Localization structure of electronic states in the quantum Hall effect

TL;DR

The paper addresses localization in the integer quantum Hall effect by extending the localization landscape to magnetic systems through the Magnetic Localization Landscape (MLL). It defines the MLL via $-\Delta u+(\eta V+\beta)u=1$ and demonstrates that the effective potential $1/u$ identifies localization wells and hills that predict eigenstate locations and energies across disorder regimes, including the edge-influenced regime near boundaries. The work shows that below the critical energy $E_c=\eta+\beta$, eigenstates localize near $1/u$ wells with energies closely tied to local minima; above $E_c$, states localize near hills with energies connected to local maxima, while boundary effects introduce an edge scale $\ell_e$. The MLL provides a robust, deterministic link between disorder, magnetic field, and localization, with potential to inform transport modeling and critique of universality in quantum Hall transitions. Overall, the MLL extends landscape theory to magnetic problems, bridging semiclassical intuition and full quantum descriptions in IQHE systems.

Abstract

We investigate the localization of electronic states in the integer quantum Hall effect using a magnetic localization landscape (MLL) approach. By studying a continuum Schrödinger model with disordered electrostatic potential, we demonstrate that the MLL, defined via a modified landscape function incorporating magnetic effects, captures key features of quantum state localization. The MLL effective potential reveals the spatial confinement regions and provides predictions of eigenstate energies, particularly in regimes where traditional semiclassical approximations break down. Numerical simulations show that below a critical energy, states localize around minima of the effective potential, while above it, they cluster around maxima-with edge effects becoming significant near boundaries. Bridging the gap between semiclassical intuition and full quantum models, the MLL offers a robust framework to understand transport and localization in disordered quantum Hall systems, and extends the applicability of landscape theory to magnetic systems.

Figures (9)

• Figure 1: Different regimes of disorder and magnetic intensity in the parameter space $(\beta,\eta)\approx (\ell_{\rm corr}^2/\ell_B^2,\ell_{\rm corr}^2/\ell_V^2)$. The regime of interest in this work is symbolized by the red hue.
• Figure 2: Typical shapes of $\abs{\psi}^2$ for localized eigenfunctions in the IQHE: (a) and (b) are wave functions localized inside the bulk, both below the critical energy, with no contribution at the edges of the domain. Type (a) correspond to a local fundamental state at the bottom of a potential well, whereas the red line in (b) is a semiclassical trajectory (a level set of the effective potential) for a higher-energy state. (c) and (d) are states above the critical energy. They look similar to (a) and (b), except for a significant contribution at the edges.
• Figure 3: Color maps of the moduli of five eigenstates of Eq. \ref{['eq:nondimensional_eigvv']} with parameters $(\beta, \eta) = (0.2 , 0.2)$, inside the first Landau level (top row, linear scale, bottom row, logarithmic scale). From left to right, the energies are about $0.334,\, 0.361,\, 0.397,\, 0.442,\, 0.462$, respectively. One observes that the first two states are localized, similarly to Figs. \ref{['fig:eigenfunction_types']}(a) and \ref{['fig:eigenfunction_types']}(b), respectively, the center state is clearly delocalized, and the last two states are localized again, albeit with a significant probability density on the edges of the domain, similar to Figs. \ref{['fig:eigenfunction_types']}(d) and \ref{['fig:eigenfunction_types']}(c), respectively. All localized states have a localization length of the order of few units, making the decay of the wave function amplitude barely visible on a linear scale.
• Figure 4: Top: Localization lengths of the first Landau level as a function of energy for $\eta=0.001$, $\beta=0.2$, for five potential realizations. The system size is $L=200$. The blue to red hue corresponds to the distance of the peak of the wave function to the boundary. The vertical dashed line materializes the energy $E_c=\eta+\beta$ which appears to be the critical energy. The localization length is assessed from the behavior of $\mathcal{A}_\psi(\varepsilon)$, assuming an exponential decay from the peak of the wave function, see Eqs. \ref{['eq:A_psi']} and \ref{['eq:bulk_fit']}. Bottom: Similar plot as above, except that the localization length is now extracted from Eq. \ref{['eq:edge_fit']} for all eigenstates of energy larger than $E_c$, theses states having a significant probability density at the edges of the domain.
• Figure 5: Top left: Original potential $V$, as defined in Eq. \ref{['eq:MLL']}, mimicking a smoothed compositional disorder. Top right: Effective potential $1/u$ for $\eta=0.2$, the shift $\beta$ being equal to zero. Bottom left: Effective potential minus the shift $\beta$, i.e., $1/u - \beta$, for $\eta=0.2$ and $\beta=0.2$. Bottom right: Effective potential minus the shift $\beta$ for $\eta=0.2$ and $\beta=1$. As $\beta$ increases, $1/u-\beta$ converges to $\eta V$.