Topological many-body phases of bosons and fermions in driven optical lattices

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Since the discovery of the quantum Hall effect, topology has emerged as a central concept for the classification of quantum matter, describing exotic quantum phases beyond the classification via symmetry breaking. The persistent interest is fueled by the discovery of a plethora of effects and by potential applications in metrology and topological quantum computation. Quantum simulation of topological matter using ultracold atoms in optical lattices has become successful using Floquet driving to engineer artificial gauge fields for charge-neutral particles, leading to the realization of paradigmatic topological models. The possibility to combine gauge fields with tunable and strong interactions is at the heart of realizing interacting topological matter such as fractional Chern insulators, and the excellent controllability of cold atom experiments promises to help establishing a comprehensive understanding of these phases.The Hamburg team pioneered e.g. Floquet driven gauge fields in optical lattices and realized for the first time a full momentum-resolved measurement of the Berry curvature, and within the Research Unit the first measurement of the Chern number via the dynamical linking number and via circular dichroism. Furthermore, the team introduced the use of machine learning techniques for an improved identification of topological phase transitions.In the second funding period, the Hamburg project will build on these techniques and will study the rich interplay of topology and interactions both in the weakly- and in the strongly-interacting regimes. We will prepare and characterize many-body phases of bosons and fermions in topological band structures both in equilibrium and via dynamics after quenches. We will implement a hexagonal superlattice, which is vital for the preparation of fractional fillings, but also allows studying topological three-band models in driven optical Kagome lattices. Furthermore, we will implement quasi-disorder and study the interplay with topology and interactions, expecting to observe both topological protection against disorder and 2d topological Anderson insulators. The Hamburg project will strongly collaborate with the theory projects and experimental projects of the Research Unit on both a direct comparison between experimental results and numerical calculations and on the identification of suitable protocols for realizing and characterizing exotic topological phases.