Time inversion symmetry in the Dirac and Schrödinger-Pauli theories

Journal Article · Fri Aug 08 00:00:00 EDT 2025 · Physics Letters. A

DOI:https://doi.org/10.1016/j.physleta.2025.130873· OSTI ID:3020180

Winkler, R. ^[1]; Zülicke, U. ^[2]

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1. Northern Illinois Univ., DeKalb, IL (United States); Argonne National Laboratory (ANL), Argonne, IL (United States)
2. Victoria Univ. of Wellington (New Zealand)

The Schrödinger-Pauli theory is generally believed to give a faithful representation of the nonrelativistic and weakly relativistic limit of the Dirac theory. However, the Schrödinger-Pauli theory is fundamentally incomplete in its account of broken time inversion symmetry, e.g., in magnetically ordered systems. Here, 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 Schrödinger-Pauli theory in the absence of spin-orbit coupling. In the Dirac theory, the Berry curvature $1/(2m^2 c^2)$ 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 Schrödinger-Pauli equation. In ferromagnetically ordered systems, the intrinsic Berry curvature yields a contribution to the anomalous Hall conductivity independent of spin-orbit coupling.

• Research Organization: Argonne National Laboratory (ANL), Argonne, IL (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division (MSE)
• Grant/Contract Number: AC02-06CH11357
• OSTI ID: 3020180
• Journal Information: Physics Letters. A, Journal Name: Physics Letters. A Vol. 558; ISSN 0375-9601
• Publisher: Elsevier
• Country of Publication: United States
• Language: English