Time Reversed: What the Schrödinger Equation Misses ✛

Two translators listen to the same symphony. One hears only the fundamental tone: a clean, predictable melody. The other catches the overtones that color every note. In quantum mechanics, two equations play the role of these translators: time-reversal symmetry is a property of the score, while the Schrödinger-Pauli equation and the Dirac equation are two ways to read it.

In magnetically ordered materials, where electron spins align into a collective pattern, this symmetry is broken. For a long time, it was thought that without spin-orbit coupling, the connection between spin and electron motion, the nonrelativistic description would not notice the breaking. It remains "deaf" to overtones, like a translator who cannot hear upper harmonics. But the authors of the study showed that the Dirac equation, even in the low-velocity limit, retains a trace of the breaking. In it, each spin block remains complex, which is the mathematical signature of the absence of time-reversal symmetry.

The key tool turned out to be the Berry curvature, a geometric phase that affects electron motion. For a nonrelativistic electron, it equals ±1/(2m²c²), independent of spin-orbit coupling. This is an intrinsic property, like the spin magnetic moment eℏ/(2m). In a ferromagnet with an imbalance of up and down spins, this curvature generates a contribution to the anomalous Hall conductivity, a transverse voltage without an external magnetic field. Numerically, the contribution is small, on the order of 10⁻³ (Ω·cm)⁻¹, but it exists, and this changes the classification of magnetic effects. The Schrödinger-Pauli equation in the same limit yields real Hamiltonians that preserve time-reversal symmetry. There, the Berry curvature appears only if you add spin-orbit coupling that depends on the potential.

Previously, the boundary between "nonrelativistic" and "requiring spin-orbit coupling" magnetic phenomena was drawn along this line. Now it is clear: the boundary is not there. Time-reversal symmetry breaking can be a direct consequence of the relativistic structure of the electron, not of an additional interaction. For theories of altermagnetism and spin groups, where symmetry classification plays a key role, this is an important refinement. Looking ahead, there is a test on realistic band structures of ferromagnets with weak spin-orbit coupling, for example in light elements. And the question of the magnetoelectric effect remains open: its microscopic theory may also not require spin-orbit coupling.

The two translators return to the score. One still hears only the fundamental tone, and his translation is predictable but incomplete. The second hears the overtones, and it is they that tell us time in this music flows in only one direction. The Dirac equation turned out to be the translator that does not lose this detail.