Proposal for detecting degenerate bands with topological invariants in optical lattices

In this paper, we present an experimental approach for simulating and detecting topological invariants using ultracold fermions confined in two-dimensional hexagonal optical lattices. We propose achieving degenerate four-band models with nontrivial 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 topological property of the valence bands. Through the analysis of tomographic results, we are able to extract and calculate the spin Chern number. Additionally, we demonstrate the robustness of the quantized topological invariants and discuss the effects of various experimental parameters.