Electron chirality and hydrodynamic helicity: Analysis in the atomic limit

Electron chirality has been proposed as a microscopic quantity that characterizes electronic handedness, yet its underlying control parameter has not been clearly identified. Furthermore, its applicability is limited to systems with spin-orbit coupling, which motivates the need for alternative measures of chirality. In this work, we explore two complementary measures of chirality: electron chirality and hydrodynamic helicity. By analyzing a minimal atomic model under chiral crystal fields, we clarify how the interplay among crystal fields, spin-orbit coupling, and electron correlation gives rise to non-zero values of chirality measures. Although electron chirality increases with both spin-orbit coupling and chiral crystal field strength, the dependence on these two factors is highly non-trivial. Particularly, when the chiral crystal field is varied continuously and the energy levels approach quasidegenerate points, the electron chirality is insensitive to spin-orbit coupling, resulting in a remarkable enhancement of chirality. In contrast, the hydrodynamic helicity, defined as a two-body pseudoscalar quantity, remains non-zero even without spin-orbit coupling, originating from electron-electron interactions. Perturbative analysis reveals distinct symmetry selection rules governing the two quantities. Our results provide fundamental insight into the origin of chiralities in electronic systems.

💡 Research Summary

This paper presents a fundamental theoretical analysis of two complementary measures for quantifying electron chirality (“handedness”) in materials: electron chirality and hydrodynamic helicity. The study is motivated by the limitations of the previously proposed electron chirality, which requires spin-orbit coupling (SOC) to be non-zero, leaving systems with weak SOC (like organic molecules) without a clear microscopic chirality metric.

The authors employ a minimal atomic model subjected to a tunable chiral crystal field, represented by four point charges placed at the vertices of a distorted cube. This setup allows them to systematically dissect the interplay between three key ingredients: chiral crystal fields, SOC, and electron-electron interactions. The two chirality measures are defined as follows:

• Electron Chirality: A one-body pseudoscalar quantity derived from relativistic quantum theory, representing the difference between right- and left-handed electron densities. In the non-relativistic limit, it is proportional to the expectation value of p·σ (momentum dotted with Pauli matrices). Its non-zero value necessitates the breaking of both inversion and time-reversal symmetries, typically achieved through the combined presence of a chiral crystal field and SOC.
• Hydrodynamic Helicity: A two-body pseudoscalar quantity inspired by fluid dynamics, defined as the dot product of the electron current density with its curl (j·(∇×j)). Crucially, this measure can originate purely from electron-electron interactions and does not inherently require SOC to be non-zero.

The core findings of the paper are:

1. Non-trivial Enhancement of Electron Chirality: While electron chirality generally increases with both SOC strength and the degree of chiral crystal field distortion, the relationship is highly non-linear. A key discovery is that when the crystal field is tuned such that electronic energy levels become nearly degenerate (quasidegenerate), the electron chirality becomes remarkably insensitive to the SOC strength while simultaneously reaching a significantly enhanced value. This suggests a powerful mechanism for amplifying chirality without relying on strong relativistic effects.
2. Distinct Origin of Hydrodynamic Helicity: In contrast, hydrodynamic helicity can attain a non-zero expectation value even in the complete absence of SOC, provided electron-electron interactions are present. Its value arises from genuine two-body electron correlations, as it vanishes under a simple mean-field (Wick decomposition) treatment. This makes it a potential candidate for describing chirality in correlation-driven systems with weak SOC.
3. Different Symmetry Foundations: Perturbative analysis reveals that the two quantities obey distinct symmetry selection rules, particularly under two-fold rotations. This confirms that they capture different physical aspects and transformation properties of chiral electronic states.

In conclusion, the paper establishes that “electron chirality” and “hydrodynamic helicity” represent distinct paradigms for quantifying electronic handedness: the former is an SOC-mediated one-body property, while the latter is an interaction-driven two-body property. This work provides essential insights for the design and interpretation of chiral materials, offering a broader toolkit to understand chirality across a wide spectrum of systems, from strongly correlated inorganic compounds to weakly relativistic organic molecules.