Payment channel network (PCN) is a promising off-chain technology for blockchain scalability, but it suffers from poor sustainability in practice. In other words, due to the imbalanced transfer in channels, the balance in one direction of channels gradually becomes exhausted until the PCN is rebalanced via a consensus-based rebalancing protocol, during which the involved channels must be suspended. This paper presents Cycle, the first off-chain protocol for a sustainable PCN. It not only keeps the PCN at a balanced level consistently but also avoids the channel freeze incurred by the rebalancing protocol, leading to minimum failed payments and sustained PCN service, respectively. Cycle achieves these benefits based on a novel idea of asynchronous rebalancing. During the normal off-chain running, the participants share the information about their payments and asynchronously rebalance the PCN following the principle that payments along circular channels can cancel each other out. To guarantee security, the protocol resolves the disputes resulting from network latency or malicious participants by a message mechanism for synchronization and a smart contract for arbitration. Moreover, to address the privacy concern during the information sharing, a truncated Laplace mechanism is designed to achieve differential privacy. Finally, we provide a proof-of-concept implementation in Ethereum, over which a real data-based simulation shows that Cycle satisfies 31% more payments than the state-of-the-art technique.