Determination of Chern Number by Measurement of Spin Polarization of Spin-Orbit Coupled Ultracold Atoms in Optical Lattices

Designing feasible and qualitative methods for estimation of topological invariants, such as the Chern number, is of great significance for experimental realization of topological quantum matter. We first show that for fermionic or bosonic two-band systems supported on Bravais lattices with 2n-fold rotation symmetry, the Chern number (mod 2n) of one of the energy bands can be inferred from the pseudo-spin polarization of the band wavefunction at the high-symmetry crystal momenta in the Brillouin zone. We leverage this result to design an experimental scheme for ultracold atomic gases that uses Zeeman spectroscopy for validation of the topological phases with Chern number 2 on a triangular lattice. For bosons, this needs to be preceded by a timed Bloch oscillation. As part of the latter result, we present the first experimental scheme to our knowledge for simulating spin-orbit coupling on a triangular optical lattice. Our results highlight Bloch oscillations and Zeeman spectroscopy as robust tools for the detection of topological order, and also open doors to new experiments based on spin-orbit coupled systems beyond square lattices.