Magnetic localization and spectral asymmetry of a charged particle at a screw dislocation

The gist Magnetic localization and spectral asymmetry of a charged particle at a screw dislocation studies how global geometric structure, specifically the screw identification, couples conserved quantities to modify particle dynamics in a uniform magnetic field.

Geometric Framework

The geometry of a spiral dislocation is formulated as a Euclidean embedding supplemented by nontrivial screw identification The ambient Cartesian coordinates are (X, Y, Z), with X = r cos φ, Y = r sin φ, and Z = βφ The induced metric follows directly from the Euclidean ambient inner product gab = ea · eb The screw parameter does not modify the radial line element or the coefficient of the ordinary azimuthal term, instead it mixes the angular and axial directions through the globally nontrivial one-form dz+β dφ On every regular simply connected patch with r > 0, the metric is locally identical to the Euclidean metric written in cylindrical coordinates The global identification provides the geometric characterization of the dislocation

Electromagnetic Coupling

The construction involves pulling back a uniform magnetic field B0 directed along the positive Z axis to the spiral-dislocation geometry The pullback of the ambient gauge potential is A = B0/2 r squared dφ The intrinsic field strength follows from the pullback of the ambient curvature twoform, F = dA = B0r dr ∧ dφ The locally measured magnetic-field magnitude is B = B0, and the screw parameter β does not alter the local magnitude of the imposed magnetic field The global screw character is encoded in the translational holonomy associated with an azimuthal circuit, where I(dZ − dz) = Iβ dφ = 2πβ

Classical Dynamics and Conserved Quantities

The Lagrangian for a charged particle on the regular region r > 0 is L = µ/2 gijx˙ ix˙j + qAix˙i The canonical momenta are pφ = l, pz = k, where l denotes the conserved canonical azimuthal momentum and k the conserved momentum conjugate to the intrinsic axial coordinate z The relation between the axial velocity and mechanical velocity is z˙ + βφ˙ = k/µ The quantity that governs the azimuthal mechanical motion is not the canonical momentum pφ by itself, but the combination Πφ − βpz The Hamiltonian can be written as H = k 2/2µ + p 2r 2µ + (l − βk - qB0/2 r 2) 2/ (4µr 4)

Radial Dynamics and Confinement

The Hamiltonian is decomposed into three distinct contributions: the inverse-square term generated by the screw-shifted angular momentum J = l − βk, a constant magnetic cross-term, and a quadratic term responsible for magnetic confinement at large radius The radial effective potential is Veff (r) = (J − ar 2) squared / (4µr 2) The product of the two turning-point radii is independent of the particle energy and is fixed by the competition between the magnetic scale a and the screw-shifted angular momentum J The stationary point r c is determined by r c = q l−βk / qB0/2, and exact cancellation of terms occurs when aJ > 0

Quantum Spectrum and Spectral Asymmetry

The reduced radial problem is quantized on L 2((0, ∞), dr) with the eigenvalue equation-1/2µ d 2dr squared + k 2/2µ + (J − ar 2) 2/ (4µr 4) u(r) = E u(r) For qB0 6= 0, the spectrum is discrete and given by En = k 2/2µ - qB0J 2/2µ + qB02/ (4µ) (n+1)/2 The screw parameter β does not occur independently in the radial eigenvalue problem; its effect is encoded through the invariant J = l − βk The spectral asymmetry is given by En(−χ) - En(χ) = 2σχ, where χ = J and σ = sgn(qB0) This separation shows that the geometric contribution enters through J, while the magnetic coupling produces the spectral asymmetry

