The creation of new topological phases of matter in periodically driven systems is now a topic of wide interest. In this talk, I shall first discuss a few examples to explain how and why periodic driving can generate intriguing topological phases that are otherwise absent without driving.  In particular, I show that the symmetry class of a system can be altered, long-range effective Hamiltonians can be synthesized, and topologically nontrivial flat bands may be induced, all by periodic driving.  I shall then focus on adiabatic pumping as a means to detect topological phase transitions in periodically driven systems. It is shown that for initial states prepared as a Wannier state of a Floquet quasi-energy band, the adiabatic pumping can be simply connected with the Chern number of the occupied band.  By contrast and much more remarkable, for general and easy-to-prepare initial states possessing coherence between different Floquet quasienergy bands, adiabatic pumping is found to be comprised by two components independent of the pumping time scale:  a weighted integral of the Berry curvature summed over all Floquet quasienergy bands,  plus an inter-band-coherence induced correction. It is stressed that the found correction is always there no matter how slowly a pumping cycle is executed.  In addition to probing topological phase transitions, adiabatic pumping is now also anticipated to be useful in manifesting coherence and decoherence effects in periodically driven systems.

References:
1.Interband-coherence-induced correction to adiabatic pumping in periodically driven systems (under review), by H.L. Wang, L.W. Zhou, and J.B. Gong.
2.Topological effects in chiral symmetric driven systems, PRB 90, 195419 (2014) (16 pages),   by D.Y.H. Ho and J.B. Gong.
3.Aspects of Floquet bands and topological phase transitions in a continuously driven superlattice, EPJB 87, 204 (2014), by L. W. Zhou, H. L. Wang, D. Y. H. Ho and J.B. Gong.
4. Kicked-Harper model vs on-resonance double kicked rotor model: From spectral difference to topological equivalence, PRE 88, 052920 (2013) (15 pages), by H.L. Wang, D.Y.H. Ho, W. Lawton, J. Wang and J.B. Gong.
5. Generating many Majorana modes via periodic driving: A superconductor model,  PRB 87, 201109 (R) (2013), by Q.J. Tong, J.H. An, J.B. Gong, H.G. Luo, and C.H. Oh.
6. Quantized adiabatic transport in momentum space, PRL 109, 010601 (2012) (Editor’s Suggestion),  by D.Y.H. Ho and J.B. Gong.