
 Phase transitions of the coherently coupled twocomponent Bose gas in a square optical lattice (arXiv:1705.02833)We investigate properties of an ultracold, twocomponent bosonic gas in a square optical lattice at unit filling. In addition to densitydensity interactions, the atoms are subject to coherent lightmatter interactions that couple different internal states. We examine the influence of this coherent coupling on the system and its quantum phases by using Gutzwiller mean field theory as well as bosonic dynamical mean field theory. We find that the interplay of strong interspecies repulsion and coherent coupling affects the Mott insulator to superfluid transition and shifts the tip of the Mott lobe toward higher values of the tunneling amplitude. In the strongly interacting Mott regime, the resulting BoseHubbard model can be mapped onto an effective spin Hamiltonian that offers additional insights into the observed phenomena. 
 Spectral functions of a timeperiodically driven FalicovKimball model: realspace Floquet DMFT study (arXiv:1704.03250)We present a systematic study of spectral functions of a timeperiodically driven FalicovKimball Hamiltonian. In the highfrequency limit, this system can be effectively described as a HarperHofstadterFalicovKimball model. Using realspace Floquet dynamical meanfield theory (DMFT), we take into account interaction effects and contributions from higher Floquet bands in a nonperturbative way. Our calculations show a high degree of similarity between the interacting driven system and its effective static counterpart with respect to spectral properties. However, as also illustrated by our results, one should bear in mind that Floquet DMFT describes a nonequilibrium steady state (NESS), while an effective static Hamiltonian describes an equilibrium state. We further demonstrate the possibility of using realspace Floquet DMFT to study edge states on a cylinder geometry. 
 Breaking of SU(4) symmetry and interplay between stronglycorrelated phases in the Hubbard model (arXiv:1612.06258,Phys. Rev. B 95, 125108(2017))We study thermodynamic properties of fourcomponent fermionic mixtures described by the Hubbard model using the dynamical meanfield theory approach. Special attention is given to the system with SU(4)symmetric interactions at half filling, where we analyze equilibrium manybody phases and their coexistence regions at nonzero temperature for the case of simple cubic lattice geometry. We also determine the evolution of observables in lowtemperature phases while lowering the symmetry of the Hamiltonian towards the twoband Hubbard model. This is achieved by varying interflavor interactions or by introducing the spinflip term (Hund's coupling). By calculating the entropy for different symmetries of the model we determine the optimal regimes for approaching the studied phases in experiments with ultracold alkali and alkalineearthlike atoms in optical lattices. 
 The infinite occupation number basis of bosons  solving a numerical challenge (arXiv:1611.10185)In any bosonic lattice system, which is not dominated by local interactions and thus "frozen" in a Motttype state, numerical methods have to cope with the infinite size of the corresponding Hilbert space even for finite lattice sizes. While it is common practice to restrict the local occupation number basis to Nc lowest occupied states, the presence of a finite condensate fraction requires the complete number basis for an exact representation of the manybody ground state. In this work we present a novel truncation scheme to account for contributions from higher number states. By simply adding a single \textit{coherenttail} state to this common truncation, we demonstrate increased numerical accuracy and the possible increase in numerical efficiency of this method for the Gutzwiller variational wave function and within dynamical meanfield theory. 
 Operatorbased derivation of phonon modes and characterization of correlations for trapped ions at zero and finite temperaturel (arXiv:1608.07235, Phys. Rev. B 94, 214305(2016))We present a selfcontained operatorbased approach to derive the spectrum of trapped ions. This approach provides the complete normal form of the lowenergy quadratic Hamiltonian in terms of bosonic phonons, as well as an effective freeparticle degree of freedom for each spontaneously broken spatial symmetry. We demonstrate how this formalism can directly be used to characterize an ion chain both in the linear and the zigzag regimes. In particular, we compute, both for the ground state and finite temperature states, spatial correlations, heat capacity, and dynamical susceptibility. Last, for the ground state, which has quantum correlations, we analyze the amount of energy reduction compared to an uncorrelated state with minimum energy, thus highlighting how the system can lower its energy by correlations. 
