  We revisit the transport of spin-degrees of freedom across an electrically
and thermally biased tunnel junction between two ferromagnets with
non-collinear magnetizations. Besides the well-known charge and spin currents
we show that a non-zero spin-quadrupole current flows between the ferromagnets.
This tensor-valued current describes the non-equilibrium transport of
spin-anisotropy relating to both local and non-local multi-particle spin
correlations of the circuit. This quadratic spin-anisotropy, quantified in
terms of the spin-quadrupole moment, is fundamentally a two-electron quantity.
In spin-valves with an embedded quantum dot such currents have been shown to
result in a quadrupole accumulation that affects the measurable quantum dot
spin and charge dynamics. The spin-valve model studied here allows fundamental
questions about spin-quadrupole storage and transport to be worked out in
detail, while ignoring the detection by a quantum dot. This physical
understanding of this particular device is of importance for more complex
devices where spin-quadrupole transport can be detected. We demonstrate that,
as far as storage and transport are concerned, the spin anisotropy is only
partly determined by the spin polarization. In fact, for a thermally biased
spin-valve the charge- and spin-current may vanish, while a pure exchange
spin-quadrupole current remains, which appears as a fundamental consequence of
Pauli's principle. We extend the real-time diagrammatic approach to efficiently
calculate the average of multi-particle spin-observables, in particular the
spin-quadrupole current. Although the paper addresses only leading order and
spin-conserving tunneling we formulate the technique for arbitrary order in an
arbitrary, spin-dependent tunnel coupling in a way that lends itself to
extension to quantum-dot spin-valve structures.
