We study the emergence of nonreciprocal and topologically nontrivial phonon transport in nano-optomechanical networks. We develop on-chip nanophotonic systems in which multiple mechanical modes are strongly coupled through radiation pressure. Through temporal modulation and retardation, suitable laser control fields can break time-reversal symmetry and introduce controlled gain and loss at will on any of the network link and nodes. We reveal the emergence of nanomechanical circulation, helical quantum Hall states, and chiral thermal transport. Exploiting optomechanical gain, we study the combination of broken time-reversal symmetry and non-Hermiticity. This leads to rich phenomenology including magnetic-field tuning of exceptional points and unidirectional phononic amplification. It promises to serve as building blocks for new bosonic topological phases in the domain of nanomechanics, which is rich in applications in sensing and signal processing.
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