Spin-gradient–induced polarization in magnetic textures: A variational continuum theory of skyrmions and spin waves

We develop a continuum variational theory describing the coupling between magnetic textures and electric polarization in systems where spatial gradients of the spin field generate an electrical response. The framework is formulated through a thermodynamically consistent Landau–Ginzburg free-energy functional that couples the spin field S , electric polarization P , and electric potential Φ. The model introduces a gradient magnetoelectric interaction of the form λ ijk P i ∂ j S k , linking polarization to spatial variations of the spin texture. Within this formulation, spatial inhomogeneities of the magnetization act as geometric sources of electric polarization, yielding the relation P = − λχ ∇ ⁡ × S for isotropic media. To elucidate the physical implications of this coupling, we analyze several illustrative boundary-value problems. In particular, a periodic spin spiral is shown to generate a corresponding polarization wave, predicting an electrical voltage signal that can serve as an experimental signature of spin–polarization coupling. A confined magnetic texture similarly produces localized polarization structures directly tied to spin curvature. The theory further admits nonlinear topological equilibrium solutions, including axisymmetric skyrmions whose spin textures generate annular polarization rings. Explicit analytical profiles for Néel- and Bloch-type skyrmions reveal distinct polarization vortices and characteristic chiral signatures associated with the topology of the spin field. Linearization around a uniform configuration yields closed-form dispersion relations for transverse and longitudinal spin waves. The analysis shows that the gradient magnetoelectric coupling selectively softens the transverse branch by renormalizing the exchange stiffness, thereby modifying the propagation of spin excitations. Phase diagrams constructed from the analytical solutions demonstrate how the effective stiffness A eff = A − λ 2 χ governs skyrmion size, polarization strength, and the transition between isolated skyrmions and skyrmion lattice phases. Together, these results establish a unified continuum framework linking magnetic topology, induced polarization, and spin-wave dynamics. The model provides a predictive tool for analyzing gradient magnetoelectric effects in multiferroic and chiral magnetic materials and suggests new strategies for electrical detection and control of complex spin textures in spintronic devices.