  Motivated by a recent experiment [J. H. Han, et. al., Phys. Rev. Lett.122,
065303 (2019)], we investigate many-body physics of interacting fermions in a
synthetic Hall tube, using state-of-the-art density-matrix
renormalization-group numerical method. Since the inter-leg couplings of this
synthetic Hall tube generate an interesting spin-tensor Zeeman field, exotic
topological and magnetic properties occur. Especially, four new quantum phases,
such as nontopological spin-vector and -tensor paramagnetic insulators, and
topological and nontopological spin-mixed paramagnetic insulators, are
predicted by calculating entanglement spectrum, entanglement entropies, energy
gaps, and local magnetic orders with 3 spin-vectors and 5 spin-tensors.
Moreover, the topologically magnetic phase transitions induced by the
interaction as well as the inter-leg couplings are also revealed. Our results
pave a new way to explore many-body (topological) states induced by both the
spiral spin-vector and -tensor Zeeman fields.
