Apparent and hidden spin splitting and polarization in crystalline solids

(a) Apparent effect X refers to cases where X is enabled by a specific global symmetry and experimentally observed as such. An example is the Rashba effect enabled by the broken inversion symmetry in non-magnets (e.g. BiTeI). In contrast, (b) “hidden effect X”, refers to the general condition where the nominal global symmetry would disallow the effect X, whereas the symmetry of the local parts of the system would allow X. A known example is hidden Rashba spin polarization in centrosymmetric systems of nonmagnets (e.g. LaOBiS₂ and BaNiS₂). Both (a) and (b) require spin-orbit coupling (SOC). Here we describe (c) apparent and (d) hidden effects that exist independent of SOC. (c) Apparent, SOC-independent spin splitting is a non-relativistic effect enabled by broken time-space reversal and spin-translation reversal symmetry. Here, magnetic order breaks spin degeneracy (e.g. MnTe) and includes even spin splitting at the Brillouin-zone center. (d) Hidden, SOC-independent spin polarization is the case where local spin polarization appears and compensated to zero (e.g. CuMnAs). We will provide examples and discussion of properties of (c) and (d). Furthermore, analogies with breaking of spatial symmetry, transforming metals to insulators and the emergence of “flat bands” (e.g. NbO₂ and LaTiO₃), will be given.