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