Helical Dirac Current with Local Coupling to a Chiral Potential

Abstract: We show that exact Dirac eigenstates in cylindrical confinement carry a definite helical conserved-current texture even in the zero orbital angular momentum channel l = 0. For the lowest confined mode, the Dirac current contains a nonvanishing azimuthal component together with longitudinal transport and exhibits opposite handedness in the two spin-resolved sectors. The structure also persists into the evanescent region. We further derive the channel-resolved matrix-element kernel generated by a static chiral scalar potential acting on the confined l = 0 Dirac modes. The resulting spin-selective coupling arises from the Dirac current texture and the scalar chiral potential, and yields a geometric selection rule in which diagonal channels vanish while off-diagonal conversion channels survive. The coupling strength is governed by an internal sampled-current overlap Jchi(k), defined as the integral from 0 to R of f(rho) times jphi_up(rho, k) times rho d rho. This quantity measures the spatial overlap between the chiral radial profile and the spin-up azimuthal Dirac-current density. The mechanism is fully local and texture-based, without external magnetic fields or spin-orbit coupling. Within standard Dirac theory, this work identifies the minimal static Dirac-geometric kernel underlying spin-selective response, establishing a baseline structure from which dynamical-medium, scattering, and transport formalisms can be systematically developed toward a complete description of spin-polarization phenomena such as CISS.

• The paper demonstrates that confinement of Dirac electrons yields an intrinsic helical current even in the absence of orbital angular momentum.
• The analysis reveals a strict selection rule where spin-preserving transitions vanish and only off-diagonal spin-flip couplings remain.
• The findings provide a foundational framework for developing quantum transport models and understanding chiral-induced spin selectivity.

Helical Dirac Current with Local Coupling to a Chiral Potential: An Analytical Summary

Introduction and Motivation

The investigation centers on the intrinsic properties of one-dimensional confined Dirac electrons and how these properties affect their interaction with locally chiral potentials. While spatial helical electron flow is typically attributed to orbital angular momentum or external electromagnetic fields, this work rigorously establishes that confinement itself—absent of orbital angular momentum ($l=0$) and in the absence of external fields—yields a conserved current with a helical (screw-like) geometry embedded in the Dirac spinor structure. The resulting current texture directly enables spin-resolved, selection-rule-defined coupling to static chiral scalar potentials, constituting a local geometric mechanism for spin selectivity in chiral environments without recourse to spin-orbit coupling.

Confined Dirac Eigenstates and Conserved Currents

The study analyzes solutions of the Dirac equation in a cylindrical geometry with a step radial potential, focusing on the lowest ($l=0$) radial mode. Despite the absence of orbital angular momentum or explicit helical phase winding in the eigenstates’ charge density, the Dirac current exhibits a nonvanishing azimuthal component in addition to its longitudinal flow. The key results are as follows:

• Vanishing Radial Current: $j_\rho^{\uparrow,\downarrow}=0$ identically.
• Longitudinal and Azimuthal Current Structure: For each spin sector, $j_z^{\uparrow}(\rho;k)$ captures forward motion along $z$, while $j_\phi^{\uparrow}(\rho;k)$ (and its opposite for $\downarrow$) encodes azimuthal circulation. Thus, the flow is intrinsically helical, yet spin-resolved: opposite spin sectors possess currents of opposite handedness, with identical longitudinal but anti-symmetric azimuthal components.
• Streamline Geometry: The local helical pitch is quantified by $d\phi/dz = j_\phi(\rho)/[\rho j_z(\rho)]$—setting a geometric pitch $Z(\rho)$ that is distinct from the de Broglie wavelength and solely determined by the local ratio of azimuthal to longitudinal current.

For representative physical parameters (e.g., a 1 nm radius tube with a 2 eV barrier and 25 meV kinetic energy), the characteristic radius of maximal azimuthal current is found at $\rho^* = 0.51$ nm, and the corresponding helical pitch is $l=0$0 nm.

