The Hall tube as a minimum model to simulate the integer quantum Hall effect is essential for exploring topological physics, while it has not been constructed in the recent developing successfully experiments on superconducting circuits. In this work, we propose a feasible experiment scheme using three legs super-conducting circuits with transmon qubits to realize a Hall tube. Then we first investigate its topological properties. Since the time-reversal, particle-hole, and chiral symmetries are all broken for the system, the Hall tube belongs to the A class of the Altland-Zirnbauer classification. We obtain the corresponding topological phase transition both numerically and analytically. Since the chirality is a key character of the quantum Hall effect, we secondly investigate the chiral physics in the Hall tube. We find the topological protected chiral edge currents and discuss its robustness. Finally, we give the possible experimental observations of the topological state and topological protected chiral edge currents.

Hall tube is an important model to simulate the quantum Hall effect. However it hasn't been realized in superconducting circuits which have emerged as a promising platform for macro-controlling quantum effect. Taking advantage of the fine tunability of superconducting circuits, the three-chain superconducting transmon qubits with periodic boundary condition are designed in this paper. For constructing a synthetic Hall tube, ac magnetic fluxes are introduced to drive each transmon qubit. The gauge field emerged in this synthetic Hall tube can be tuned independently by properly choosing the driving phases. Then the ground-state chiral currents are discovered in this synthetic Hall tube, which are Meissner current on <i>xy</i> plane (<i>xy</i>-M), vortex current on <i>xy</i> plane (<i>xy</i>-V), vortex current on <i>xz</i> plane (<i>xz</i>-V), and vortex current on both <i>xy</i> and <i>xz</i> planes (DV). For distinguishing these chiral currents, four order parameters <i>J</i>_C||, <i>J_AB</i> (<i>J_BC</i>), and <i>J_CA</i> are defined. Then the ground-state quantum phase diagrams are mapped out. The emergence of the different quantum phases is due to the competition between the coupling strengths <i>t</i> and <i>t_CA</i>. The Meissner and vortex currents emerging in this synthetic Hall tube also emerge in type II superconductor, which can generate an opposite field to weaken the influence of the applied field. Thus this synthetic Hall tube can be used as a diamagnet. At last we consider the influence of the imperfections in device fabrication. We proof when the strength of the imperfection is not large enough, the quantum phase diagrams shown in this paper remain valid. Moreover, the possible experimental observations of the ground-state chiral currents are addressed. The ground state of this synthetic Hall tube can be generated by applying microwave pulses. Then the corresponding density matrix can be constructed by the quantum state tomography. After constructing the density matrix, the order parameters can be obtained by calculating the trace. These results enrich the quantum currents in Hall tube and provide a new route to explore novel quantum phases.