Luis Foà Torres
This article is about Floquet Topological Insulators. Topological Insulators (TIs) are a recent family of exotic materials with a few intriguing properties. They are insulators in the bulk but their edges support propagating states bridging the bulk band-gap, much like the states emerging in the Quantum Hall effect discovered by Klaus von Klitzing in 1980 but in the case of TIs this happens without a magnetic field thanks to spin-orbit interaction. Recently, a new way of achieving properties akin those found in quantum Hall systems started to gain attention: using a time-dependent potential to harness tunable topological properties.
Floquet Topological Insulators [1,2] can be made out of normal materials which develop properties akin those in the quantum Hall regime, for example, upon illumination with a laser field. In “Floquet chiral edge states in graphene”  we offered the first analytical solution for the topological states as well as numerics proving their chirality [A simple analysis for bilayer graphene can be found in ]. This study has been followed up in  with a few additional twists.
More recently, we tackled the multiterminal conductance of such systems and found more surprises: Notably, the usual connection between counting the number/chirality of the edge states and the Hall conductance breaks down, not all the chiral edge states contribute equally to the Hall response as in the case of time-dependent systems .
As a side note, one of the simplest possible systems showing Floquet topological states and transitions between trivial and non-trivial band-structure is the driven Su-Schrieffer-Heeger model analyzed in . Besides the creation of Floquet topological states, the SSH model allows to study the interplay between the native topology and the driving. The driving a 3D topological insulator was studied in .
The field of Floquet Topological Insulators is just starting to flourish.
 T. Oka and H. Aoki, “Photovoltaic Hall effect in graphene”, Phys. Rev. B 79, 081406(R) (2009). link
 N. H. Lindner, Gil Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells”, Nature Physics 7, 490–495 (2011). link
 P. M. Perez-Piskunow, G. Usaj, C. A. Balseiro, L. E. F. Foa Torres, “Floquet chiral edge states in graphene”, Physical Review B, 89, 121401(R) (2014). link pdf
 E. Suárez Morell and L. E. F. Foa Torres, “Radiation effects on the electronic structure of bilayer graphene”, Physical Review B 86, 125449 (2012). link [pdf]
 G. Usaj, P. M. Perez-Piskunow, L. E. F. Foa Torres, C. A. Balseiro, “Irradiated graphene as a tunable Floquet topological insulator”, Physical Review B 90, 115423 (2014). [link]
 H. L. Calvo, H. M. Pastawski, S. Roche, and L. E. F. Foa Torres, “Tuning laser-induced band gaps in graphene”, Appl. Phys. Lett. 98, 232103 (2011).[pdf]
 L. E. F. Foa Torres, P. M. Perez-Piskunow, C. A. Balseiro, and G. Usaj, “Multiterminal Conductance of a Floquet Topological Insulator”, Physical Review Letters, 113, 266801 (2014). [pdf]
 V. Dal Lago, M. Atala, and L. E. F. Foa Torres, “Floquet topological transitions in a driven one-dimensional topological insulator”, Physical Review A 92, 023624 (2015).
 H. L. Calvo et al., Physical Review B 91, 241404(R) (2015).
The quest for new materials: Can any material become topological?
A related talk: Using light as a topological switch, ICTP, Trieste, January 2019.