Tuning and stabilising topological states, such as Weyl semimetals, Dirac
semimetals, or topological insulators, is emerging as one of the major topics
in materials science. Periodic driving of many-body systems offers a platform
to design Floquet states of matter with tunable electronic properties on
ultrafast time scales. Here we show by first principles calculations how
femtosecond laser pulses with circularly polarised light can be used to switch
between Weyl semimetal, Dirac semimetal, and topological insulator states in a
prototypical 3D Dirac material, Na3Bi. Our findings are general and apply to
any 3D Dirac semimetal. We discuss the concept of time-dependent bands and
steering of Floquet-Weyl points (Floquet-WPs), and demonstrate how light can
enhance topological protection against lattice perturbations. Our work has
potential practical implications for the ultrafast switching of materials
properties, like optical band gaps or anomalous magnetoresistance. Moreover, we
introduce Floquet time-dependent density functional theory (Floquet-TDDFT) as a
general and robust first principles method for predictive Floquet engineering
of topological states of matter.Comment: 21 pages, 4 figure