Recent theoretical work has established the presence of hidden spin and orbital textures in non-magnetic materials with inversion symmetry. Here, we propose that these textures can be detected by nuclear magnetic resonance (NMR) measurements carried out in the presence of an electric field. In crystals with hidden polarizations, a uniform electric field produces a staggered magnetic field that points to opposite directions at atomic sites related by spatial inversion. As a result, the NMR resonance peak corresponding to inversion partner nuclei is split into two peaks. The magnitude of the splitting is proportional to the electric field and depends on the orientation of the electric field with respect to the crystallographic axes and the external magnetic field. As a case study, we present a theory of electric-field-induced splitting of NMR peaks for $^{77}$Se, $^{125}$Te and $^{209}$Bi in Bi$_2$Se$_3$ and Bi$_2$Te$_3$. In conducting samples with current densities of $simeq 10^6, {rm A/cm}^2$, the splitting for Bi can reach $100, {rm kHz}$, which is comparable to or larger than the intrinsic width of the NMR lines. In order to observe the effect experimentally, the peak splitting must also exceed the linewidth produced by the Oersted field. In Bi$_2$Se$_3$, this requires narrow wires of radius $lesssim 1, mu{rm m}$. We also discuss other potentially more promising candidate materials, such as SrRuO$_3$ and BaIr$_2$Ge$_2$, whose crystal symmetry enables strategies to suppress the linewidth produced by the Oersted field.
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