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.

Figure 1

Microreversibility operating on a two-terminal $\mathcal{S}$ matrix in the absence (top) and in the presence of superconductivity (bottom). The scattering amplitudes $a$ and $b$ depend on spin indices, $\sigma =\uparrow (+1)$, $\downarrow (-1)$, quasiparticle indices, $\mu =e(+1)$, $h(-1)$, and terminal indices 1 and 2. Time reversal implies inverting the particle's momentum, spin, and quasiparticle isospin as well as magnetic fields and fluxes.

Figure 2

Sketch of a four-terminal spin-Hall (a) and inverse spin Hall (b) experiment. (a) In the spin Hall effect, an electric current (blue arrow) generates transverse spin currents (red) via the action of spin-orbit coupling. (b) In the inverse spin Hall effect, spin accumulations inject a spin current (blue arrow) which, in the presence of spin-orbit coupling, generates a transverse electric current (red).

Figure 3

An Andreev interferometer: two-terminal hybrid system with two normal metallic/superconducting contacts with tunable superconducting phase difference ${\phi }_{\mathrm{SC}}$. The latter, together with a systematic delay (indicated by the extra length $\ell$) towards one of the superconducting contacts can break PHS and generate finite thermoelectric effects.[10] The particular symmetry class in which the system falls is given by how much magnetic field $\mathcal{H}$ and how much spin-orbit coupling there is in the dot, see Table .

Figure 4

Venn diagram summarizing the Onsager relations we derived. Here, $\alpha ,\beta \in \{0,x,y,z\}$, $n_0=0$, and $n_x=n_y=n_z=1$. Any system obeys the relations given in all boxes it is within. For instance, the Onsager relations for a generic system with time-reversal symmetry (TRS) read $G_{ij}^{\left(\alpha \beta \right)}={(-1)}^{n_{\alpha }+n_{\beta }}{G}_{ji}^{\left(\beta \alpha \right)}$ and $B_{ij}^{\left(\beta 0\right)}{T}_0={(-1)}^{n_{\beta }}{\Gamma }_{ji}^{\left(0\beta \right)}$, since TRS requires that any external magnetic field $\mathcal{H}=0$. If that system also had SRS, then the only nonzero coefficients are those with repeated upper indices ($\alpha =\beta$ for $G$ and $\beta =0$ for $B$) for which $G_{ij}^{\left(\beta \beta \right)}=G_{ji}^{\left(\beta \beta \right)}$ and $B_{ij}^{\left(00\right)}{T}_0={\Gamma }_{ji}^{\left(00\right)}$. If the system contains no Andreev reflection (i.e., no superconductors), one additionally has[44] $B_{ij}^{\left(00\right)}\left(\mathcal{H}\right)=B_{ji}^{\left(00\right)}(-\mathcal{H})$. In the case of PHS and SLS, the extra symmetry relations derived in the text are summarized in Table .