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.