Intertwined space-time symmetry, orbital magnetism, and dynamical Berry connection in a circularly shaken optical lattice

We study the circular shaking of a two dimensional optical lattice, which is essentially (2+1) space-time lattice exhibiting periodicities in both spatial and temporal dimensions. The near-resonant considered here dynamically couples low-lying $s$ band first excited $p$ bands by transferring photon frequency. intertwined symmetries are further uncovered to elucidate degeneracy spectrum solved with generalized Bloch-Floquet theorem. Setting chirality explicitly breaks time reversal symmetry lifts $p_\pm = p_x \pm ip_y$ orbitals, leading local circulation orbital magnetism, i.e imbalanced occupation $p_\pm$ orbitals. Moreover, dynamics Berry connection revealed evolution curvature polarization, have physical observable effects experiments. Interestingly, found characterized universal phase shift, governed screw rotational involving fractional translation time. These findings suggest that present lattice-shaking scheme provides versatile platform for investigation physics symmetry-protected dynamics.