Dirac and topological phonons with spin-orbital entangled orders

We propose to study novel quantum phases and excitations for a 2D spin-orbit (SO) coupled bosonic $p$-orbital optical lattice based on the recent experiments. The orbital and spin degrees of freedom with SO coupling compete and bring about nontrivial interacting quantum effects. We develop a self-consistent method for bosons and predict a spin-orbital entangled order for the ground phase, in sharp contrast to spinless high-orbital systems. Furthermore, we investigate the Bogoliubov excitations, showing that the Dirac and topological phonons are obtained corresponding to the predicted different spin-orbital orders. In particular, the topological phonons exhibit a bulk gap which can be several times larger than the single-particle gap of $p$-bands, reflecting the enhancement of topological effect by interaction. Our results highlight the rich physics predicted in SO coupled high-orbital systems and shall attract experimental efforts in the future.