When we work on the topological insulator (protected by time reversal symmetry ), it is often said that the SOC is the main ingredient because it is a way to open a gap when the band inversion present. I want to ask what is the mechanism cause the gap to be opened ? More precisely, I found that SOC will cause the splitting of energy band. Let’s say if the system contain both time reversal and inversion symmetry, each energy band must be 2-fold degenerate. However the SOC term don’t violate these symmetries, where is the splitting come from ?