Floquet engineering, the control of quantum systems using periodic driving, is an old concept in condensed matter physics, dating back to ideas such as the inverse Faraday effect. There is a renewed interest in this concept owing to the rapid developments in laser and ultrafast spectroscopy techniques and discovery and understanding of various "quantum materials" hosting interesting exotic quantum properties. Here, starting from a nontechnical introduction with emphasis on the Floquet picture and effective Hamiltonians, we review the recent applications of Floquet engineering in ultrafast, nonlinear phenomena in the solid state. In particular, Floquet topological states, application to ultrafast spintronics, and to strongly correlated electron systems are overviewed. arXiv:1804.03212v1 [cond-mat.str-el] 9 Apr 2018 2 Floquet Kubo and TKNN formulae Floquet Keldysh greens function α VH Jα Floquet engineering of band topology New states, new functions Laser-assisted hopping Floquet topological states Floquet-Schrieffer-Wolff transf. Magneto-electric coupling Correlated electrons driven by electric fields Basics of Floquet engineering Floquet theorem Floquet picture Effective Hamiltonians Floquet many-body methods (e.g. DMFT) Quantum field theory Heisenberg-Euler effective Lagrangian (=Loschmidt echo) Switching (optical memory & Mott RRAM) Carrier generation Breakdown and field induced states Heating and thermalization Higher harmonics generation Frequency conversion, heterodyne Usage of metamaterial -and near field techniques .. ground state spin excitation charge excitation energy 0 frequency Ω Field F Schwinger limit avalanche E sp E ch E ch h, J ex , J χ effective interactions current F Nonlinear transport ultrafast spintronics avalanche conductor (hot plasma) Mott ins. Ω Quantum materials Mott insulator Topological bands Berry curvature TKNN formula Topological invariants b r e a k d o w n (nonlinear) photo-absorption Schwinger -limit E sp Floquet engineering in ultrafast spintronics (E sp~Ω

We construct a systematic high-frequency expansion for periodically driven quantum systems based on the Brillouin-Wigner (BW) perturbation theory, which generates an effective Hamiltonian on the projected zero-photon subspace in the Floquet theory, reproducing the quasienergies and eigenstates of the original Floquet Hamiltonian up to desired order in 1/omega, with omega being the frequency of the drive. The advantage of the BW method is that it is not only efficient in deriving higher-order terms, but even enables us to write down the whole infinite series expansion, as compared to the van Vleck degenerate perturbation theory. The expansion is also free from a spurious dependence on the driving phase, which has been an obstacle in the Floquet-Magnus expansion. We apply the BW expansion to various models of noninteracting electrons driven by circularly polarized light. As the amplitude of the light is increased, the system undergoes a series of Floquet topological-to-topological phase transitions, whose phase boundary in the high-frequency regime is well explained by the BW expansion. As the frequency is lowered, the high-frequency expansion breaks down at some point due to band touching with nonzero-photon sectors, where we find numerically even more intricate and richer Floquet topological phases spring out. We have then analyzed, with the Floquet dynamical mean-field theory, the effects of electron-electron interaction and energy dissipation. We have specifically revealed that phase transitions from Floquet-topological to Mott insulators emerge, where the phase boundaries can again be captured with the high-frequency expansion

Van der Waals interfaces can be formed by layer stacking without regard to lattice constants or symmetries of individual building blocks. We engineered the symmetry of a van der Waals interface of tungsten selenide and black phosphorus and realized in-plane electronic polarization that led to the emergence of a spontaneous photovoltaic effect. Spontaneous photocurrent was observed along the polar direction and was absent in the direction perpendicular to it. The observed spontaneous photocurrent was explained by a quantum-mechanical shift current that reflects the geometrical and topological electronic nature of this emergent interface. The present results offer a simple guideline for symmetry engineering that is applicable to a variety of van der Waals interfaces.

Just below Eq. (30) of our paper, we assigned "an order 1 for H (t ) and the order k + 1 for (k) (t ) and F (k) ." Although this procedure reproduces the correct high-frequency series, the assignment should be corrected to "an order 1 for H (t ), the order k for (k) (t ), and the order k + 1 for F (k) " in order for the superscript (k) to represent the order in 1/ω. In accordance with this correction, the k sum for (t ) in Eq. (30) should start from k = 1: Namely, Eq. (30) should readWith this, one can indeed derive Eqs. (32b) and (32c) and (34b)-(34d) correctly. These typographical errors do not alter any conclusions drawn in the paper.

Scalar spin chirality, a three-body spin correlation that breaks time-reversal symmetry, is revealed to couple directly to circularly polarized laser. This is shown by the Floquet formalism for the periodically driven repulsive Hubbard model with a strong-coupling expansion. A systematic derivation of the effective low-energy Hamiltonian for a spin degree of freedom reveals that the coupling constant for scalar spin chirality can become significant for a situation in which the driving frequency and the on-site interaction are comparable. This implies that the scalar chirality can be induced by circularly polarized lights, or that it can be used conversely for probing the chirality in Mott insulators as a circular dichroism