Geometrical control of topology with orbital angular momentum modes

We study how the topological properties of a one-dimensional staggered lattice, loaded into states with orbital angular momentum $l=1$, can be controlled simply by tuning the relative angle between sites. The original system under consideration can be depicted as a Creutz ladder model when unwrapping the different state circulations in a synthetic dimension. Depending on the hopping strengths of the chain, different topological regimes may be accessed by changing the ladder angle, as determined by the value of the winding number of the chain. We analytically and numerically explore the different available regimes, and determine the number of topologically protected edge states that exist in each case. We also study the emergence of band inversion across topological transitions and show that it agrees with the winding number calculations, thus serving as an additional topological marker. Then, we propose a realistic experimental implementation in a photonic waveguide system, where the topological transition manifests as a sudden change of the behavior of the propagation of light in the system.