  The Yao-Lee model is an example of exactly solvable spin-orbital models that
are generalizations of the original Kitaev honeycomb model with extra local
orbital degrees of freedom. Similar to the Kitaev model, both spin and orbital
degrees of freedom are effectively represented using sets of three-flavored
Majorana fermions. The Yao-Lee model exhibits a quantum spin liquid ground
state with gapped and immobile $Z_2$ fluxes and three-fold degenerate itinerant
Majorana fermions. Our work demonstrated that by introducing different
time-reversal symmetry (TRS) breaking fields one can split the degeneracy of
Majorana fermions and close the gap for some of the bands, thus changing its
topology. We calculated a comprehensive topological phase diagram for the
Yao-Lee model by considering various combinations of TRS-breaking fields. This
investigation revealed the emergence of distinct topological regions, each
separated by nodal lines, signifying an evolution in the model's topological
properties. We also investigated the impact of vacancies in the system. Our
findings revealed that while vacancies modify the low-energy spectrum of the
model, their presence has a limited impact on the topological properties of the
model, at least for small enough concentrations.
