Proposal for realizing and probing topological crystalline insulators in optical lattices

We develop a lattice model which exhibits topological transitions from $Z_2$ topological insulators to mirror symmetry-protected topological crystalline insulators by introducing additional spin-orbit coupling terms. The topological phase is characterized by the mirror winding number, defined within the mirror symmetry invariant subspace, which ensures the protection of gapless edge states and zero-energy corner states under specific boundary conditions. Additionally, we propose a feasible scheme using ultracold atoms confined in a stacked hexagonal optical lattice with Raman fields to realize the two-dimensional topological crystalline insulators. Detection of the mirror winding number in these systems can be achieved by implementing a simple quench sequence and observing the evolution of the time-of-flight patterns.