 Interaction effects have been a subject of contemporary interest in topological phases of matter. But in the presence of interactions, the accurate determination of topological invariants in their general form is difficult due to their dependence on multiple integrals containing Green's functions and their derivatives. Here we employ the recently proposed "effective topological Hamiltonian" approach to explore interactioninduced topological phases in the timereversalinvariant HofstadterHubbard model. Within this approach, the zerofrequency part of the selfenergy is sufficient to determine the correct topological invariant. We combine the topological Hamiltonian approach with the local selfenergy approximation within HartreeFock and dynamical mean field theory (DMFT), and present the resulting phase diagram in the presence of manybody interactions. We investigate the emergence of quantum spin Hall (QSH) states for different interaction strengths by calculating the Z2 invariant. The interplay of strong correlations and a staggered potential also induces magnetic longrange order with an associated first order transition. We present results for the staggered magnetisation (m_{s}), staggered occupancy (n_{s}) and double occupancy across the transition. 
 In this work we study the formation and dynamics of polarons in a system with a few impurities in a lattice immersed in a BoseEinstein condensate (BEC). This system has been experimentally realized using ultracold atoms and optical lattices. Here we consider a twoband model for the impurity atoms, along with a Bogoliubov approximation for the BEC, with phonons coupled to impurities via both intra and interband transitions. We decouple this Fröhlichlike term by an extended twoband LangFirsov polaron transformation using a variational method. The new effective Hamiltonian with two (polaron) bands differs from the original Hamiltonian by modified coherent transport, polaron energy shifts and induced longrange interaction. A Lindblad master equation approach is used to take into account residual incoherent coupling between polaron and bath. This polaronic treatment yields a renormalized interband relaxation rate compared to Fermi's Golden Rule. For a strongly coupled twoband Fröhlich Hamiltonian, the polaron is tightly dressed in each band and can not tunnel between them, leading to an interband selftrapping effect. 
 Condensation versus Longrange Interaction: Competing Quantum Phases in Bosonic Optical Lattice Systems at Nearresonant Rydberg Dressing (arXiv:1509.06292)Recent experiments have shown that (quasi)crystalline phases of Rydbergdressed quantum manybody systems in optical lattices (OL) are within reach. While conventional neutral atomic OL gases lack strong longrange interactions, these arise naturally in Rydberg systems, due to the large polarizability of Rydberg atoms. Combined with the bosonic character of the systems considered in our work, a wide range of quantum phases have been predicted. Among them are a devil's staircase of lattice incommensurate density wave phases as well as more exotic supersolid lattice order. High experimental tunability opens up a wide range of parameters to be studied. Guided by results in the "frozen" gas limit, we study the ground state phase diagram at finite hopping amplitudes and in the vicinity of resonant Rydberg driving. Simulations within realspace bosonic dynamical meanfield theory (RBDMFT) yield an extension of the devil's staircase into the supersolid regime where the competition of condensation and interaction leads to a sequence of crystalline phases. 
 Effects of anisotropy in simple lattice geometries on manybody properties of ultracold fermions in optical lattices (arXiv:1505.02733, Phys. Rev. A 92, 043623 (2015))We study the effects of anisotropic hopping amplitudes on quantum phases of ultracold fermions in optical lattices described by the repulsive FermiHubbard model. In particular, using dynamical meanfield theory (DMFT) we investigate the dimensional crossover between the isotropic square and the isotropic cubic lattice. We analyze the phase transition from the antiferromagnetic to the paramagnetic state and observe a significant change in the critical temperature: Depending on the interaction strength, the anisotropy can lead to both a suppression or increase. We also investigate the localization properties of the system, such as the compressibility and double occupancy. Using the local density approximation in combination with DMFT we conclude that density profiles can be used to detect the mentioned anisotropydriven transitions. 