Conclusion

The analysis demonstrates that nontrivial global geometry can leave a direct quantum imprint even in the absence of local curvature The exact solvability then makes it possible to disentangle this global contribution from magnetic confinement and to identify separately its effects on localization, level spacing, and spectral asymmetry The screw parameter β links these two levels through J = l − βk, which converts the global translational holonomy of the defect into a shift of the conserved angular-momentum sector that directly controls the radial dynamics The ambient uniform magnetic field then converts the global momentum shift into finite radial confinement and orientation-selective compensation The spiral dislocation thus provides a particularly transparent realization in which local geometry, global topology, and electromagnetic dynamics remain sharply distinguishable while jointly determining the observable phase-space structure

CREDIT AUTHORSHIP CONTRIBUTION

Claudio Furtado: Conceptualization, Methodology, Formal Analysis, Writing - Original Draft, Investigation, Visualization, Writing - Review and Editing Omar Mustafa: Conceptualization, Methodology, Formal Analysis, Writing - Original Draft, Investigation, Visualization, Writing - Review and Editing Hassan Hassanabadi: Conceptualization, Methodology, Formal Analysis, Writing - Original Draft, Investigation, Visualization, Writing - Review and Editing Abdullah Guvendi: Conceptualization, Methodology, Formal Analysis, Writing - Original Draft, Investigation, Visualization, Writing - Review and Editing

DATA AVAILABILITY

This manuscript has no associated data

CONFLICTS OF INTEREST STATEMENT

No conflict of interest declared by the authors

FUNDING

The work by C. Furtado is supported by the CNPq (project PQ Grant 1A No. 311781/2021-7)

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1. Improve molecular dynamics simulations by incorporating the screw-modified conserved momentum J = l − βk into force calculations, enabling accurate modeling of particle motion in locally Euclidean geometries where global identification couples axial and azimuthal quantities. This allows the AI to predict trajectory shifts dictated by the defect's topological holonomy rather than relying solely on local curvature potentials.

2. Develop a quantum chemistry module capable of solving the reduced radial problem described by Eq. (104) for charged particles, specifically targeting systems where the inverse-square oscillator is exactly solvable. This system can accurately predict the discrete energy spectrum, including orientation-dependent shifts derived from Eq. (122), which are crucial for characterizing electronic states near topological defects.

3. Enhance materials science models by using the spectral asymmetry relation of Eq. (176), εn(−χ) − εn(χ) = 2σχ, to predict how structural defects, represented by the screw parameter β and magnetic field orientation, affect observable spectral signatures in condensed matter systems. This enables the AI to distinguish between physical observables that might otherwise appear identical under local approximations.

4. Create a topological defect analysis tool that uses the invariant J = l − βk as a primary descriptor for global geometric information, allowing the AI to classify material states based on conserved quantities rather than just local metric properties. This facilitates the identification of topological invariants that survive in the locally flat description of regular regions.

5. Implement a phase-space reconstruction algorithm that maps continuous dynamical invariants (E, l, k) and discrete topological information (n) to identify the helicoidal family of lifted trajectories described in Eq. (90). This allows the AI to visualize how global screw structure influences observable motion through the conserved combination J = l − βk.

We study a charged particle moving outside a spiral (screw) dislocation in a uniform magnetic field. Although the geometry is locally Euclidean, its global screw identification couples the conserved axial and azimuthal quantities through the combination J= -βk After eliminating the cyclic coordinates, the radial motion is governed by an inverse-square term determined by J and a harmonic confining term generated by the magnetic field. The corresponding reduced quantum problem is exactly solvable for q B 0 not equal to0, yielding an equally spaced spectrum with an orientation-dependent energy shift. The radial probability density depends on J, and opposite values of the screw-modified quantum number have identical spatial profiles but different energies for a fixed magnetic-field orientation. The radial size scales as sqrt r 2 n proportional to q B 0-1/2, with the magnetic length setting the localization scale. At q B 0=0, the confining term vanishes, and the problem reduces to a scale-free inverse-square system. Thus, the screw parameter enters the dynamics through the global coupling of the conserved axial and azimuthal quantities, whereas the magnetic field provides radial confinement and its orientation determines the spectral asymmetry.