Optical Hall response in spin-orbit coupled metals: Comparative study of magnetic cluster monopole, quadrupole, and toroidal orders

The optical Hall response is theoretically studied for spin-orbit coupled metals with ferroic orders of cluster-type magnetic multipoles. We find that different magnetic multipoles give rise to distinct spectra in the optical Hall conductivity. In the cases of monopole and quadrupole orders, the optical Hall response appears predominantly in high- and low-energy regions, which correspond to the energy scales of electron correlation and kinetic energy, respectively, while the response is dispersed and rather weak in the case of toroidal order. By decomposing the spectra into different interband contributions, we reveal selection rules stemming from the interplay between the antisymmetric spin-orbit coupling and the underlying multipoles. Our results suggest that the optical Hall measurement is useful to detect and distinguish the cluster-type magnetic multipole orders.

Figure 1

Schematics of (a) a perspective view and (b) a top view of the layered square-cluster lattice. In (a), the transfer integrals $t_1, t_2$, and $t_z$ are shown. $a$ and $c$ are the lattice constants in the $xy$ plane and in the $z$ direction, respectively. In (b), $\tilde{a}$ represents the size of the square cluster. The dotted square indicates a unit cell (a square cluster) with four sublattices $\alpha , \beta , \gamma$, and $\delta$. The gray arrows denote the directions of ${\mathit{D}}_l$ at each site; see Eqs. (7) and (8). (c)–(e) Schematics of cluster-type magnetic multipoles composed of four magnetic dipoles (red arrows): (c) monopole (d) quadrupole, and (e) toroidal.

Figure 2

Electronic band structures of the model in Eq. (1) in the presence of the ferroic order of monopole (dashed blue), quadrupole (dash-dotted red), and toroidal (black solid). The inset shows the first Brillouin zone and the green lines indicate the symmetric lines on which the electronic band structures are presented. The results are obtained with $a=c=1, \tilde{a}=0.35, t_1=1.25, t_2=0.75, t_z=1, D=0.5, M=8$, and $\mathit{B}=(0.5,0,0)$.

Figure 3

Optical Hall conductivity ${\sigma }^{yz}\left(\omega \right)$ as a function of the energy $\omega$ in the presence of the multipole orders of (a) monopole, (b) quadrupole, and (c) toroidal. The green solid (black dashed) line indicates the real (imaginary) part of ${\sigma }^{yz}\left(\omega \right)$. The results are computed at the electron filling of $n_{\text{e}}=0.1$ with $k_{\text{B}}T=0.1$ and $\delta =0.02$ in Eqs. (13) and (14). The other model parameters are common to those in Fig. 2.

Figure 4

Integrated intensities of $\left|\text{Re}{\sigma }^{yz}\left(\omega \right)\right|$ as functions of the electron filling $n_{\text{e}}$ in the presence of (a) monopole, (b) quadrupole, and (c) toroidal orders [see Eqs. (15) and (16)]. The parameters except for $n_{\text{e}}$ are common to those in Fig. 3.

Figure 5

Histogram of the integrated intensities of the interband contributions. The blue and red bars represent the low- and high-energy intensities, $I_{m,n}^{\mathrm{low}}$ in Eq. (17) and $I_{m,n}^{\mathrm{high}}$ in Eq. (18), respectively, in the presence of (a) and (b) monopole, (c) and (d) quadrupole, and (e) and (f) toroidal orders. The results are shown for $n=1$ in (a), (c), and (e), while $n=2$ in (b), (d), and (f). The parameters are common to those in Fig. 3.

Figure 6

Dominant interband contributions ${\sigma }_{m,n}^{yz}\left(\omega \right)$ in the presence of (a) monopole, (b) quadrupole, and (c) toroidal orders. The parameters are common to those in Fig. 3.

Figure 7

Schematic picture of the energy levels. The eightfold degeneracy in a four-site cluster is lifted successively by ${\mathcal{H}}_{\text{MF}}, {\mathcal{H}}_t$, and ${\mathcal{H}}_{\text{Zeeman}}$.

Figure 8

Schematics of (a) a perspective view and (b) a top view of the layered honeycomb lattice. In (a), the transfer integrals $t$ and $t_z$ are shown. $a$ and $c$ are the lattice constants in and out of the plane, respectively. In (b), the dotted hexagon indicates the six-sublattice magnetic unit cell. The gray arrows denote the directions of ${\mathit{D}}_l$ at each sublattice; see Eqs. (7) and (A1). (c)–(e) Schematics of cluster-type magnetic multipoles composed of six magnetic dipoles (red arrows): (c) monopole (d) quadrupole-type, and (e) toroidal.

Figure 9

Optical Hall conductivities (a), (c), and (e) ${\sigma }^{yz}\left(\omega \right)$ and (b), (d), and (f) ${\sigma }^{xz}\left(\omega \right)$ as functions of the energy $\omega$ in the presence of the multipole orders of (a) and (b) monopole, (c) and (d) quadrupole-type, and (e) and (f) toroidal. The green solid (black dashed) line indicates the real (imaginary) part of the optical Hall conductivities. The results are obtained at $a=c=1, t=t_z=1, D=0.5, M=8, k_{\text{B}}T=0.1$, and $\delta =0.02$. The electron filling is set at $n_{\text{e}}=0.04$. The magnetic field is applied along the $x$ and $y$ directions for ${\sigma }^{yz}\left(\omega \right)$ and ${\sigma }^{xz}\left(\omega \right)$ as $\mathit{B}=(0.5,0,0)$ and $\mathit{B}=(0,0.5,0)$, respectively.

Figure 10

Histogram of the integrated intensities of the interband contributions ${\sigma }_{m,n}^{yz}\left(\omega \right)$. The blue and red bars represent the low- and high-energy intensities, $I_{m,n}^{\mathrm{low}}$ in Eq. (17) and $I_{m,n}^{\mathrm{high}}$ in Eq. (18), respectively, in the presence of (a) and (b) monopole, (c) and (d) quadrupole-type, and (e) and (f) toroidal orders. The results are shown for $n=1$ in (a), (c), and (e), while $n=2$ in (b), (d), and (f). The parameters are common to those in Fig. 9.