 Recent experiments in ultracold atoms and photonic analogs have reported the implementation of artificial gauge fields in lattice systems, facilitating the realization of topological phases. Motivated by such advances, we investigate the Haldane honeycomb lattice tightbinding model, for bosons with local interactions at the average filling of one boson per site. We analyze the groundstate phase diagram and uncover three distinct phases: a uniform superfluid (SF), a chiral superfluid (CSF), and a plaquette Mott insulator with local current loops (PMI). Nearestneighbor and nextnearestneighbor currents distinguish CSF from SF, and the phase transition between them is first order. We apply bosonic dynamical meanfield theory and exact diagonalization to obtain the phase diagram, complementing numerics with calculations of excitation spectra in strong and weak coupling perturbation theory. The characteristic density fluctuations, current correlation functions, and excitation spectra are measurable in ultracold atom experiments.

 We investigate the groundstate properties of BoseBose mixtures with Rashbatype spinorbit (SO) coupling in a square lattice. The system displays rich physics from the deep Mott insulator (MI) all the way to the superfluid (SF) regime. In the deep MI regime, exotic spinordered phases arise due to the effective DzyaloshinskiiMoriya type of superexchange interactions. By employing the nonperturbative bosonic dynamical meanfield theory (BDMFT), we numerically study and establish the stability of these magnetic phases against increasing hopping amplitude. We show that as hopping is increased across the MI to SF transition, exotic superfluid phases with magnetic textures emerge. In particular, we identify an exotic spinspiral magnetic texture with spatial period 3 in the superfluid close to the MISF transition. 
 QuasiParticle Theory for the Higgs Amplitude Mode (arXiv:1401.4466)We present a generalized quasiparticle theory for bosonic lattice systems, which naturally contains all relevant collective modes, including the Higgs amplitude in the strongly correlated superfluid. It provides a systematic framework for efficiently calculating observables beyond the Gutzwiller approximation and for including external perturbations, as well as higher order decay and interactions in terms of quasiparticle operators. It allows for the construction of an alternative path integral approach in terms of quasiparticle coherent states. 
 We create an artificial graphene system with tunable interactions and study the crossover from metallic to Mott insulating regimes, both in isolated and coupled twodimensional honeycomb layers. The artificial graphene consists of a twocomponent spin mixture of an ultracold atomic Fermi gas loaded into a hexagonal optical lattice. For strong repulsive interactions, we observe a suppression of double occupancy and measure a gapped excitation spectrum. We present a quantitative comparison between our measurements and theory, making use of a novel numerical method to obtain Wannier functions for complex lattice structures. Extending our studies to timeresolved measurements, we investigate the equilibration of the double occupancy as a function of lattice loading time. 
 We propose and theoretically investigate a hybrid system composed of a crystal of trapped ions coupled to a cloud of ultracold fermions. The ions form a periodic lattice and induce a band structure in the atoms. This system combines the advantages of scalability and tunability of ultracold atomic systems with the high fidelity operations and detection offered by trapped ion systems. It also features close analogies to natural solidstate systems, as the atomic degrees of freedom couple to phonons of the ion lattice, thereby emulating a solidstate system. Starting from the microscopic manybody Hamiltonian, we derive the low energy Hamiltonian including the atomic band structure and give an expression for the atomphonon coupling. We discuss possible experimental implementations such as a Peierlslike transition into a perioddoubled dimerized state. 
