Time Reversed: What the Schrödinger Equation Loses ✛
Context
This work stands apart: the nearest neighbors in the archive explore quantum computers, time crystals, and gravitons, while here the focus is a fundamental question about how two theories describe magnetism. Much has already been done: electronic structure calculations for magnetic materials, theories of orbital magnetization, and symmetry-based classification of bands. But the question of when exactly time-reversal symmetry is broken in the nonrelativistic limit remained contentious. Some studies suggested that without spin-orbit coupling, magnetic order does not break the symmetry; others argued that it does, but only in a relativistic description. This paper brings clarity by directly comparing the two equations.
Methods
The authors take a two-dimensional model of an electron in an exchange field that mimics magnetic order. They write the Dirac equation as two independent blocks for spin up and spin down, then compute for each block the orbital magnetic moment and Berry curvature using modern theory of orbital magnetization. In parallel, they write the Schrödinger-Pauli equation in the same limit and check whether it preserves time-reversal symmetry. Comparing the two approaches reveals where exactly the information about symmetry breaking is lost.
Results
In the Dirac equation, each spin block turns out to be complex, even when the exchange field tends to zero. This means that time-reversal symmetry is always broken, regardless of spin-orbit coupling. In the nonrelativistic limit, the Berry curvature for an electron is ±1/(2m²c²), an intrinsic property like the spin magnetic moment eℏ/(2m). In a ferromagnet with an imbalance of spin-up and spin-down electrons, there arises a contribution to the anomalous Hall conductivity proportional to (e²/ℏ)(ℏ/mc)²Δn/2. Numerically, this contribution is small: for typical parameters, it gives a conductivity on the order of 10⁻³ (Ω·cm)⁻¹, which is orders of magnitude smaller than experimental values. But the size is not the point; the fact that the contribution exists without spin-orbit coupling is. The Schrödinger-Pauli equation in the same limit yields real Hamiltonians that preserve time-reversal symmetry. There, Berry curvature appears only if you add spin-orbit coupling that depends on the potential.
Implications
This work differs from its neighbors on the archive in that it does not build a new model or numerical method, but rather uncovers a fundamental difference between two equations. Neighbors investigate specific systems, such as superconductors, time crystals, and quantum computers, whereas here the question is about the very language of description. This is the first explicit demonstration that time-reversal symmetry breaking in magnets may not be a consequence of spin-orbit coupling, but a direct consequence of the relativistic structure of the electron. For theories that divide magnetic phenomena into 'nonrelativistic' and 'requiring spin-orbit coupling,' this is an important clarification: the boundary lies not where it is usually drawn.
Future development
The topic could develop toward more realistic models: three-dimensional crystals, noncollinear magnetic order, and electron interactions. It would be interesting to check how much this intrinsic contribution to the anomalous Hall conductivity shows up in real materials where spin-orbit coupling is weak, such as light ferromagnets. Also open is the question of the magnetoelectric effect: the authors mention that its microscopic theory could be constructed without spin-orbit coupling, but details have not yet been published.
Impact
The result will affect electronic structure calculations for magnetic materials, especially those where spin-orbit coupling is considered small. It is also important for theories of altermagnetism and spin groups, where the classification of symmetries plays a key role.
Next steps
Test the prediction on realistic band structures of ferromagnets using relativistic calculations. Construct a full microscopic theory of the magnetoelectric effect without spin-orbit coupling.
Key open problems
In condensed matter physics, the question remains open as to which magnetic phenomena truly require spin-orbit coupling and which arise from deeper principles. This work shows that time-reversal symmetry breaking can be embedded in the relativistic kinematics of the electron, forcing a reclassification of magnetic effects. The connection to the unresolved problem of fully describing the anomalous Hall effect in real materials is clear: the intrinsic contribution found here is small, but it sets a lower bound that cannot be ignored.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The authors examine a two-dimensional model of an electron in an exchange field that mimics magnetic order. They write the Dirac equation as two independent blocks for spin-up and spin-down and compute for each block the orbital magnetic moment and Berry curvature using the modern theory of orbital magnetization. In parallel, they write the Schrödinger-Pauli equation in the same limit and check whether it preserves time-reversal symmetry. The comparison shows that in the Dirac equation each spin block is complex, and time-reversal symmetry is always broken, whereas the Schrödinger-Pauli equation yields real Hamiltonians and preserves the symmetry if no spin-orbit coupling is added.
Where its strength is
The work uncovers a fundamental difference between the two equations in describing time-reversal symmetry breaking in the nonrelativistic limit. It shows that the Berry curvature 1/(2m²c²) is an intrinsic property of the nonrelativistic electron, analogous to the spin magnetic moment, which opens the way to revisiting the role of spin-orbit coupling in magnetic phenomena.
Where it could go next
- It might be interesting to see how this intrinsic contribution to the Berry curvature appears in magnetically ordered systems with nonstandard symmetry, for example in superconductors with twisted symmetry, where effects related to time-reversal symmetry breaking are already being studied.based on: Superconductor with Twisted Symmetry arXiv:2602.22793
- Closely related is work on magnetic ripples in a relativistic context, which considers the interaction of spins with gravitational fields. Comparing with those results might help see how the intrinsic Berry curvature behaves in curved spacetime.based on: Magnetic Ripples: How Relativity Plays with Spins arXiv:2511.01690
- In our archive there are works discussing the connection between time-reversal symmetry and fundamental questions of time, such as how a timeless world acquires a past and a future. One could check whether the discovered difference between the equations offers a new perspective on the origin of the arrow of time in quantum systems.based on: How a Timeless World Gains a Past and a Future arXiv:2605.17935
Worth trying
- Calculate the contribution of the intrinsic Berry curvature to the anomalous Hall conductivity for specific materials with known band structures and compare it with experimental data.
- Check how this contribution changes when going from the two-dimensional model to three-dimensional systems with different crystal lattice symmetries.
- Investigate whether the intrinsic Berry curvature manifests in other transport phenomena, such as the spin Hall effect or thermoelectric effects.
Why this may matter
Understanding that time-reversal symmetry breaking is embedded in the relativistic structure of the electron itself could change the interpretation of many magnetic phenomena previously attributed solely to spin-orbit coupling. This may lead to a revision of theoretical models and to the search for new materials with anomalous Hall properties.
Nearby works in our archive
- How Time Can Run Backwardssimilarity 0.53
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The archive is sparse around this work: the closest review is only 0.53, against a usual 0.64. It looks like few have drilled nearby.
In the limit of zero momentum, the orbital magnetic moment in the Dirac equation exactly equals the spin magnetic moment eℏ/(2m), even though spin is not explicitly in the equation. This coincidence is a consequence of spin being 'sewn into' the structure of the four-component wave function.
In Greg Egan's novel 'Schild's Ladder,' physicists use alternative mathematical structures to describe a world without familiar symmetries. Here we have a similar plot: the same physics looks different depending on the chosen equation, and the loss of symmetry is not a property of the world but an artifact of the approximation.
Key numbers
- Berry curvature at zero momentum: 1/(2m²c²) ≈ 1.5×10⁻⁶ Ų
- Spin magnetic moment: eℏ/(2m) ≈ 9.27×10⁻²⁴ J/T
- Estimated anomalous Hall conductivity: ~10⁻³ (Ω·cm)⁻¹
- Ratio of Berry curvature to square of Compton wavelength: ~10⁻⁶