Capital to topological insulators, the bulk-boundary correspondence ties a topological invariant computed from the bulk (extended) states with those at the boundary, which are hence robust to disorder. Here we put forward a different ordering unique to non-Hermitian lattices whereby a pristine system becomes devoid of extended states, a property which turns out to be robust to disorder. This is enabled by a peculiar type of non-Hermitian degeneracy where a macroscopic fraction of the states coalesce at a single point with a geometrical multiplicity of 1.

In the presence of a circularly polarized mid-infrared radiation graphene develops dynamical band gaps in its quasienergy band structure and becomes a Floquet insulator. Here, we analyze how topologically protected edge states arise inside these gaps in the presence of an edge. Our results show that the gap appearing at $\hslash \Omega /2$, where $\hslash \Omega$ is the photon energy, is bridged by two chiral edge states whose propagation direction is set by the direction of the polarization of the radiation field. Therefore, both the propagation direction and the energy window where the states appear can be controlled externally. We present both analytical and numerical calculations that fully characterize these states. This is complemented by simple topological arguments that account for them and by numerical calculations for the case of the semi-infinite sample, thereby eliminating finite-size effects.

We report on the emergence of laser-induced chiral edge states in graphene ribbons. Insights on the nature of these Floquet states is provided by an analytical solution which is complemented with numerical simulations of the transport properties. Guided by these results we show that graphene can be used for realizing nonequilibrium topological states with striking tunability: while the laser intensity can be used to control their velocity and decay length, changing the laser polarization switches their propagation direction.

We report on simulations of the dc conductance and quantum Hall response of a Floquet topological insulator using Floquet scattering theory. Our results reveal that laser-induced edge states lead to quantum Hall plateaus once imperfect matching with the nonilluminated leads is lessened. The magnitude of the Hall plateaus, however, is not directly related to the number and chirality of all the edge states at a given energy, as usual. Instead, the plateaus are dominated by those edge states adding to the time-averaged density of states. Therefore, the dc quantum Hall conductance of a Floquet topological insulator is not directly linked to topological invariants of the full Floquet bands.

Could a laser field lead to the much sought-after tunable band gaps in graphene? By using Floquet theory combined with Green's functions techniques, we predict that a laser field in the mid-infrared range can produce observable band gaps in the electronic structure of graphene. Furthermore, we show how they can be tuned by using the laser polarization. Our results could serve as a guidance to design optoelectronic nanodevices.