  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.
