1. 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 solid-state systems, as the atomic degrees of freedom couple to phonons of the ion lattice, thereby emulating a solid-state system. Starting from the microscopic many-body Hamiltonian, we derive the low energy Hamiltonian including the atomic band structure and give an expression for the atom-phonon coupling. We discuss possible experimental implementations such as a Peierls-like transition into a period-doubled dimerized state.
2. We consider a spinful and time-reversal invariant version of the Hofstadter problem which can be realized in ultracold atoms. In these experiments, an additional staggered potential and spin-orbit coupling are available. Without interactions, the system exhibits various phases such as topological and normal insulator, metal as well as semi-metal phases with two or even more Dirac cones. Using a combination of real-space dynamical mean-field theory and analytical techniques, we discuss the effect of on-site interactions and determine the corresponding phase diagram. Specifically, we investigate the stability of topological insulator phases in the presence of strong interactions.
3. We study Bose-Bose mixtures on a triangular lattice, where geometric frustration arises for asymmetric hopping. We map out a rich ground state phase diagram including xy-ferromagnetic, spin-density wave, superfluid, and supersolid phases.