Essential role of the anisotropic magnetic dipole in the anomalous Hall effect

We theoretically investigate the anomalous Hall effect (AHE) that requires neither a net magnetization nor an external magnetic field in collinear antiferromagnets. We show that such an emergent AHE is essentially caused by a ferroic ordering of the anisotropic magnetic dipole (AMD), which provides an effective coupling between ordered magnetic moments and electronic motion in the crystal. We demonstrate that the AMD is naturally induced by the antiferromagnetic ordering, in which the magnetic moments have a quadrupole spatial distribution. In view of the ferroic AMD ordering, we analyze the behavior of the AHE in the orthorhombic lattice system, where the AHE is largely enhanced by the large coupling between the AMD and the spin-orbit interaction. From these findings, the AMD can be used as a descriptor in general to investigate the ferromagnetic-related physical quantities in antiferromagnets including noncollinear ones, which are detectable by using the x-ray magneto-circular dichroism.

Figure 1

Schematic pictures of the spin polarization of the AMD (a) $M_z^{\prime }$ and (b) its ${\sigma }_x$ component in real space $[\mathit{r}=(x,y,z\left)\right]$. In (a) and (b), the arrows represent the spin polarization at each $\mathit{r}$ and the colors denote the spatial distribution of ${\sigma }_z$ and ${\sigma }_x$ components ($Q_u$ and $Q_{zx}$) in Eq. (3).

Figure 2

(a) Orthorhombic lattice structure consisting of four sublattices (A–D) with hopping parameters ($t_a,t_b,t_c,t_a^{\prime },t_c^{\prime },\alpha$). (b) Collinear antiferromagnetic structure, which can be decomposed into the cluster electric quadrupole $Q_{zx}$ and spin ${\sigma }_x$.

Figure 3

(a) The $\mu$ dependence of ${\sigma }_{xy}^{\mathrm{A}}$ with fixed $\alpha =0.2$ for $h=0.5$ (red square) and $h=3$ (blue circle). (b) Staggered magnetization $m_x^{\mathrm{AFM}}$ corresponding to (a). (c, d) Contour plots of (c) ${\sigma }_{xy}^{\mathrm{A}}$ and (d) $m_x^{\mathrm{AFM}}$ in the plane of $\mu$ and $h$ at $\alpha =0.2$. (e, f) The same plot as (a, b) with fixed $h=2$ for $\alpha =0.1$ (red square) and $\alpha =0.7$ (blue circle). (g, h) The same plot as (c, d) in the plane of $\mu$ and $\alpha$ at $h=2$. In (c) and (g), the dashed and solid lines represent the eigenvalues at R and U points in the Brillouin zone as shown in the inset of Fig. 4.

Figure 4

(a) Electronic band structure along the symmetry lines of the Brillouin zone in the top-right inset at $\alpha =0.2$ and $h=3$. The bottom-left inset indicates the schematic picture of an effective flux $\varphi$ (thick orange arrow) and the effective imaginary hopping (green arrows on the bond) under the AMD. (b) DOS with the same parameters as (a). The red curves stand for the total DOS, while the blue dashed (green dotted) curves represent the DOS at $k_y=0$ ($k_y=\pi$).