  Gauge fields play important roles in condensed matter, explaining for example
nonreciprocal and topological transport phenomena. Establishing gauge
potentials for phonon transport in nanomechanical systems would bring quantum
Hall physics to a new domain, which offers broad applications in sensing and
signal processing, and is naturally associated with strong nonlinearities and
thermodynamics. In this work, we demonstrate a magnetic gauge field for
nanomechanical vibrations in a scalable, on-chip optomechanical system. We
exploit multimode optomechanical interactions, which provide a useful resource
for the necessary breaking of time-reversal symmetry. In a dynamically
modulated nanophotonic system, we observe how radiation pressure forces mediate
phonon transport between resonators of different frequencies, with a high rate
and a characteristic nonreciprocal phase mimicking the Aharonov-Bohm effect. We
show that the introduced scheme does not require high-quality cavities, such
that it can be straightforwardly extended to explore topological acoustic
phases in many-mode systems resilient to realistic disorder.
