Usually, topological photonic structures are based on spatial lattices, where the sites are identified by their location in space. Nevertheless, we can realize different kinds of lattices, which may take forms other than a spatial arrangement of sites – these are synthetic dimensions. In this work we induce in a synthetic space an effective magnetic field for photons, implemented by a dynamically-invariant (static) potential. Usually, artificial gauge fields are induced by modulation during propagation. Implementing propagation-invariant systems with artificial gauge fields is important because the Hamiltonian describing this system remains constant both in  time and along the propagation axis, which makes the system robust to variations, making it appealing for experiments. Thus far, however, such systems were demonstrated only for gauge fields in real-space, never in synthetic space, while in synthetic space all suggestions for implementing artificial magnetic fields on photons required modulation of the system either in time or along the propagation axis which plays the role of time in static systems. To the best of our knowledge, thus far no evolution-invariant system facilitating the implementation of artificial magnetic fields in synthetic dimensions has ever been proposed. 

Here, we introduce a Propagation-Invariant two-dimensional photonic lattice that creates an effective magnetic field in a synthetic dimension, and we simulate a non – trivial propagation at the interface edge of two coupled  systems in the synthetic space.