Gyrotropic magnetic effects in chiral metals

We consider two conjugate transport effects occuring in chiral metals as the low-frequency limit of natural optical activity (optical gyrotropy). One occurs in the clean limit where $\omega$ is small compared to the minimum energy for interband transitions, but large compared to the scattering rate $1/\tau$. It consists of a dissipationless current induced by a magnetic field, $J_i = \alpha'_{ij}B_j$, and is different from the chiral magnetic effect requiring a static ${\bf B}$ and an electric-field pulse ${\bf E}\parallel {\bf B}$. In the inverse effect a magnetization is generated by a dissipative current, $M_i =(1/\omega)\alpha''_{ji}E_j$, with ${\bf E}$ the field driving the current and $\omega \ll 1/\tau$, as discussed by Yoda {\it et al.}, Sci. Rep.~{\bf 5}, 12024 (2015). The low-frequency gyrotropic responses $\alpha'$ and $\alpha''$ in the clean and dirty limits can be combined into a complex tensor $\alpha=\alpha'+i\alpha''$ given by the Fermi-surface integral of the total (orbital plus spin) intrinsic magnetic moment of the Bloch electrons, with a prefactor proportional to $1-i\omega\tau$. Without spin-orbit coupling, only the orbital moment contributes.