  We present the appearance of nearly flat band states with nonzero Chern
numbers in a two-dimensional "diamond-octagon" lattice model comprising two
kinds of elementary plaquette geometries, diamond and octagon, respectively. We
show that the origin of such nontrivial topological nearly flat bands can be
described by a short-ranged tight-binding Hamiltonian. By considering an
additional diagonal hopping parameter in the diamond plaquettes along with an
externally fine-tuned magnetic flux, it leads to the emergence of such nearly
flat band states with nonzero Chern numbers for our simple lattice model. Such
topologically nontrivial nearly flat bands can be very useful to realize the
fractional topological phenomena in lattice models when the interaction is
taken into consideration. In addition, we also show that perfect band
flattening for certain energy bands, leading to compact localized states can be
accomplished by fine-tuning the parameters of the Hamiltonian of the system. We
compute the density of states and the wavefunction amplitude distribution at
different lattice sites to corroborate the formation of such perfectly flat
band states in the energy spectrum. Considering the structural homology between
a diamond-octagon lattice and a kagome lattice, we strongly believe that one
can experimentally realize a diamond-octagon lattice using ultracold quantum
gases in an optical lattice setting. A possible application of our lattice
model could be to design a photonic lattice using single-mode laser-induced
photonic waveguides and study the corresponding photonic flat bands.
