Electric-field-induced magnetic toroidal moment and nonlinear magnetoelectric effect in antiferromagnetic olivines

Beyond conventional electric and magnetic monopoles, electric and magnetic toroidal monopoles, which are rank-0 multipoles distinguished by opposite parities under spatial inversion and time reversal, can exist in nature. The recent observation of electric-field-induced directional dichroism in antiferromagnetic olivine Co$_2$SiO$_4$ has provided the first concrete example of a magnetic toroidal monopole; however, its microscopic origin remains elusive. Here, we propose a minimal spin model that incorporates magnetoelectric coupling via the $d$-$p$ hybridization mechanism and analyze it within the mean-field approximation. The model qualitatively reproduces the experimentally observed temperature dependence of the dielectric constant and its pronounced sensitivity to the direction of the applied electric field. Furthermore, it elucidates the temperature evolution of the magnetic toroidal monopole and the strong electric-field-direction dependence of the magnetic toroidal moment. Our calculations also predict a second-order nonlinear magnetoelectric response, consistent with the symmetry classification of Co$_2$SiO$_4$ as an altermagnet. Additionally, we demonstrate that the same framework is applicable to other antiferromagnetic olivines with analogous magnetic order, indicating the robustness and generality of the toroidal-type magnetoelectric response in this material family.