Generalized Onsager reciprocal relations of charge and spin transport

TL;DR

This work addresses how charge and spin transport reciprocity can be preserved when time-reversal symmetry is broken but a combined OT symmetry remains. By a symmetry analysis of spin and charge displacement operators and a Kubo-formalism derivation, the authors obtain a generalized Onsager relation $\sigma^{\rm sc}_{\mu\nu}(\omega) = (-1)^{I_{\mu\nu}}\sigma^{\rm cs}_{\nu\mu}(\omega)$, with $\mathcal{O}^{-1}\hat{r}_{\mu}\mathcal{O} = (-1)^{I_{\mu}}\hat{r}_{\mu}$ and $\mathcal{O}^{-1}\hat{s}_{z}\mathcal{O} = (-1)^{I_{s}}\hat{s}_{z}$, and a classification of $\mathcal{O}$ into three categories. The framework is exemplified and tested in a 2D spin–orbit-coupled quantum-gas model possessing $\mathcal{PT}$ and $\tilde{D}_4$ symmetry, with ground-state phases showing distinct OT-symmetry constraints on the cross-couplings. Explicit calculations for noninteracting Fermi gases and Bose gases (via Bogoliubov theory) verify the predicted relations, including phase-dependent longitudinal behavior and robust transverse antisymmetry, at finite temperature and with dissipation. The paper also outlines experimental quench protocols in trapped cold atoms to measure center-of-mass responses and test the generalized reciprocity, highlighting the potential for observing novel spin–charge transport phenomena and extending Onsager-type relations to other cross-coupled transport processes. These results provide a principled path to exploit hidden symmetries in magnetically ordered or spin-orbit–coupled systems, including antiferromagnets, for spintronics applications and beyond.

Abstract

In spin-orbit-coupled systems the charge and spin transport are generally coupled to each other, namely a charge current will induce a spin current and vice versa. In the presence of time-reversal symmetry $T$, the cross-coupling transport coefficients describing how one process affects the other are constrained by the famous Onsager reciprocal relations. In this paper, we generalize the Onsager reciprocal relations of charge and spin transport to systems that break the time-reversal symmetry but preserve a combined symmetry of $T$ and some other symmetry operation $O$. We show that the symmetry or antisymmetry of the cross-coupling transport coefficients remains in place provided that the operator $O$ meets certain conditions. Among many candidate systems where our generalized Onsager relations apply, we focus on a conceptually simple and experimentally realized model in cold atomic systems for explicit demonstration and use these relations to predict highly non-trivial transport phenomena that can be readily verified experimentally.

Figures (9)

• Figure 1: Illustration of the symmetry operations in the $\tilde{D}_4$ group.
• Figure 2: Verification of the generalized Onsager relations for the noninteracting Fermi gas. The conductivity tensors are in units of $\rho_u A$ where $\rho_u = 2\rho \pi^2 /k_L^2$ is the atom number per unit cell and $A$ is the area of the system. The parameters are $\rho_u=4$, $V_0=4.0E_r$ and $M_0=2.0E_r$; a spectral broadening $\eta = 0.1E_r$ is used in plotting.
• Figure 3: Verification of the generalized Onsager relations for the Bose gas in perpendicularly magnetized phase (phase I). Here $V_0 = 4E_r$, $M_0= 2.5E_r$, $\rho g_{\uparrow\uparrow} = 0.25 E_r$ and $g_{\uparrow\downarrow} /g_{\uparrow\uparrow}= 0.9954$ (experimental ratio), where $\rho$ is the average atomic density and $E_r = k_L^2/2m$ is the recoil energy. A spectral broadening of $\eta = 0.1E_r$ is again used.
• Figure 4: Verification of the generalized Onsager relations for the Bose gas in the in-plane magnetized phase (phase II). Here the parameters are $V_0=4.0E_r$, $M_0=1.0E_r$, $\rho g_{\uparrow\uparrow}=0.25E_r$, $\rho g_{\uparrow\downarrow}=0.2E_r$ and the spectral broadening $\eta = 0.1E_r$.
• Figure 5: Illustration of two experimental quench protocols that can be used to generate a charge force [protocol (a)] and a spin force [protocol (b)] in harmonically trapped quantum gases.