  Existence of nontrivial topological phases in a tight binding Haldane-like
model on the depleted Lieb lattice is reported. This two-band model is
formulated by considering the nearest-neighbor, next-nearest-neighbor and
next-next-nearest-neighbor hopping terms along with complex phase which breaks
the time reversal symmetry of this semi-metallic system. Topological feature of
this model is studied along with the presence of sublattice symmetry breaking
staggered onsite energy. Combined effect of these two broken symmetries is
found crucial for an additional transition between nontrivial and trivial
phases. System exhibits two types of phase transitions, say, between two
nontrivial phases and nontrivial to trivial phases. Nonzero Chern numbers,
existence of Hall plateau and symmetry protected edge states confirm the
presence of the nontrivial phases. This two-band system hosts four different
types of phases where two are topological. Additionally topological properties
of stacked bilayer of the depleted Lieb lattices is also studied with similar
Haldane-like Hamiltonian. This four-band system is found to host Chern
insulating phases, with higher values of Chern numbers supported by in-gap edge
states.
