Direct Landau-level computation of the crossed-field stress response

Derive the expectation value of the mixed lowest-Landau-level stress-tensor component \(\langle T_{a\phi}\rangle\) directly for charged fermions in crossed \(E_\theta,B_r\) fields, in order to test the gradient corrections to the covariance-based drift-lift argument.

Background

In the axial-metric/polar-gauge perturbation sector, the lowest-Landau-level fermions inhabit field-line worldsheets that tilt under transverse electric fields and metric perturbations. The paper adopts a drift lift for the Casimir stress tensor and derives the associated polarisation currents from force balance and charge conservation.

The authors emphasize that consistency conditions do not uniquely determine the constitutive relation for the mixed stress component $T_{a\phi}$. Instead, the drift lift is selected using locality, covariance, local Lorentz transformations, and the index-theorem structure of the lowest Landau level. A direct microscopic Landau-level calculation would therefore provide an independent test of this constitutive choice and of the omitted gradient corrections of order $l/\sqrt q$.