Local Geometric Coupling to Chiral Scalar Potentials

The central analytic development addresses coupling of the described helical Dirac spinor structure to a static chiral scalar potential with a spatial screw phase ($l=0$1 modulation). Key derivations and findings:

• Matrix-Element Kernel: The channel-resolved matrix element $l=0$2 is formalized. The potential supports only $l=0$3 angular harmonics.
• Spinor Structure and Selection Rule: The diagonal (spin-preserving) matrix elements vanish identically: $l=0$4. This stringent geometric selection rule arises because the diagonal spinor overlaps cannot compensate the potential’s angular screw phase, resulting in complete suppression of handedness-preserving transitions at first order.
• Surviving Off-Diagonal Transitions: Only off-diagonal (spin-flip, handedness-converting) transitions remain,

$l=0$5

$l=0$6

where $l=0$7 measures the spatial overlap between the chiral potential’s radial profile and the azimuthal Dirac-current density. The longitudinal delta functions strictly enforce momentum conservation in the coupling.

• Absence of Spin-Orbit Coupling or Magnetic Fields: Notably, these effects occur even with a pure scalar potential, without any explicit spin-orbit terms or external magnetic field, distinguishing this mechanism from standard CISS models reliant on spin-orbit interaction.

Implications and Theoretical Consequences

The implications for physical theory and potential applications are significant:

• Minimal Dirac-Geometric Kernel: The work rigorously derives the minimal local-geometric interaction kernel for spin-selective response of confined Dirac electrons in chiral environments. It sets a strictly enforced selection rule architecture—vanishing diagonal elements and spin-selective off-diagonal matrix elements—dictated entirely by internal current geometry and external potential symmetry.
• Foundational Baseline for CISS: With the demonstrated coupling structure, this framework is positioned as a fundamental ingredient for future dynamical models of the chiral-induced spin selectivity (CISS) effect, and as a starting point for nonperturbative treatments (e.g., T-matrix, transport theory).
• Distinction from Charge-Density Models: The coupling is mediated via the conserved current rather than mere charge density. Observable response thus hinges on the geometric and phase structure of the spinor current, not solely on the occupation or density of eigenmodes.
• Generalization to Chiral Environments: The theoretical structure derived here generalizes to chiral environments beyond the depicted cylindrical geometry, provided the crucial local overlap with a nonvanishing azimuthal Dirac-current density is preserved.

Strong Numerical Results and Claims

• Intrinsic Helical Current at $l=0$8: The confined exact Dirac eigenstate carries a significant helical current texture (with pitch $l=0$9 nm for tabulated parameters), without any orbital angular momentum.
• Strict Selection Rule: Diagonals of the channel-resolved matrix element kernel are identically zero for a screw-phase chiral scalar potential, irrespective of dynamical details, establishing robust analytic control over spin selectivity at the matrix-element level.

Future Directions

Further developments are naturally motivated:

• Complete Transport and Scattering Theory: The present work stops at the static matrix-element level; full dynamical models for transmission, reflection, and spin-polarization observables in CISS systems should build directly on the kernel derived herein.
• Ab Initio Modeling for Biophysical and Nanoelectronic Systems: Given the direct connection to spin transport phenomena in helical organic molecules, conserving current-based geometric coupling models may be pivotal in resolving ongoing debates regarding the microscopic origin of CISS [see, e.g., Naaman & Waldeck, Annual Review of Physical Chemistry (2015)].
• Beyond First-Order Perturbation: Higher-order, nonperturbative, or strongly coupled regimes can be addressed by promoting the Dirac-geometric selection structure into interacting field-theoretic or quantum transport frameworks.
• Dynamic, Time-Dependent Coupling: Extension to dynamic chiral potentials or time-varying constraining geometries could yield novel regimes for spintronic device operation or quantum control.

Conclusion

This work establishes—at the analytic and kernel level—a minimal geometric mechanism for spin-selective response of confined Dirac electrons in chiral environments, reliant solely on the spatial structure of the Dirac-conserved current and scalar chiral potentials. A strict selection rule suppresses diagonal (spin-preserving) transitions, while off-diagonal (spin-flipping) transitions are supported with explicit geometric and symmetry control. This framework supplies a robust, physically transparent baseline for future dynamical, transport, and device-theoretic studies of spin selectivity in chiral systems and sets a new standard for the minimal ingredients required in analytic modeling of CISS-like phenomena.

Reference: "Helical Dirac Current with Local Coupling to a Chiral Potential" (2606.17618)