Measuring the Chern number with weakly interacting spin-orbit-coupled Bose gases in optical lattices

The quantification of topological invariants for tight-binding Hamiltonians with certain crystalline point group symmetries is well-studied, and in particular, the Chern number has been shown to be expressible in terms of the eigenvalues of the symmetry operator at the high-symmetry points of the Brillouin zone. In recent years, it has become possible to utilize this relation in cold atom experiments to measure the Chern number by loading spinor BECs with spin-orbit coupling onto 2D optical lattices. Spin polarization measurements are used to quantify the topological invariant, although so far it has been limited to Chern number 1. A recent experimental proposal based on Chern insulators with additional spin-space symmetries has paved the way for extracting higher Chern number with better resolution by performing polarization measurements only at those high-symmetry points for an ideal BEC. In this work, we go beyond the assumption that the BEC is non-interacting and study the effects of interactions that are weak enough not to destroy the condensation but can still be comparable or even stronger than the band gaps of the spin-orbit coupling Hamiltonian. Our framework is based on a mean-field treatment including fluctuations. Aside from quantifying the deviations in Chern number measurements from the non-interacting limit, we are able to investigate other interesting effects arising from the competition between the interactions, spin-orbit coupling, and the optical lattice potential under realistic experimental conditions.