Vison crystals, chiral and crystalline topological phases in Yao-Lee model

We perform a variational analysis in single and bilayer Yao-Lee model as a function of the external magnetic field (B) and Dzyaloshinskii−Moriya interaction (DMI). We uncover a rich phase diagram where a variety of vison crystals are stabilized. For in-plane B, these phases are mostly gapless and gapped for out-of-plane B. The gapped phases are further classified by a Chern number that determines the number of chiral edge modes. We also elucidate crystalline topological phases protected by mirror symmetry.