Proposal for implementing Stiefel-Whitney insulators in an optical Raman lattice

The Stiefel-Whitney insultor is a two-dimensional topological insulator protected by parity-time ($\mathrm{PT}$) symmetry. With a vanishing Chern number, the topology in this system is characterized by second Stiefel-Whitney class. We propose a feasible scheme to realize a four-band Stiefel-Whitney insultor with spin-orbit coupled ultracold atoms in an optical Raman lattice. Four selected spin states are coupled by carefully designed Raman lasers to generate the desired spin-orbit interactions with spacetime inversion symmetry. We map out a phase diagram with respect to the experimental parameters, where a large topological phase region exists. We further present two distinct detection methods to resolve the non-Abelian band topology, in both equilibrium and dynamical ways. The detection relies on the spin textures extracted from the time-of-flight imaging, showing the tomographic signatures in the ground states and long-time averaged patterns on certain submanifolds via a bulk-surface duality. Our work paves a realistic way to explore novel topology inside real Berry bundles with quantum matters.