  The band structure of a crystal may have points where two or more bands are
degenerate in energy and where the geometry of the Bloch state manifold is
singular, with consequences for material and transport properties. 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). Here, we probe
band-structure singularities through the non-Abelian transformation produced by
transport directly through the singular points. We prepare atoms in one Bloch
band, accelerate them along a quasi-momentum trajectory that enters, turns, and
then exits the singularities at linear and quadratic touching points of a
honeycomb lattice. Measurements of the band populations after transport
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.
