  In this paper, we present a novel experimental approach for simulating and
detecting topological invariants using ultracold fermions confined in
two-dimensional hexagonal optical lattices. We propose achieving two-fold
degenerate four-band models with non-trivial topologies in both the AII and A
classes by introducing additional inertial forces, Raman processes, or periodic
driving. By implementing various quench sequences and observing the evolution
of the time-of-flight pattern, we can gather comprehensive information about
the ground states and determine the topology of the valence bands. Through the
analysis of tomographic results, we are able to extract and calculate the spin
Chern number for both spin-conserving and spin-nonconserving cases.
Additionally, we demonstrate the robustness of the quantized topological
invariants and discuss the effects of various experimental parameters.
