We show that a propagating electron wave in the lowest cylindrical channel carries a spin-resolved helical conserved current, and connect this current texture to its local chiral coupling. In a weakly relativistic positive-energy Dirac projection, the two sectors have the same azimuthally symmetric charge density and opposite circulation superposed on longitudinal flow, even though their nonrelativistic orbital harmonic is $l=0$. The current is radially distributed and extends into the evanescent tail. A scalar screw potential couples these modes through a derived vertex expressed by the \emph{chiral current overlap}, $\mathcal J_χ$. Both spatial textures determine the coupling: the environmental radial profile weights the electronic circulation, while the screw harmonic selects correlated changes of spin sector and longitudinal momentum. For a uniform static harmonic, elastic energy conservation makes the first-order spin-flip amplitude vanish within the same transverse mode. Environmental energy exchange is therefore required to activate these selected channels. The same current overlap supplies the spatial kernel for heliciton-assisted inelastic scattering in a companion study and for a subsequent transport treatment of chirality-induced spin selectivity (CISS).
Comments
6 pages, 1 figure. Revised version with expanded discussion of the helical current, chiral coupling, and physical interpretation
We show that a propagating electron wave in the lowest cylindrical channel carries a spin-resolved helical conserved current, and connect this current texture to its local chiral coupling. In a weakly relativistic positive-energy Dirac projection, the two sectors have the same azimuthally symmetric charge density and opposite circulation superposed on longitudinal flow, even though their nonrelativistic orbital harmonic is l=0. The current is radially distributed and extends into the evanescent tail. A scalar screw potential couples these modes through a derived vertex expressed by the \emph{chiral current overlap}, Jχ. Both spatial textures determine the coupling: the environmental radial profile weights the electronic circulation, while the screw harmonic selects correlated changes of spin sector and longitudinal momentum. For a uniform static harmonic, elastic energy conservation makes the first-order spin-flip amplitude vanish within the same transverse mode. Environmental energy exchange is therefore required to activate these selected channels. The same current overlap supplies the spatial kernel for heliciton-assisted inelastic scattering in a companion study and for a subsequent transport treatment of chirality-induced spin selectivity (CISS).
We develop a quantized screw-mode mechanism for chirality-induced spin selectivity (CISS). The corresponding quantum, termed a heliciton, is a screw-symmetric environmental excitation with phase φ−qz, longitudinal momentum ℏq, and energy ℏΩq. Its absorption and emission convert the static local chiral vertex developed in our preceding work into an inelastic resonant scattering process. In first Born approximation, absorption maps ↑,k to the ↓,k+q sideband, whereas emission maps ↓,k to the ↑,k−q sideband. The two outputs share the sampled-current overlap Jχ(k) but differ in ladder factors, final momenta, and detunings. With spectral factors S+ and S−, Psb=[S−−O(T)S+]/[S−+O(T)S+], where O(T)=exp[−ℏΩq/(kBT)]. An isolated emission or absorption resonance yields Psb≃+1 or −1, respectively, in the resolved sideband sector. Reversing the screw handedness interchanges the spin identities of the two sidebands while leaving their spectral and occupation weights unchanged, and therefore reverses Psb at every temperature. At high temperature, absorption and emission have nearly equal occupation weights; at low temperature, absorption is exponentially suppressed while spontaneous emission remains. Liquid-nitrogen temperature can already produce a pronounced asymmetry for higher-Ωq modes. Thus a spatially resolved Dirac wave with spin-dependent helical conserved current couples locally to a heliciton and produces thermally weighted spin- and momentum-resolved sidebands without an ad hoc spin-dependent potential.
We show that a propagating Dirac electron with intrinsic spin generically carries a real--space helical conserved current, even in the absence of orbital angular momentum. Using exact Dirac eigenstates in cylindrical confinement, we demonstrate that this helical structure possesses definite handedness, persists into evanescent regions, and is characterized by a geometric helix pitch independent of the longitudinal de~Broglie wavelength. This intrinsic helical geometry enables a local geometric coupling between a propagating electron and a chiral environment, yielding chirality--dependent spin selectivity through current geometry rather than through a spin--orbit coupling term.
