenberg model. For spintronics and spin-caloritronics applications, it is however insufficient to only consider the magnetic degrees of freedom; one should also include the coupling of the spins to the elastic lattice vibrations, i.e., the phonons. Besides giving an overview on techniques used throughout the thesis, the introductory Ch. 1 provides a discussion of the microscopic Hamiltonian used to model the coupled spin-phonon system in the subsequent chapters.
The topic of Ch. 2 are the consequences of the magnetoelastic coupling on the low-energy magnon excitations in YIG. Starting from the microscopic spin-phonon Hamiltonian, we rigorously derive the magnon-phonon hybridization and scattering vertices in a controlled spin wave expansion. For the experimentally relevant case of thin YIG films at room temperature, these vertices are then used to compute the magnetoelastic modes as well as the magnon damping. In the course of this work, the damping of magnons in this system was also investigated experimentally using Brillouin light scattering spectroscopy. While comparison to the experimental data shows that the magnetoelastic interactions do not dominate the total magnon relaxation in the experimentally accessible regime, we are able to show that the spin-lattice relaxation time is strongly momentum dependent, thereby providing a microscopic explanation of a recent experiment.
In the final Ch. 3, we investigate a different phenomenon occurring in thin YIG films: Room temperature condensation of magnons. Prior work attributed this condensation process to quantum mechanics, i.e., it was interpreted as Bose-Einstein condensation. However, this is not satisfactory because at room temperature, the magnons in YIG behave as purely classical waves. In particular, the quantum Bose-Einstein distribution reduces to the classical Rayleigh-Jeans distribution in this case. In addition, the effective spin in YIG is very large. Therefore we start from the hypothesis that the room temperature magnon condensation is actually a new example of the kinetic condensation of classical waves, which has so far only been observed by imaging classical light in a photorefractive crystal. To distinguish this classical condensation from the quantum mechanical Bose-Einstein one, we refer to it as Rayleigh-Jeans condensation. To prove our claim, we consider the classical equations of motion of the coupled spin-phonon system. By eliminating the phonon degrees of freedom, we microscopically derive a non-Markovian stochastic Landau-Lifshitz-Gilbert equation (LLG) for the classical spin vectors. We then use this LLG to perform numerical simulations of the magnon dynamics, with all parameters fixed by experiments. These simulations accurately reproduce all stages of the magnon time evolution observed in experiments, including the appearance of the magnon condensate at the bottom of the magnon spectrum. In this way we confirm our initial hypothesis that the magnon condensation is a classical Rayleigh-Jeans condensation, which is unrelated to quantum mechanics.