Generating and detecting topological phases with higher Chern number

Topological phases with broken time-reversal symmetry and Chern number |C|>=2 are of fundamental interest, but it remains unclear how to engineer the desired topological Hamiltonian within paradigm spin-orbit-coupled particles hopping only between nearest neighbours a static lattice. We show that higher arise when spin-orbit coupling satisfies combination spin spatial rotation symmetries. leverage this result both construct minimal two-band tight binding Hamiltonians exhibit |C|=2,3 phases, one energy bands can be inferred from particle polarization at high-symmetry crystal momenta in Brillouin zone. Using these insights, we provide detailed experimental scheme for specific realization time-reversal-breaking phase |C|=2 ultracold atomic gases on triangular lattice subject coupling. The directly measured using Zeeman spectroscopy; fermions amplitudes via time flight, while bosons is preceded by short Bloch oscillation. Our results pathway detection novel gases.