Toroidal order in metals without local inversion symmetry

Toroidal order, given by a composite of electric and magnetic orders, manifests itself not only in unusual magnetism but also in anomalous transport and magnetoelectric effects. We report our theoretical results on the influence and stability of toroidal order in metals on the basis of a microscopic model. We consider an effective single-band Hubbard-type model with a site-dependent antisymmetric spin-orbit coupling, which is derived from a four-band tight-binding model including atomic spin-orbit coupling, off-site hybridization between orbitals with different parities, and an odd-parity crystalline electric field. For this single-band model on a layered honeycomb lattice, we investigate the electronic structure, magnetotransport, and magnetoelectric effect in a toroidal ordered state with a vortexlike magnetic structure. The ferroic order of the microscopic toroidal moments acts as an effective gauge field for electrons, which modulates the electronic band structure with a shift of the band bottom in momentum space. In addition, the site-dependent antisymmetric spin-orbit coupling gives rise to highly anisotropic Hall responses. The most salient feature is two different types of magnetoelectric response: one is a magnetic order with net toroidal magnetization induced by an electric current perpendicular to the planes, and the other is a uniform transverse magnetization induced by an electric current within the planes. We examine the ground state of the effective model by the mean-field approximation, and show that the toroidal order is stabilized by strong electron correlations at low electron density. We also discuss the temperature dependence of the magnetoelectric effects associated with spontaneous toroidal ordering. Implications for experiments are also presented.

Figure 1

Schematic picture of the toroidal moment, which is defined by the vector product of the position from the inversion center and the magnetic moment. See Eq. (3).

Figure 2

Schematic pictures of lattice structures with local inversion symmetry breaking: (a) One-dimensional zigzag chain, (b) honeycomb lattice, and (c) 1/5-depleted square lattice. Circles indicate the inversion centers, while triangles represent the lattice sites at which the inversion symmetry is broken.

Figure 3

(a) Schematic picture of a projection of the layered honeycomb lattice onto the $xy$ plane. The thin (blue) arrows show the magnetic pattern assumed in the toroidal ordered state [Eq. (18)], and the thick (green) arrows indicate the specific directions of the odd-parity crystalline electric field [Eq. (17)]. The dashed hexagon represents the six-sublattice unit cell with the sublattice indices $(l=0,1,...,5).a$ is the lattice constant in the $xy$ plane. (b) Schematic picture of the layered honeycomb lattice. The red arrows in the $z$ direction represent the toroidal magnetizations. $c$ is the lattice constant in the $z$ direction.

Figure 4

Energy dispersion along the $k_z$ direction of the Hamiltonian in Eq. (16) with Eqs. (17) and (18) for (a) $D=0,M_{\mathit{T}}=0$, (b) $D=0,M_{\mathit{T}}=8$, (c) $D=0.5,M_{\mathit{T}}=0$, and (d) $D=0.5,M_{\mathit{T}}=8$. The dashed red lines in (d) indicate the band bottoms in the $k_z$ direction.

Figure 5

Electron density dependence of the Hall conductivity for (a) $M_{\mathit{T}}=0$ and (b) $M_{\mathit{T}}=8$. The data are obtained at $D=0.5$ in the magnetic field $H=0.5$ applied in the $x$ direction. We take the broadening factor $\delta =0.01$ and temperature $T=0.1$.

Figure 6

Electron density dependence of (a) the toroidal-current correlation $K_{\mathit{T}z}$ and (b) the magnetization-current correlation $K_{xy}$. The insets of (a) and (b) show their $M_{\mathit{T}}$ dependences at $n_{\mathrm{e}}=0.1$. The data are obtained for $D=0.5,\delta =0.01$, and $T=0.1$. Schematic pictures for the magnetoelectric responses are shown in the bottom panels. In each panel, the left and right pictures show the alignment of the magnetic moments before and after the electric current is applied. In the rightmost one, the arrow in the center of hexagon indicates a net uniform magnetization induced by the electric current.

Figure 7

Summary of the magnetoelectric effects under toroidal or flux orders. In the table, “toroidal” and “flux” indicate the underlying magnetic orders for which the magnetic response in the top row is induced by the electric current in the left column. “Any” represents paramagnetic or any magnetically ordered state, including the toroidal and flux orders. Schematic pictures of the toroidal and flux orders are shown at the bottom; the arrows indicate the magnetic moments.

Figure 8

Ground-state phase diagram of the model in Eq. (23) on a layered honeycomb lattice obtained by the mean-field calculations. The data are taken at $D=3$. Schematic pictures of the ordering patterns are shown in the bottom panel. The arrows represent magnetic moments. “Other MO” represents other complicated magnetically ordered states. Phase 1 corresponds to the toroidal ordered state. There exist no flux-type orders.

Figure 9

Temperature dependences of (a) the magnitude of the toroidal magnetic moments $m_{\mathit{T}}$, (b) $K_{\mathit{T}z}$, and (c) $K_{xy}$ at $U=40,D=3$, and $n_{\mathrm{e}}=0.05$. The inset of (a) shows the $U$ dependence of $T_c$ for $n_{\mathrm{e}}=0.05$. $K_{\mathit{T}z}$ in a broader temperature range is shown in the inset of (b). The shaded temperature region indicates the toroidal ordered phase and the vertical dotted lines show the transition temperature $T_c$.

Figure 10

(a) Schematic energy diagram in the derivation of the effective single-band model in Eq. (16) from the four-band model in Eq. (8). The dashed circle represents the energy levels considered in the effective single-band model.