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.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Let
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