 Motivated by the recent progress in engineering artificial nonAbelian gauge fields for ultracold fermions in optical lattices, we investigate the timereversalinvariant HofstadterHubbard model. We include an additional staggered lattice potential and an artificial Rashbatype spinorbit coupling term available in experiment. Without interactions, the system can be either a (semi)metal, a normal or a topological insulator, and we present the nonAbelian generalization of the Hofstadter butterfly. Using a combination of realspace dynamical meanfield theory (RDMFT), analytical arguments, and MonteCarlo simulations we study the effect of strong onsite interactions. We determine the interacting phase diagram, and discuss a scenario of an interactioninduced transition from normal to topological insulator. At halffilling and large interactions, the system is described by a quantum spin Hamiltonian, which exhibits exotic magnetic order due to the interplay of Rashbatype spinorbit coupling and the artificial timereversalinvariant magnetic field term. We determine the magnetic phase diagram: both for the itinerant model using RDMFT and for the corresponding spin model in the classical limit using MonteCarlo simulations. 
 We theoretically investigate the thermodynamics of an interacting inhomogeneous twocomponent Fermi gas in an optical lattice. Motivated by a recent experiment by L. Hackerm\"uller et al., Science, 327, 1621 (2010), we study the effect of the interplay between thermodynamics and strong correlations on the size of the fermionic cloud. We use dynamical meanfield theory to compute the cloud size, which in the experiment shows an anomalous expansion behavior upon increasing attractive interaction. We confirm this qualitative effect but, assuming adiabaticity, we find quantitative agreement only for weak interactions. For strong interactions we observe significant nonequilibrium effects which we attribute to a dynamical arrest of the particles due to increasing correlations. 
 We investigate the zero temperature quantum phases of a BoseBose mixture on a triangular lattice using Bosonic Dynamical Mean Field Theory (BDMFT). We consider the case of total filling one where geometric frustration arises for asymmetric hopping. We map out a rich ground state phase diagram including xyferromagnetic, spindensity wave, superfluid, and supersolid phases. In particular, we identify a stripe spindensity wave phase for highly asymmetric hopping. On top of the spindensity wave, we find that the system generically shows weak charge (particle) density wave order. 
 We numerically simulate strongly correlated ultracold bosons in a highfinesse optical cavity by means of Bosonic Dynamical Mean Field Theory. The complete phase diagram is established, which contains two novel selforganized quantum phases, supersolid and checkerboard solid, in addition to conventional phases such as superfluid and Mott insulator. At finite but low temperature, thermal fluctuations are found to enhance the buildup of the selforganized phases. We demonstrate that cavitymediated longrange interactions can give rise to stable supersolid and checkerboard solid phases even in the regime of strong swave scattering. In the presence of a harmonic trap, we discuss coexistence of these selforganized phases, as relevant to experiment. 
 We consider a spinful and timereversal invariant version of the Hofstadter problem which can be realized in cold atom experiments. In these experiments, an additional staggered potential and a Rashbatype hopping are available. Without interactions, the system exhibits various phases such as topological and normal insulator, metal as well as semimetal phases with two or even more Dirac cones. Using a combination of realspace dynamical meanfield theory and analytical techniques, we discuss the effect of onsite interactions and determine the corresponding phase diagram. In particular, we investigate the semimetal to antiferromagnetic insulator transition and the stability of different topological insulator phases in the presence of strong interactions. We compute spectral functions which allow us to study the edge states of the strongly correlated topological phases.  
 Since the experimental realization of synthetic gauge fields for neutral atoms, the simulation of topologically nontrivial phases of matter with ultracold atoms has become a major focus of cold atom experiments. However, several obvious differences exist between cold atom and solid state systems, for instance the finite size of the atomic cloud and the smooth confining potential. In this article we show that sharp boundaries are not required to realize quantum Hall or quantum spin Hall physics in optical lattices and, on the contrary, that edge states which belong to a smooth confinement exhibit additional interesting properties, such as spatially resolved splitting and merging of bulk bands and the emergence of robust auxiliary states in bulk gaps to preserve the topological quantum numbers. In addition, we numerically validate that these states are robust against disorder. Finally, we analyze possible detection methods, with a focus on Bragg spectroscopy, to demonstrate that the edge states can be detected and that Bragg spectroscopy can reveal how topological edge states are connected to the different bulk bands.  
