Electron chirality and hydrodynamic helicity: Analysis in the atomic limit

TL;DR

The paper analyzes two complementary measures of electronic handedness: electron chirality $\tau^Z(\bm r)$, a relativistic one-body quantity that requires spin–orbit coupling and chiral crystal fields, and hydrodynamic helicity $H(\bm r)$, a two-body pseudoscalar that can arise from electron–electron interactions even without SOC. Using a minimal atomic model with chiral crystal-field configurations, the authors dissect how crystal fields, SOC, and interactions generate these chiralities, revealing that electron chirality is enhanced near quasi-degenerate points and can become SOC-insensitive, while hydrodynamic helicity scales linearly with interaction strength and is governed by $C_2$ symmetry, with no divergence near degeneracies. The work highlights distinct symmetry-based mechanisms for chiral electronic phenomena and provides design principles for materials with chiral transport and optical responses, while connecting the electron chirality to Berry-connection concepts in the absence of SOC. These insights may extend to ionic crystals and molecular systems, offering a framework to understand and engineer electronic handedness in complex materials.

Abstract

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.

Figures (15)

• Figure 1: (a) Schematic figure for the atomic model. (b) Configurations of crystal fields. The cross symbols indicate the nuclear potential at the origin, and the electronic cloud is illustrated as a gray shaded area. The red, orange, and yellow circles represent the charges of the crystal fields.
• Figure 2: (a) Energy diagram of the lowest three eigenstates. (b) Enlarged view of energy diagram of the first and second excited state near the point P $[(t, s) = (0.0,0.505) \to (0.0,0.51)]$ for several values of $\lambda_{\mathrm{SOC}}$. (c) Enlarged view of energy diagram of the first and second excited state near the point F $[(t, s) = (0.51, 0.0) \to (0.49, 0.0)]$.
• Figure 3: Electron chirality in the ground state. The point where the electron chirality vanishes is denoted as Q.
• Figure 4: (a) Electron chirality in the first excited state. (b) Enlarged view of the electron chirality of the first excited state near the peak [$(t,s)=(0.0,0.49)\rightarrow(0.0,0.55)$]. The inset shows the distribution of the full width at half maximum $\Delta S$ for each SOC strength. (c) Enlarged view of the first excited-state electron chirality near the F point [$(t,s)=(0.51,0.0)\rightarrow(0.49,0.0)$].
• Figure 5: (a) Positions of the peak points P (green line) and zero points Q (orange line) on the $t\mathchar'-s$ plane of electron chirality in a symmetric logarithmic scale. (b) $L$-dependence of peak height at point P for electron chirality. (c) $C_2$ eigenvalues for first and second excited states.