Gyrotropic Magnetic Effect and the Magnetic Moment on the Fermi Surface

The current density ${\mathit{j}}^{\mathbf{B}}$ induced in a clean metal by a slowly-varying magnetic field $\mathbf{B}$ is formulated as the low-frequency limit of natural optical activity, or natural gyrotropy. Working with a multiband Pauli Hamiltonian, we obtain from the Kubo formula a simple expression for ${\alpha }_{ij}^{\mathrm{GME}}=j_i^{\mathbf{B}}/B_j$ in terms of the intrinsic magnetic moment (orbital plus spin) of the Bloch electrons on the Fermi surface. An alternate semiclassical derivation provides an intuitive picture of the effect, and takes into account the influence of scattering processes in dirty metals. This “gyrotropic magnetic effect” is fundamentally different from the chiral magnetic effect driven by the chiral anomaly and governed by the Berry curvature on the Fermi surface, and the two effects are compared for a minimal model of a Weyl semimetal. Like the Berry curvature, the intrinsic magnetic moment should be regarded as a basic ingredient in the Fermi-liquid description of transport in broken-symmetry metals.

Figure 1

(a) Chiral magnetic effect in a $T$-broken Weyl semimetal in a static $\mathbf{B}$ field. The left- and right-handed Weyl nodes are at the same energy ${\epsilon }_L={\epsilon }_R$, but the enclosing Fermi pockets are not in chemical equilibrium (${\mu }_L\ne {\mu }_R$) due to the application of an $\mathbf{E}\parallel \mathbf{B}$ pulse, and this drives the current [Eq. (3)]. (b) Gyrotropic magnetic effect. $P$ symmetry is now broken along with $T$, leading to ${\epsilon }_L\ne {\epsilon }_R$. The Fermi pockets are in chemical equilibrium, ${\mu }_L={\mu }_R={\epsilon }_F$, and an oscillating $\mathbf{B}$ field drives the current [Eq. (17)]. The bottom of each panel shows the Fermi pockets, and the arrows represent the Fermi velocities.