Quasilocal transport in the integer quantised Hall effect: adiabatic separation, local Ohm's law, and Joule heating of the incompressible strips

Abstract

We reassess the semiclassical route to integer quantum Hall transport around one idea: the separation of fast, quantised cyclotron motion from a slow, self-consistently screened guiding-centre landscape. Bohr--Sommerfeld quantisation of the canonical action fixes the Landau spectrum; smooth confinement turns it into Born--Oppenheimer surfaces with adiabatic parameter $\ell |\nabla V|/\hslash {\omega }_c$; screening supplies the self-consistent field. Transport is quasilocal, $R_H=\int {\rho }_H{j}_{y}dx/\int j_{y}dx$ with $j_y\propto 1/{\rho }_l$, so within the regularised local Ohm model the incompressible strips attract the current by weight and the computed Joule-power profile follows the current profile. In linear response the intrinsic electrostatic AC correction of the frozen-profile model is reactive, $\mathrm{Im}{R}_H\propto \omega$, with $\mathrm{Re}{R}_H-R_H^{\mathrm{DC}}\propto {\omega }^2$ below $10^{-9}$ up to $2$~MHz. With the current fed back into the electrostatics at $1$--$5\mu$A, where the Hall drop is one to five cyclotron gaps, the driven boundary condition breaks the reflection symmetry of the bar, the Hall drop concentrates in one strip (interchanged to $10^{-9}$ under current reversal), and the plateau narrows from its low-field flank while, for the GaAs parameters studied, its high-field edge stays at $\nu (0)=2$. A local transverse thermal closure inspired by Akera, evaluated as a post-process on the frozen profile with two phonon-loss laws, indicates that the narrow current-carrying strips of the low-field flank may heat by of order the lattice temperature; with the activated-heat-capacity closure the cold strip loses its stationary solution at model-dependent currents of $0.3$--$2\mu$A. The thermal conclusions are bounded by the loss law and by the frozen profile, and are presented as estimates.

25 pages, 15 Figures