 Advantages of massimbalanced ultracold fermionic mixtures for approaching quantum magnetism in optical lattices (arXiv:1203.4658, Phys. Rev. Lett. 109, 205303 (2012))We study magnetic phases of twocomponent mixtures of ultracold fermions with repulsive interactions in optical lattices in the presence of hopping imbalance. Our analysis is based on dynamical meanfield theory (DMFT) and its realspace generalization at finite temperature. We study the temperature dependence of the transition into the ordered state as a function of the interaction strength and the imbalance parameter in two and three spatial dimensions. We show that below the critical temperature for Neel order massimbalanced mixtures also exhibit a chargedensity wave, which provides a directly observable signature of the ordered state. For the trapped system, we compare our results obtained by realspace DMFT to a localdensity approximation. We calculate the entropy for a wide range of parameters and identify regions, in which massimbalanced mixtures have clear advantages over balanced mixtures for the purpose of obtaining and detecting quantum magnetism.  
 We theoretically investigate finitetemperature thermodynamics and demagnetization cooling of twocomponent BoseBose mixtures in a cubic optical lattice, by using bosonic dynamical mean field theory (BDMFT). We calculate the finitetemperature phase diagram, and remarkably find that the system can be heated from the superfluid into the Mott insulator at low temperature, analogous to the Pomeranchuk effect in 3He. This provides a promising manybody cooling technique. We examine the entropy distribution in the trapped system and discuss its dependence on temperature and an applied magnetic field gradient. Our numerical simulations quantitatively validate the spingradient demagnetization cooling scheme proposed in recent experiments.  
 AndersonHubbard model with box disorder: Statistical dynamical meanfield theory investigation (arXiv:1106.4028, Phys. Rev. B 84, 115113 (2011))Strongly correlated electrons with box disorder in highdimensional lattices are investigated. We apply the statistical dynamical meanfield theory, which treats local correlations nonperturbatively. The incorporation of a finite lattice connectivity allows for the detection of disorderinduced localization via the probability distribution function of the local density of states. We obtain a complete paramagnetic ground state phase diagram and find correlationinduced as well as disorderinduced metalinsulator transitions. Our results qualitatively confirm predictions obtained by typical medium theory. Moreover, we find that the probability distribution function of the local density of states in the metallic phase strongly deviates from a lognormal distribution as found for the noninteracting case.  
 Effective multibody induced tunneling and interactions in the BoseHubbard model of the lowest dressed band of an optical lattice (arXiv:1108.6047)We construct the effective lowestband BoseHubbard model incorporating interactioninduced onsite correlations. The model is based on ladder operators for local correlated states, which deviate from the usual Wannier creation and annihilation, allowing for a systematic construction of the most appropriate singleband lowenergy description in form of the extended BoseHubbard model. A formulation of this model in terms of ladder operators not only naturally contains the previously found effective multibody interactions, but also contains multibody induced single particle tunneling, pair tunneling and nearestneighbor interaction processes of higher orders. An alternative description of the same model can be formulated in terms of occupationdependent BoseHubbard parameters. These multiparticle effects can be enhanced using Feshbach resonances, leading to corrections which are well within experimental reach and of significance to the phase diagram of ultracold bosonic atoms in an optical lattice. We analyze the energy reduction mechanism of interacting atoms on a local lattice site and show that this cannot be explained only by a spatial broadening of Wannier orbitals on a single particle level, which neglects correlations.  
 We theoretically address magnetic ordering at zero and finite temperature in both homogeneous and trapped BoseBose mixtures in optical lattices. By using Bosonic Dynamical MeanField Theory, we obtain the phase diagram of the homogeneous twocomponent BoseHubbard model in a threedimensional (3D) cubic lattice, which features competing magnetic order of XYferromagnetic and antiferromagnetic type in addition to the Mott and superfluid states. We show that these magnetic phases persist also in the presence of a harmonic trap.  
