Onsager relations in coupled electric, thermoelectric, and spin transport: The tenfold way

Hamiltonian systems can be classified into ten classes, in terms of the presence or absence of time-reversal, particle-hole, and sublattice/chiral symmetries. We construct a quantum coherent scattering theory of linear transport for coupled electric, heat, and spin transport; including the effect of Andreev reflection from superconductors. We derive a complete list of the Onsager reciprocity relations between transport coefficients for coupled electric, spin, thermoelectric and spin caloritronic effects. We apply these to all ten symmetry classes, paying special attention to specific additional relations that follow from the combination of symmetries, beyond microreversibility. We discuss these relations in several illustrative situations. We show the reciprocity between spin-Hall and inverse spin-Hall effects, and the reciprocity between spin-injection and magnetoelectric spin currents. We discuss the symmetry and reciprocity relations of Seebeck, Peltier, spin Seebeck, and spin Peltier effects in systems with and without coupling to superconductors.