Spontaneous spin momentum locking and anomalous Hall effect in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>BiFeO</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>

The studies for the discovery of new nontrivial degrees of freedom of Bloch electrons are at the forefront of condensed matter physics. The focus remains on the spin-orbit coupling (SOC)-induced pseudospin valley state and its interplay with time inversion asymmetry. Here we propose a distinct magnetic order-protected, momentum-dependent spin-splitting valley (SSV) state in polar bulk ${\mathrm{BiFeO}}_{3}$. The octahedral distortion-induced spatial asymmetry of the superexchange interaction path breaks spin degeneracy and antiferromagnetic time-reversal symmetry. The SSV polarization is caused by crystal symmetry-related intrinsic SOC and leads to nonzero anomalous Hall conductivity (AHC) and magneto-optical effects. The anomalous Hall conductivity and magneto-optical effects are expected to be most prominent if magnetic and electric dipoles are orthogonal. Due to the ferroelectric nature of ${\mathrm{BiFeO}}_{3}$, such unconventional magneto-optical effect and AHC in fully compensated magnetic order offers straightforward tunability under an electric field.