  The spin transport in a strongly interacting spin-pump nano-device is studied
using the time-dependent variational-matrix-product-state (VMPS) approach. The
precession magnetic field generates a dissipationless spin current through the
quantum dot. We compute the real time spin current away from the equilibrium
condition. Both transient and stationary states are reached in the simulation.
The essentially exact results are compared with those from the Hartree-Fock
approximation (HFA). It is found that correlation effect on the physical
quantities at quasi-steady state are captured well by the HFA for small
interaction strength. However the HFA misses many features in the real time
dynamics. Results reported here may shed light on the understanding of the
ultra-fast processes as well as the interplay of the non-equilibrium and
strongly correlated effect in the transport properties.
