Unified Exact Cylindrical Dirac Modes and Symmetry-Resolved Quantum Geometry

Relativistic fermion systems with axial symmetry—such as electron vortex beams, twisted-particle scattering, and rotating matter—are often described using diverse internal bases for the Dirac equation. This paper seeks to unify these disparate representations by deriving a general, exact cylindrical solution family and identifying the underlying symmetry that organizes this solution space.

The authors employ a generalized series-expansion method to solve the first-order Dirac equation in cylindrical coordinates. By avoiding the premature selection of an internal basis, they construct a two-dimensional solution space parameterized by a complex constant. They then introduce a conserved transverse operator, $K = \beta(\Sigma \times p)_z$, which commutes with the Hamiltonian and resolves the residual degeneracy into two distinct, symmetry-adapted branches. The paper further characterizes these branches using quantum-geometric tensors, including Berry curvature and the quantum metric.

The study demonstrates that commonly used modes—including spin-polarized vortex modes, separation modes, and helicity-adapted states—are all specific members of a single, unified cylindrical Dirac family. The conserved operator $K$ provides a basis-independent resolution of this family into two branches. These branches exhibit opposite Berry curvatures and identical quantum metrics, satisfying the two-level metric–curvature relation. Additionally, the authors derive symmetry constraints on branch conversion, showing that a transverse radial gradient can induce transitions between these branches, while longitudinal profiles preserve them.

This framework provides a rigorous, unified language for describing relativistic cylindrical fermions. By connecting mode classification to quantum geometry and symmetry-resolved dynamics, the paper allows researchers to map between different physical representations (e.g., spin vs. helicity) and predict how these states evolve under external perturbations. This is particularly relevant for the precise control and manipulation of relativistic electron beams and the study of twisted-particle scattering.