  Recent ab initio calculations and experiments reported
insulating-semimetallic phase transitions in multilayer phosphorene under a
perpendicular dc field, pressure or doping, as a possible route to realize
topological phases. In this work, we show that even a monolayer phosphorene may
undergo Lifshitz transitions toward semimetallic and topological insulating
phases, provided it is rapidly driven by in-plane time-periodic laser fields.
Based on a four-orbital tight-binding description, we give an
inversion-symmetry-based prescription in order to apprehend the topology of the
photon-renormalized band structure, up to the second order in the
high-frequency limit. Apart from the initial band insulating behavior, two
additional phases are thus identified. A semimetallic phase with massless Dirac
electrons may be induced by linear polarized fields, whereas elliptic polarized
fields are likely to drive the material into an anomalous quantum Hall phase.
