Hall tube with a tunable flux is an important geometry for studying quantum Hall physics, but its experimental realization in real space is still challenging. Here, we propose to realize a synthetic Hall tube with tunable flux in a one-dimensional optical lattice with the synthetic ring dimension defined by atomic hyperfine states. We investigate the effects of the flux on the system topology and study its quench dynamics. Utilizing the tunable flux, we show how to realize topological charge pumping, where interesting charge flow and transport are observed in rotated spin basis. Finally, we show that the recently observed quench dynamics in a synthetic Hall tube can be explained by the random flux existing in the experiment.
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