Direct Geometric Probe of Singularities in Band Structure of an Optical Honeycomb Lattice

In certain real-space lattice geometries, such as in some solid-state crystals, the associated band structure can exhibit band degeneracies. In special cases these degeneracies, or band “touching points,” are associated with a singular Bloch state manifold with unusual consequences for material and transport properties. So far, ultracold atoms in optical lattices have been used to characterize such points only indirectly, e.g., by detection of an Abelian Berry phase, and only at singularities with linear dispersion (Dirac points). We probe the band structure directly by evolving Bose-condensed atoms over trajectories in momentum space such that they pass through band touching points. We vary the angle between the incoming and the outgoing ray with respect to the touching point and use this technique to probe linear and quadratic band touching points of a honeycomb lattice. Measurements of the band populations after this transport scheme lets us identify the winding numbers of these singularities to be 1 and 2, respectively. Our work opens the study of quadratic band touching points in ultracold-atom quantum simulators, and also provides a novel method for probing other band geometry singularities.