 We study the physics of a threecomponent Fermi gas in an optical lattice, in the presence of a strong threebody constraint arising due to threebody loss. Using analytical and numerical techniques, we show that an atomic color superfluid phase is formed in this system and undergoes phase separation between unpaired fermions and superfluid pairs. This phase separation survives well above the critical temperature, giving a clear experimental signature of the threebody constraint.  
 We study a system of ultracold fermionic polar molecules in a twodimensional square lattice interacting via both the longranged dipoledipole interaction and a shortranged onsite attractive interaction. Singlet superfluid, charge density wave, and supersolid phases are found to exist in the system. We map out the zero temperature phase diagram and find that the supersolid phase is considerably stabilized by the dipoledipole interaction and thus can exist over a large region of filling factors. We study the melting of the supersolid phase with increasing temperature, map out a finite temperature phase diagram of the system at fixed filling, and determine the parameter region where the supersolid phase can possibly be observed in experiments.  
 We report the first detection of the Higgstype amplitude mode using Bragg spectroscopy in a strongly interacting condensate of ultracold atoms in an optical lattice. In contrast to the Bogoliubov sound mode, the amplitude mode (which is an additional collective mode) only appears in the presence of a lattice. By the comparison of our experimental data with a spatially resolved, timedependent dynamic Gutzwiller calculation, we obtain good quantitative agreement. This allows for a clear identification of the amplitude mode, showing that it can be detected with full momentum resolution by going beyond the linear response regime. A systematic shift of the sound and amplitude modes' resonance frequencies due to the finite Bragg beam intensity is observed.  
 We study the phase diagram of an SU(3)symmetric mixture of threecomponent ultracold fermions with attractive interactions in an optical lattice, including the additional effect on the mixture of an effective threebody constraint induced by threebody losses.We address the properties of the system in D > 2 by using dynamical meanfield theory and variational Monte Carlo techniques. The phase diagram of the model shows a strong interplay between magnetism and superfluidity. In the absence of the threebody constraint (no losses), the system undergoes a phase transition from a color superfluid (cSF) phase to a trionic phase, which shows additional particle density modulations at halffilling. Away from the particle–hole symmetric point the cSF phase is always spontaneously magnetized, leading to the formation of different cSF domains in systems where the total number of particles of each species is conserved. This can be seen as the SU(3) symmetric realization of a more general tendency for phase separation in threecomponent Fermi mixtures. The threebody constraint strongly disfavors the trionic phase, stabilizing a (fully magnetized) cSF also at strong coupling. With increasing temperature we observe a transition to a nonmagnetized SU(3) Fermi liquid phase.  
 Creating exotic condensates via quantum phase revival dynamics in engineered lattice potentials (arXiv:1012.5100, Phys. Rev. A 84, 023631 (2011))
In the field of ultracold atoms in optical lattices a plethora of phenomena governed by the hopping energy J and the interaction energy U have been studied in recent years. However, the trapping potential typically present in these systems sets another energy scale and the effects of the corresponding time scale on the quantum dynamics have rarely been considered. Here we study the quantum collapse and revival of a lattice BoseEinstein condensate in an arbitrary spatial potential, focusing on the special case of harmonic confinement. Analyzing the time evolution of the single particle density matrix, we show that the physics arising at the (temporally) recurrent quantum phase revivals is essentially captured by an effective single particle theory. This opens the possibility to prepare exotic nonequilibrium condensate states with a large degree of freedom by engineering the underlying spatial lattice potential.  
 We investigate a BoseFermi mixture in a threedimensional optical lattice, trapped in a harmonic potential. Using generalized dynamical meanfield theory, which treats the BoseBose and BoseFermi interaction in a fully nonperturbative way, we show that for experimentally relevant parameters a peak in the condensate fraction close to the point of vanishing BoseFermi interaction is reproduced within a singleband framework. We identify two physical mechanisms contributing to this effect: the spatial redistribution of particles when the interspecies interaction is changed and the reduced phase space for strong interactions, which results in a higher temperature at fixed entropy.  
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