Time inversion symmetry in the Dirac and Schr\"odinger-Pauli theories

The Schr\"odinger-Pauli theory is generally believed to give a faithful representation of the nonrelativistic and weakly relativistic limit of the Dirac theory. However, the Schr\"odinger-Pauli theory is fundamentally incomplete in its account of broken time inversion symmetry, e.g., in magnetically ordered systems. In the Dirac theory of the electron, magnetic order breaks time inversion symmetry even in the nonrelativistic limit, whereas time inversion symmetry is effectively preserved in the Schr\"odinger-Pauli theory in the absence of spin-orbit coupling. In the Dirac theory, the Berry curvature $1/(2m^2c^2)$ is thus an intrinsic property of nonrelativistic electrons similar to the well-known spin magnetic moment $e\hbar/(2m)$, while this result is missed by the nonrelativistic or weakly relativistic Schr\"odinger-Pauli equation. In ferromagnetically ordered systems, the intrinsic Berry curvature yields a contribution to the anomalous Hall conductivity independent of spin-orbit coupling.