We investigate the Dirac oscillator in a helically twisted spacetime endowed with a uniform axial torsion. Starting from an orthonormal coframe, we compute the Levi--Civita spin connection explicitly and separate the geometric contribution from the axial contortion. Retaining the matrix β in the radial Moshinsky coupling, we show that the second-order problem is the ordered product Π^+Π^− rather than the square of a single operator. The resulting radial dynamics is a coupled, self-adjoint two-component system in which the spin connection supplies the correct cylindrical radial operator, while the off-diagonal metric generates the helical combination m/r−ωk and a Coulomb-like geometric term. A finite-element solution reproduces the planar Dirac-oscillator spectrum in the flat limit and reveals asymmetric dependence on the longitudinal momentum, avoided level crossings, and a supersymmetric zero mode at E=Mc2. The axial torsion and longitudinal momentum preserve this zero mode, whereas the helical twist lifts it quadratically. Sector-resolved thermodynamic functions are obtained from the relativistic bound-state spectrum. The explicit spinors further determine longitudinal vector and axial currents, and a Witten-index analysis identifies the helical twist as the deformation that removes the protected zero mode.
Cylindrical Dirac modes underlie constructions in rotating QCD matter, boost-invariant Dirac-field quantization in heavy-ion physics, and high-energy twisted-particle scattering. The corresponding complete spinor frames can be regarded as alternative bases, but their equivalence does not determine the global behavior of eigenlines selected by conserved observables. Within the positive-energy doublet of the free massive Dirac Hamiltonian, we compare three conserved resolutions: K, which couples spin to transverse momentum; Km, a mass-dependent operator derived from the transverse Dirac Hamiltonian; and helicity. On the common regular domain away from the momentum axis, explicit smooth, single-valued SU(2) transformations relate all three splittings. Although globally SU(2)-equivalent on this common domain, they exhibit three distinct global extension behaviors. The K projectors have azimuth-dependent polar limits and do not extend continuously to the axis. For nonzero mass, the Km projectors extend smoothly over the enclosed momentum ball and define Chern-trivial eigenlines. The helicity projectors are smooth on every nonzero momentum sphere, but their eigenlines carry opposite unit Chern numbers and cannot extend through the enclosed origin. The parent positive-energy Dirac connection Abelianizes exactly in the K eigenlines on fixed-azimuth meridians, whereas its full three-dimensional curvature has noncommuting components. We obtain the azimuthal Wilson loop in closed form and derive the exact conversion probability between the two K branches under purely geometric positive-energy transport.
Field-Free Transverse Aharonov--Bohm Phase Gate for an Orbital l-Qubit
The Aharonov--Bohm (AB) effect is usually read out through phase differences associated with spatially distinct electron paths. We show that confined orbital modes provide a same-path alternative: a core-confined magnetic flux writes opposite propagation phases on the co-propagating modes ∣±l⟩ of a straight annular electron guide while the transported electron-wave support remains field free. In a spin-resolved Dirac treatment, the phase is carried by the overlap of the field-free vector potential Aφ with the mode's azimuthal conserved-current texture. The spin-dependent radial-gradient current becomes a boundary term that cancels when the complete finite-wall evanescent tail is retained, leaving the spin-independent orbital phase Δφln∝lΦLint⟨ρ−2⟩ln/vz. The matched ∣±l⟩ modes therefore realize a same-path Rz(2δl) gate, with differential internal-mode readout and common-mode phase rejection. For a=75nm, R=95nm, Lint=1mm, Ez=10meV, and ∣l∣=10, the gate angle is 2.315rad/G and Rz(π) occurs at 1.357G. Finite-barrier, mode-spacing, disorder-mismatch, and readout-visibility checks quantify the main implementation constraints. More broadly, the result connects a mode-resolved AB energy shift to a measurable propagation operation and shows how the spatially distributed conserved current of a Dirac wave can become an operational quantum-control resource.
Exact cylindrical solutions of the free Dirac equation provide natural single-particle modes for a broad class of axially symmetric relativistic fermion systems, including electron vortices, twisted-particle scattering, rotating matter, and cylindrical field quantization, but are commonly formulated in different internal bases. We derive the general regular positive-energy cylindrical solution at fixed energy, transverse and longitudinal momenta, and total angular momentum, and show how the commonly used spin-polarized, separation, helicity, and transverse-helicity modes are embedded in the resulting two-dimensional solution space. A conserved transverse operator resolves the residual doublet into two symmetry-defined branches. Their parameter-dependent eigenspaces admit a gauge-invariant quantum-geometric characterization, with opposite Berry curvatures, a common quantum metric, and saturation of the two-level metric--curvature relation. We further derive symmetry constraints on branch conversion and the corresponding reduced two-state dynamics. The resulting framework connects mode classification, symmetry resolution, quantum geometry, and dynamics across distinct cylindrical Dirac settings.