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Band crossings observed in a wide range of condensed matter systems arerecognized as a key to understand low-energy fermionic excitations that behaveas massless Dirac particles. Despite rapid progress in this field, theexploration of non-equilibrium topological states remains scarce and it haspotential ability of providing a new platform to create unexpected masslessDirac states. Here we show that in a semiconductor quantum-well driven by acw-laser with linear polarization, the optical Stark effect conducts bulk-bandcrossing, and the resulting Floquet-Dirac semimetallic phase supports anunconventional edge state in the projected one-dimensional Brillouin zone undera boundary condition that an electron is confined in the directionperpendicular to that of the laser polarization. Further, we reveal that thisedge state mediates a transition between topological and non-topological edgestates that is caused by tuning the laser intensity. We also show that theproperties of the edge states are strikingly changed under a different boundarycondition. It is found that such difference originates from that nearlyfourfold-degenerate points exist in a certain intermediate region of the bulkBrillouin zone between high-symmetry points.

Edge states of Floquet-Dirac semimetal in a laser-driven semiconductor quantum-well