Spin in Uniform Gravity, Hidden Momentum, and the Anomalous Hall Effect

We review the recent discussion of the absence of spin Hall effect in a uniform gravitational field, pointing out differences from the anomalous spin Hall effect in ferromagnetics despite a similar form of the Hamiltonian.

💡 Research Summary

The paper by Czarnecki and Gao investigates whether a spin‑dependent transverse deflection—analogous to a spin Hall effect—can arise for neutral Dirac particles in a uniform gravitational field. The authors begin by recalling that spin‑orbit coupling in condensed‑matter systems gives rise to Hall‑type phenomena, most famously the anomalous Hall effect (AHE) described by Karplus and Luttinger (KL) and later understood in terms of Berry curvature. Recent work (Wang, PRD 2024) claimed a “gravitational spin Hall effect” (SHE) in which particles with spin polarized along ±y acquire a transverse velocity proportional to għ/4mc².

To address this claim, the authors first introduce the concept of hidden momentum, originally discovered in electromagnetic systems, and adapt it to a gravitational context. Using a simple classical model—a rectangular pipe with circulating masses—they show that relativistic time‑dilation (γ‑factor) differences between the upper and lower streams generate a non‑vanishing x‑component of momentum even when the net linear momentum is zero. This hidden momentum can be written compactly as p_hidden = L × g / c², where L is the total angular momentum of the circulating system. The key point is that the canonical momentum p and the mechanical velocity v are no longer directly proportional; the hidden term must be accounted for.

Next, the authors analyze a Dirac wave packet in a uniform gravitational field using a Foldy‑Wouthuysen (FW) transformation that is exact to all orders in 1/c but only linear in the gravitational potential V = 1 − gz/c². The resulting FW Hamiltonian (Eqs. 8‑9) contains a spin‑orbit term (għ/4mc²)(σ_x p_y − σ_y p_x) that appears in the velocity operator but not in the canonical momentum. By writing down Hamilton’s equations, they find

 m dx/dt = ∂H/∂p_x = V p_x − (għ/4c²)σ_y.

Thus, if one prepares an initial state with ⟨p⟩ = 0, the particle already possesses a non‑zero mechanical velocity due to hidden momentum. The “at rest” condition used in the earlier claim is therefore incomplete; a true rest state requires an additional spin‑dependent phase that cancels the hidden term. When this correction is applied, the transverse velocity vanishes at order O(g). The authors further show, using the same FW framework, that any spin‑dependent transverse drift can only appear at O(g²) or higher for a broad class of wave packets. This constitutes a rigorous no‑go theorem for a linear‑order gravitational SHE.

The paper then contrasts this result with the KL mechanism in ferromagnets. In the KL picture, the Hamiltonian consists of a kinetic term, a spin‑orbit term proportional to M × ∇U (where U(r) is the periodic crystal potential), and an electric‑field coupling −eE·r. The periodic potential yields Bloch eigenstates ϕ_{n,k} and a non‑trivial Berry curvature that generates an intrinsic Hall conductivity σ_H. The crucial ingredient is the lattice periodicity, which provides matrix elements ⟨n,k|H′′₁|n′,k′⟩ that are proportional to the interband Berry connection J_{nn′}(k). In a uniform gravitational field, however, the natural basis is plane waves; there is no periodic potential, and therefore no analogue of H′′₁. Consequently, the intrinsic Hall pathway is absent, and the transverse conductivity σ_xy vanishes. The authors emphasize that while the Hamiltonians in the gravitational and ferromagnetic cases look formally similar, the underlying physics is fundamentally different because the lattice provides the “Berry curvature engine” that is missing in free‑particle gravity.

In the concluding section, the authors summarize three main points: (1) a spinning object in a uniform gravitational field acquires hidden momentum p_hidden = L × g / c², which modifies the velocity–momentum relation; (2) when the hidden momentum is properly accounted for, no O(g) transverse drift (i.e., no gravitational spin Hall effect) occurs for neutral Dirac particles; (3) the anomalous Hall effect in ferromagnets relies on crystal periodicity, and this essential ingredient is absent in the uniform‑gravity scenario, explaining why the two phenomena diverge despite superficial similarities.

The paper also notes that the literature search was assisted by ChatGPT‑5 and that figures were generated with Asymptote, acknowledging NSERC funding. Overall, the work provides a clear theoretical resolution to the recent claim of a gravitational spin Hall effect, reinforcing the importance of hidden momentum and the role of lattice structure in Hall phenomena.