  We propose an intrinsic nonlinear spin magnetoelectric effect in magnetic
materials, offering the potential of all-electric control of spin degree of
freedom in centrosymmetric magnets, which reside outside of the current
paradigm based on linear spin response. We reveal the band geometric origin of
this effect in the momentum and magnetization space Berry connection
polarizabilities, and clarify its symmetry characters. As an intrinsic effect,
it is determined solely by the material's band structure and represents a
material characteristic. Combining our theory with first-principles
calculations, we predict sizable nonlinear spin magnetoelectricity in
single-layer MnBi$_{2}$Te$_{4}$, which can be detected in experiment. Our
theory paves the way for exploring rich nonlinear spintronic effects and novel
device concepts based on them.
