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 ℓ|∇ V|/ℏωc; screening supplies the self-consistent field. Transport is quasilocal, RH=∫ρHjy dx/∫ jy dx with jy∝1/ρl, 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, Im RH∝ω, with Re RH−RHDC∝ω2 below 10−9 up to $2$~MHz. With the current fed back into the electrostatics at $1−−5\,μ$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 ν(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 $0.3−−2\,μ$A. The thermal conclusions are bounded by the loss law and by the frozen profile, and are presented as estimates.
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BibTeX
@misc{siddiki2026quasilocal,
title = {{Quasilocal transport in the integer quantised Hall effect: adiabatic separation, local Ohm's law, and Joule heating of the incompressible strips}},
author = {A. Siddiki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15231},
eprint = {2609.15231},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15231},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Quasilocal transport in the integer quantised Hall effect: adiabatic separation, local Ohm's law, and Joule heating of the incompressible strips
AU - A. Siddiki
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15231
UR - https://arxiv.org/abs/2609.15231
AB - 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 ℓ|∇ V|/ℏω_c; screening supplies the self-consistent field. Transport is quasilocal, R_H=∫ρ_Hj_y dx/∫ j_y dx with j_y∝1/ρ_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, Im R_H∝ω, with Re R_H−R_H^DC∝ω^2 below 10^−9 up to $2$~MHz. With the current fed back into the electrostatics at $1−−5\,μ$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 ν(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 $0.3−−2\,μ$A. The thermal conclusions are bounded by the loss law and by the frozen profile, and are presented as estimates.
ER -