We study the charge and spin currents transported by the elementary
excitations of the one-dimensional (1D) Hubbard model and derive the
corresponding current spectra. We present results both for
finite-size systems and in the thermodynamic limit. This includes
finding the couplings of both the low-energy and finite-energy
(string) excitations to external charge and spin probes. At zero
magnetic field the general structure of the charge-spin separation
survives at all energy scales and the effective charges of both the
low-energy and finite-energy charge excitations are studied as
functions of the on-site Coulomb interaction U, electronic density
n, and applied magnetic field H. In some limits the effective
charge of the low-energy excitations equals that of the electrons,
whereas that of the finite-energy charge-string excitations of
rapidity length γ is found to be 2γ times the
electronic charge. At U = ∞ the spin excitations do not
contribute to spin transport, whereas the low-energy charge
excitations feel an effective flux given by
(ϕ↑N↑-ϕ↓N↓)/(N↑ + N↓),
where Nσ is the number of electrons of spin σ and
ϕσ is a spin-dependent flux. This reveals that at zero
magnetic field and U = ∞ there is no spin transport, while at
finite magnetic field the low-energy charge excitations also carry
spin. In the U>>t limit the spin is carried both by holons and
spinons. Finally, we find that the charge- and spin-current spectra
can be derived from a semi-classical approach.