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

Andrzej Czarnecki, Ting Gao

TL;DR

The paper investigates whether a uniform gravitational field can induce a spin Hall–like transverse response for Dirac particles. It combines a hidden-momentum argument with a Foldy–Wouthuysen analysis to demonstrate that spin-dependent transverse motion is absent at linear order in the gravitational field $g$, with any such effects arising only at $O(g^{2})$ for broad states. It contrasts this with the Karplus–Luttinger anomalous Hall effect in crystals, which relies on Bloch-band structure and lattice-induced interband matrix elements, a mechanism absent in free-space gravity. The results clarify that a gravitational spin Hall effect does not exist in a uniform field and highlight the essential role of lattice periodicity for Hall transport phenomena.

Abstract

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.

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

TL;DR

The paper investigates whether a uniform gravitational field can induce a spin Hall–like transverse response for Dirac particles. It combines a hidden-momentum argument with a Foldy–Wouthuysen analysis to demonstrate that spin-dependent transverse motion is absent at linear order in the gravitational field , with any such effects arising only at for broad states. It contrasts this with the Karplus–Luttinger anomalous Hall effect in crystals, which relies on Bloch-band structure and lattice-induced interband matrix elements, a mechanism absent in free-space gravity. The results clarify that a gravitational spin Hall effect does not exist in a uniform field and highlight the essential role of lattice periodicity for Hall transport phenomena.

Abstract

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.

Paper Structure

This paper contains 7 sections, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. What constitutes a Hall effect?
  3. Spinning particle in gravitational field
  4. The hidden momentum
  5. Trajectory in a uniform gravitational field
  6. Comparison with Hall effects in Ferromagnetics
  7. Summary

Figures (2)

  • Figure 1: Masses circulating in a frictionless pipe bent into a rectangle. The rectangle is placed in the constant gravitational field with field strength $g$ near the Earth.
  • Figure 2: Trajectories for particles with spin $\sigma_y=+1$ (left), $\boldsymbol{\sigma}=0$ (middle), and $\sigma_y=-1$ (right) when (a) the trajectories experience a spin-dependent deflection; and (b) the trajectories are independent of the spin. The initial velocity is along the $z$-direction (vertical) and is chosen to be the same for all particles for the ease of comparison.