Ultracold fermions provide a new testbed for dynamical phases of quantum matter forbidden by equilibrium thermodynamics.

SO (SO) interactions give a spin-dependent correctionrso to the position operator, referred to as the anomalous position operator. We study the contributions ofrso to the spin-Hall effect (SHE) in quasi two-dimensional (2D) semiconductor quantum wells with strong band structure SO interactions that cause spin precession. The skew scattering and side-jump scattering terms in the SHE vanish, but we identify two additional terms in the SHE, due torso, which have not been considered in the literature so far. One term reflects the modification of spin precession due to the action of the external electric field (the field drives the current in the quantum well), which produces, viarso, an effective magnetic field perpendicular to the plane of the quantum well. The other term reflects a similar modification of spin precession due to the action of the electric field created by random impurities, and appears in a careful formulation of the Born approximation. We refer to these two effects collectively as anomalous spin precession and we note that they contribute to the SHE to the first order in the SO coupling constant even though they formally appear to be of second order. In electron systems with weak momentum scattering, the contribution of the anomalous spin precession due to the external electric field equals 1/2 the usual side-jump SHE, while the additional impurity-dependent contribution depends on the form of the band structure SO coupling. For band structure SO coupling linear in wave vector the two anomalous spin precession contributions cancel. For band structure SO coupling cubic in wave vector, however, they do not cancel, and the anomalous spin precession contribution to the SHE can be detected in a high-mobility 2DEG with strong SO coupling. In 2D hole systems both anomalous spin precession contributions vanish identically.