Fourth-order closure obstruction and chiral nonlocality in circular kinetic magnetotransport

Abstract

Closing an angular moment hierarchy at the stress level omits a definite back-action from higher Fermi-surface harmonics. For a circular two-dimensional Fermi surface, streaming changes angular momentum by one, so the shortest omitted sequence, $1\to 2\to 3\to 2\to 1$, adds a fourth-order term to the current eigenvalue, $\Lambda (q)={\gamma }_1+\nu q^2-{\kappa }_4{q}^4+\cdots$, with $\nu =v_F^2/(4{\gamma }_2)$ and ${\kappa }_4={\nu }^2/{\gamma }_3$. We call this missing operator term the fourth-order closure obstruction. Its gradient expansion is controlled when $\nu q^2/{\gamma }_3\ll 1$. Circular symmetry carries the same coefficient into a radial bi-Laplacian within each conserved angular-momentum block, and retaining $m=3$ exactly, without a gradient expansion, amplifies higher radial modes monotonically. At zero field, positive collision rates exclude real-wave-number poles and response zeros; an equal-rate tail gives a square-root completion. A magnetic field makes the coefficient chiral, produces a Hall sign reversal, and enhances it when the $m=3$ harmonic is long lived. In the collisionless high-field limit, the complete hierarchy becomes a Bessel pole--zero ladder, while finite closures form rational approximants to it. The result separates a controlled low-gradient coefficient from its geometry- and field-dependent finite-wave-number completion.