Magnetic toroidal monopoles from relativistic polarization responses to magnetic field gradients

The magnetic toroidal monopole, a time-reversal-odd scalar, has attracted attention through its characteristic responses, such as electric-field-induced nonreciprocal directional dichroism observed in Co$_2$SiO$_4$. However, its evaluation in crystalline solids remains unresolved, as it cannot be defined within conventional multipole expansions or thermodynamic formulations. In this paper, we propose a theoretical framework to evaluate the magnetic toroidal monopole in periodic crystals based on the response of relativistic electric polarization to a magnetic field gradient. By incorporating the magnetic-field-gradient correction to the relativistic polarization, we derive an explicit expression for the magnetic toroidal monopole beyond symmetry arguments. The resulting expression is formulated in terms of geometric quantity such as Berry curvatures and orbital magnetic moment defined in an extended parameter space spanning momentum, magnetic field, and electric field. We further perform model calculations for an antiferromagnetic system hosting a magnetic toroidal monopole and confirm that the proposed quantity is finite. These results provide a practical route to characterize magnetic toroidal monopoles in crystalline solids and clarify their quantum geometric nature.