  Photonic topological insulators supporting unidirectional topologically
protected edge states represent attractive platform for realization of
disorder- and backscattering-immune transport of edge excitations in both
linear and nonlinear regimes. In many realizations of topological insulators
structured periodic materials are used, since they may admit specific Dirac
degeneracy in the spectrum, around which unidirectional edge states appear
under the action of physical effects breaking time-reversal symmetry. While
properties of the edge states at unclosed interfaces of two bulk media with
different topology are known, the existence of the edge states in practical
finite-dimensional topological insulators fully immersed in nontopological
environment remains largely unexplored. In this work using as an example
realistic polariton topological insulators built from small-size honeycomb
arrays of microcavity pillars, we illustrate how topological properties of the
system build up upon gradual increase of its dimensionality. To account for
dissipative nature of polariton condensate forming in the array of microcavity
pillars, we consider the impact of losses and resonant pump leading to rich
bistability effects in this system. We describe the mechanism in accordance
with which trivial-phase pump "selects" and excites specific nonlinear
topological edge states circulating along the periphery of the structure in the
azimuthal direction dictated by the direction of the external applied magnetic
field. We also show the possibility of utilization of vortex pump with
different topological charges for selective excitation of different edge
currents.
