Extreme-Value Criticality and Gain Decomposition at the Integer Quantum Hall Transition

Abstract

Extreme-value fluctuations at quantum critical points remain poorly understood in the presence of strong correlations and openness. At the integer quantum Hall transition in the open Chalker--Coddington network, we show that the maximal wave-function amplitude separates into a global gain and an intrinsic extreme component, $|ψ|_{\max}=A\,|\tildeψ|_{\max}$. We introduce extreme-moment scaling for $|ψ|_{\max}$ and observe an approximately parabolic exponent function $τ_{\max}(q)$ over moderate $q$, while $\ln|ψ|_{\max}$ displays an almost Gaussian bulk over the studied sizes. The gain factor is close to log-normal and largely controls the raw extremes. Gain normalization reorganizes the statistics: $\tildeτ_{\max}(q)$ changes qualitatively and $|\tildeψ|_{\max}$ does not support a single-parameter generalized extreme-value collapse under standard centering/scaling in the accessible size window. Extreme observables thus provide a robust probe of correlated criticality in open quantum systems.

Figures (8)

• Figure 1: Chalker--Coddington (CC) network on a 2D square lattice. (a) The wave function lives on directed links and propagates along arrows, forming closed loops with alternating circulation on neighboring plaquettes. At each node, a local $2\times2$ scattering matrix mixes two incoming into two outgoing amplitudes; $\bullet$ and $\circ$ indicate the two node orientations. (b) Open CC network with an external point contact attached to link $c$.
• Figure 2: Parabolic extreme-moment exponent. (a) Measured $\Delta_q^{\max}$ for $|q|\lesssim 1$. The dashed curve shows the prediction $\Delta_q^{\max}\approx -\gamma q^2$, while the solid curve is an unconstrained best parabolic fit. (b) $\ln \mathbb{E}\!\left[p^{\,2q}\right]$ versus $\ln L$ (equivalently, the moments of the shifted variable entering the fit). Data from bottom to top correspond to $L=64,128,\ldots,2048$; ensemble sizes are of order $\mathcal{O}(10^7)$ for $L<1000$ and $\mathcal{O}(10^6)$ for $L>1000$. (c) Representative extractions of $\Delta_q^{\max}$ from the slopes of linear fits of $\ln \mathbb{E}\!\left[p^{\,2q}\right]$ versus $\ln L$.
• Figure 3: Near-log-normal statistics of the raw maxima. (a) Probability densities of $\ln |\psi|_{\max}$ shown in terms of the standardized variable $X=(\ln |\psi|_{\max}-\mu)/\sigma$, where $\mu$ and $\sigma$ are the mode and standard deviation of $\mathrm{PDF}(\ln |\psi|_{\max})$, respectively. The curves approach a normal distribution. (b) Two diagnostics of Gaussianity, $\sqrt{2\pi}-(\sigma h)^{-1}$ and the excess kurtosis (with $h$ the peak height of the PDF), remain close to zero; the larger scatter at the largest $L$ is due to finite ensemble size. (c) Skewness versus $L$. The data are well fitted by $aL^{-b}$ (solid line) with $b=0.314\pm0.002$ and $a=1.218\pm0.016$, while the more general form $aL^{-b}+c$ gives $c\approx 3\times 10^{-4}$ (dash-dotted line), consistent with vanishing skewness as $L\to\infty$.
• Figure 4: Scaling exponents for gain-normalized maxima $|\tilde{\psi}|_{\max}$. The exponent $\tilde{\tau}_{\max}(q)$ (solid line) is dominated by a linear contribution $\approx -\nu q$ (dashed line). Inset: the nonlinear remainder $\tilde{\Delta}_q^{\max}$ shows a small but systematic parabolic term and a cubic correction, generating an asymmetry between $q$ and $-q$. The fitted coefficients are $\gamma_2=-0.005\pm2.54\times10^{-5}$ and $\gamma_3=-0.0014\pm8.06\times10^{-6}$; higher-order terms are also present but subleading in the plotted range.
• Figure C1: Legendre-transform estimate of $\alpha_q$ from $\mathcal{A}_q(L)$ defined in Eq. (\ref{['eq:appD_A_def']}). Dashed/solid curves are finite-size extrapolation forms from Eq. (\ref{['eq:appD_A_L']}).