SL ( 2 , C ) non-Abelian gauge fields in a photonic molecule array

Engineering the topological properties of the system has been a longstanding subject in physics. Here, we propose a scheme to simulate topological materials with a non-Abelian gauge field in synthetic space-frequency dimensions where the symmetry between spin-flipped hoppings can be levitated by using coupled photonic molecules under dynamic modulations. The frequency-split supermodes in these photonic molecules can serve the pseudospin degree of freedom, which can further be connected along the frequency axis in an independent way, offering the unique opportunity to explore topological physics with imbalanced spin-flipped hoppings leading to a complex generalization of the conventional SU(2) non-Abelian gauge field, i.e., an SL $\left(2,C\right)$ gauge field. By varying the spin-flipped hopping terms, we theoretically show the existence of a variety of Dirac semimetal transitions and the rotation of the pseudospin projection for the edge states throughout the Brillouin zone. Our proposal is experimentally feasible and therefore provides a versatile platform for the study of topological materials under non-Abelian gauge fields in photonics.