Tudor Petrescu

In the first part of the talk, we introduce two–dimensional lattice tight–binding models that realize the quantum anomalous Hall effect (QAHE). For a Kagome lattice whose degrees of freedom are photons in microwave resonators, we discuss protocols to access the local Berry curvature and the Chern number of Bloch bands from the semiclassical dynamics of wavepackets. We proceed to Haldane’s model for QAHE on the honeycomb lattice, but with repulsively interacting bosons at unit filling, and uncover a rich phase diagram containing a Mott insulator with plaquette currents, as well as chiral and normal superfluids. The elementary excitations in both models have nonzero topological invariants which can be probed with current experiments.

In Part II, we turn to quasi one–dimensional lattices in a uniform magnetic field, which have been recently realized with ultracold atoms. For bosons at low magnetic flux, the ground state can sustain the Meissner effect. The Meissner currents persist inside of a Mott insulating phase at half–filling. Whenever flux and density are commensurate, the ground state is a low–dimensional precursor of Laughlin’s trial state for the fractional quantum Hall effect at filling $\nu = 1/(2m)$. We then enumerate two–dimensional generalizations, and a variety of analogous topological phases of spinful fermions. We finally discuss observables that discern such states in current experiments, including local probes such as currents and particle number fluctuations.