= Quantum Hall plateau
The <degeneracy of a Landau level> is $eBA/h$ per resolved internal branch in <area> $A$. The <filling factor> $f=n_sh/(eB)$ varies continuously. Additional <localized eigenstates> need not carry bulk transport, allowing an integer plateau over an interval of $f$; a <mobility gap> need not be an empty <spectral gap>.<Quantized Hall Conductance in a Two-Dimensional Periodic Potential by D. J. Thouless and others>[p. 405, gaps and localized tails]{ref}
\Image[/quantum-hall-plateaus-and-tensor-inversion.png]
{title=Chosen quantum Hall staircase and longitudinal response}
The plot is an original phenomenological construction, not measured data or a calculation of <Anderson localization>. It stipulates dimensionless transverse <sheet conductivity> $k(f)=1+\tfrac12\sum_{m=1}^5[1+\tanh((f-m-1/2)/w)]$ and longitudinal <sheet conductivity> $s(f)=a\sum_{m=1}^5\operatorname{sech}^2((f-m-1/2)/w)$, with chosen $w=0.06$, $a=0.08$. Both are measured in units $e^2/h$. The <quantum Hall conductivity tensor inversion> produces $|\rho_{yx}|/R_K=k/(s^2+k^2)$ and $\rho_{xx}/R_K=s/(s^2+k^2)$. Regions near $k=N$, $s=0$ yield plateaus near $R_K/N$; the model alone does not explain their microscopic robustness. Each step represents one resolved branch, without assuming unresolved <spin> or valley multiplicities.
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