  The bulk-boundary correspondence of the second-order topological insulator
(SOTI) has been well established, but a universal transport signature for open
systems is still absent. For a variety of SOTIs induced by applying in-plane
magnetic fields in Z$_2$-invariant first-order TIs, rotating this magnetic
field features the spin pump mechanism while maintaining the SOTI phase. We
demonstrate that, this spin pump can generate quantized pure spin current when
tuning the magnetic field strength, which corresponds to the formation of
topological corner states characterizing SOTI in two-dimensional (2D) systems.
Quantized spin pump is discovered in various 2D and 3D SOTI models evolved from
Z$_2$-invariant TIs, which is robust against disorder and universally
independent of system parameters including Fermi energy, system size, magnetic
field strength, and pumping frequency. These findings suggest that this
universal quantized spin pump can characterize the bulk-boundary-transport
correspondence of SOTIs. Quantized spin pump can also be realized by combining
pseudo spin such as the orbital degree of freedom with the rotating magnetic
field, which could be achieved in higher-order photonic or acoustic topological
systems. Such a quantized spin pump is promising as an accurate and stable
single-spin source.
