  The topological properties of non-Hermitian Hamiltonian is a hot topic, and
the theoretical studies along this research line are usually based on the
two-level non-Hermitian Hamiltonian (or, equivalently, a spin-$1/2$
non-Hermitian Hamiltonian). We are motivated to study the geometrical phases of
a three-level Lieb lattice model (or, equivalently, a spin-$1$ non-Hermitian
Hamiltonian) with the complex hopping and flat band in the context of a
polariton condensate, with the emphasis on the higher spin degree of freedom on
topological properties of non-Hermitian Hamiltonian. The topological invariants
are calculated by both winding numbers in the Brillouin zone and the
geometrical phase of Majorana stars in the Bloch sphere. Besides, we provide an
intuitive way to study the topological phase transformation in high dimensions,
and the flat band offers a platform to define the high spin topological phase
transition on the Bloch sphere. According to the trajectories of the Majorana
stars, we calculate the geometrical phases of the Majorana stars, and we find
they have a jump when the parameters change from the trivial phase to the
topological phase. Besides, the correlation phase of Majorana stars will rise
along with the increase of the imaginary parts of the hopping energy.
