Intertwined Space-Time Symmetry, Orbital Magnetism and Dynamical Berry Curvature in a Circularly Shaken Optical Lattice

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