Localisation and Topological Signatures by Periodic Twisting

We theoretically explore a generalisation of the Aubry-André model in two dimensions formed by superimposing two square optical lattice potentials. Motivated by the rich localisation and topological signatures emerging at different static twist angles between the two lattices, we consider the effects of periodic twisting by continuously rotating. There are two key different symmetry classes of potentials, and we analyse localisation properties contrasting with previously established results in the cases of disorder-induced and dynamical localisation. Due to the absence of spatial translational symmetry, the topological signatures in the system that may arise from the breaking of time reversal symmetry are described using a local Chern marker and the Bott index. We find that there is a zoo of different states with non-trivial topological markers including symmetric ring configurations which are stabilised, and later destroyed, by the presence of strong localisation in the system. The system also hosts a variety of transport signatures, some of which are incredibly robust and localised despite the disordered nature of the driving.