The Electron's Heel: What Dirac Knew and Pauli Missed ⚡ экспресс
The Schrödinger-Pauli equation, usually taken as the exact non-relativistic approximation of the Dirac theory, proves incomplete when describing systems with broken time-reversal symmetry, such as magnetic crystals. Within the Dirac framework, magnetic order breaks T-symmetry even when spin-orbit coupling is neglected, whereas in the Schrödinger-Pauli equation without spin-orbit coupling this symmetry is preserved. A consequence is the existence of an intrinsic Berry curvature ~1/(2m²c²), which is a fundamental characteristic of non-relativistic electrons on par with the spin magnetic moment eℏ/(2m). In ferromagnets, this curvature contributes to the anomalous Hall conductivity independently of spin-orbit coupling — a contribution that the standard Schrödinger-Pauli formalism overlooks.
In a chunk of magnet, an electron behaves like a ship with a heel: even in dead calm, its trajectory curves to the side. This is the anomalous Hall effect—motion with a transverse drift without an external magnetic field.
Previously, it was thought that it all came down to how the electron's own rotation (spin) affects its motion. But the theories of Schrödinger and Pauli missed a deeper reason. The full description by Dirac in fundamental physics shows that the electron itself possesses a built-in path curvature (Berry curvature). It’s as if a vessel had an asymmetric hull and couldn’t sail straight.
The magnitude of the heel is given by the constant 1/(2m²c²), where m is the electron mass, c is the speed of light. In magnetically ordered crystals, where all spin-"compasses" point the same way, time symmetry is broken. This creates an additional electric field—that very anomalous signal. Surprisingly, such curvature exists even for a stationary electron. Practically, it promises electronics without bulky magnets: current can be controlled using the material’s internal curvature.
🎯 Discovered over a century ago, the effect long remained a mystery; only recently was it found that the answer lies hidden in Dirac’s equations—and this "heel" is fundamental even for stationary particles.