Topological Floquet engineering using two frequencies in two dimensions

Using two-frequency driving in two dimensions opens up new possibilities for Floquet engineering, which range from controlling specific symmetries to tuning the properties of resonant gaps. In this work, we study two-band lattice models subject two-tone and analyze resulting effective band structures both numerically analytically. On one hand, extend methodology Sandholzer et al. [Phys. Rev. Res. 4, 013056 (2022)] find competing topological phases a simple Bravais when drives at $1\ensuremath{\omega}$ $2\ensuremath{\omega}$ interfere. other explore driving-induced symmetry breaking hexagonal lattice, either inversion or time-reversal can be tuned independently via modulation. Possible applications our work include simpler generation bands ultracold atoms realisation nonlinear Hall effects as well Haldane's parity anomaly inversion-symmetric parent lattices.