Topology Colloquially, by a topological classification we mean that we can assign an integer property to quantum mechanical system which depends on its global strucuture only. In other words, no local probe or small local change to the system parameters can change this integer: it is topologically protected. Traditional phases that were classified in such a topological framework include the integer and fractional quantum Hall effects and the more recently discovered topological insulators and superconductors. All of these equillibrium systems have a finite gap between the ground state and the first excited state. This gap enables a controlled derivation of a topological field theory for the low energy behavior. From that perspective, gapless or driven systems seem not to be classifiable by a topological index. Recent developments have shown indications of the contrary, however. In one particular case of strongly interacting lattice bosons in a weak magnetic field we could show that thegapless superfluid phase is intersected by lines of topological transitions where the Hall conductivity jumps by an integer value. We plan to extend this exciting new direction of using standard tools known from the description of gapped topologic matter to the deeper understanding of gapless or driven quantum phases. Out of equilibrium The use of cold atoms in simulating interesting open questions of condensed matter physics has seen a tremendous success in recent years. One key aspect is their almost perfect isolation from the environment. What proofs to be an asset in implementing a “designed” Hamiltonian turns out to be a major obstruction to the investigation of the resulting physics. In our research we focus on the development of the theoretical framework for new investigative tools adapted to the cold atoms setup. We are mainly interested in strongly correlated lattice systems where we showed how to extract coherence porperties from a bosonic or fermionic Mott insulator. Moreover, we pointed out how the “Higgs mode” close to a quantum phase transition can be measured. Recent experiments verified our predictions. We make use of a broad range of analytical tools, however, if necessary we develop new approaches to tackle the manybody problems at hand. Another consequence of the high degree of isolation is the ability to study pure nonequilibrium dynamics. In the past we studied mainly onedimensional systems, in particular coupled tubes of interacting quantum liquids that undergo quantum quenches or slow LandauZener dynamics. We plan to extend these studies by investigating the nonequilibrium dynamics of strongly correlated systems, in particular their topological properties. We believe that by focusing on the peculiarities of new engineered quantum systems like cold atoms or microstructured solid state devices like NV centers in diamonds, coupled microwave cavities, etc., we can make new discoveries in a developing field of topological phases out of equilibrium. Frustrated lattices Spins fixedly arranged on lattices where not all interactions can be fulfilled simultaneously have been shown to host intriguing quantum phases . In our research we are interested in the behavior of itinerant particles on such frustrated lattices. Due to destructive interference resulting from the underlying frustration the motion of the particles can be completely quenched. Deprived of their kinetic energy, the behavior of the particles is dominated by more complex processes such as interactions or disorder. What new physics can emerge from such a situation is the main question we address with our research. Weakly interacting bosons tend to condense in the lowest single particle state. On frustrated lattices, like the kagome net, the bosons do not find such a simple low energy state. In the absence of interactions this leads to a localized valence bond solid. When interactions are turned on, the particles delocalize and condense, a delocalized state entirely stabilized by repulsive interactions. In addition, also disorder turns out to have the quite counterintuitive effect of delocalizing the eigenstates. We are currently working on trying to understand the combined effect of both disorder and interaction on frustrated lattices. This seems to be an intractable problem as the interplay of disorder and interaction is an outstanding open theoretical problem. However, the peculiar structure of the frustrated geometry has a chance to give way to a controlled approach. Our research is aimed at finding interesting new stable phases of matter. The recent developments in the design of complicated lattice structures for cold atoms further motivates our work. In particular it has been demonstrated that one can implement an optical kagome lattice.
