  We consider a two-dimensional homogeneous ensemble of cold bosonic atoms
loaded inside two optical cavities and pumped by a far-detuned external laser
field. We examine the conditions for these atoms to self-organize into
triangular and honeycomb lattices as a result of superradiance. By collectively
scattering the pump photons, the atoms feed the initially empty cavity modes.
As a result, the superposition of the pump and cavity fields creates a
space-periodic light-shift external potential and atoms self-organize into the
potential wells of this optical lattice. Depending on the phase of the cavity
fields with respect to the pump laser, these minima can either form a
triangular or a hexagonal lattice. By numerically solving the dynamical
equations of the coupled atom-cavity system, we have shown that the two stable
atomic structures at long times are the triangular lattice and the honeycomb
lattice with equally-populated sites. We have also studied how to drive atoms
from one lattice structure to another by dynamically changing the phase of the
cavity fields with respect to the pump laser.
