Measurement of the inter-band Berry connection in honeycomb and kagome optical lattices

The optical response of solid-state materials is related to the topological classification of their band structures. Multigap topology is an active area of study into topological invariants, such as the patch Euler class, which can be determined from the inter-band Berry connection. Here, we use a degenerate Fermi gas of spin-polarized ⁴⁰K atoms to measure the Berry connection between the ground and excited bands in honeycomb and kagome optical lattices. The ground band is filled with ⁴⁰K and perturbed by shaking the lattice, equivalent to optical excitations in a solid-state material. Diffraction and band mapping images are used to measure the magnitude of excitation at all quasimomenta within the first Brillouin zone. Excitations from linear and circular shaking allow us to extract the x and y components of the Berry connection as well as their relative phase. We infer the existence of a Dirac string from the direction of the Berry connection around the K-point in a honeycomb lattice. These methods will be used to characterize the nontrivial topology of the kagome ground band manifold by measuring the Euler class of the 2^nd and 3^rd bands, it could also be used for braiding of band touching points.