  We investigate theoretically a topological vortex phase transition induced by
a superradiant phase transition in an atomic Bose-Einstein condensate driven by
a Laguerre-Gaussian optical mode. We show that superradiant radiation can
either carry zero angular momentum, or be in a rotating Laguerre-Gaussian mode
with angular momentum. The conditions leading to these two regimes are
determined in terms of the width for the pump laser and the condensate size for
the limiting cases where the recoil energy is both much smaller and larger than
the atomic interaction energy.
