  The recently-introduced concept of "synthetic dimensions" allows for the
realization of higher-dimensional topological phenomena in lower-dimensional
systems. In this work we study the complementary aspect that synthetic
dimensions provide a natural route to topological states in mesoscopic hybrid
devices. We demonstrate this for the current induced into a closed
one-dimensional Aharonov-Bohm ring by the interaction with a dynamic mesoscopic
magnet. The quantization of the magnetic moment provides a synthetic dimension
that complements the charge motion around the ring. We present a direct mapping
that places the combined ring-magnet system into the class of quantum Hall
models, and demonstrate that topological features, combined with the magnet's
anisotropy, can lead to clear signatures in the persistent current of the
single-particle ground state.
