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arXiv:2603.15290 · Physical review. B./Physical review. B

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

Wei-Han Li, Abbas Ali Saberi

cond-mat.dis-nncond-mat.stat-mechphysics.data-anquant-phSubmitted Mar 16, 2026• Announced Mar 17, 2026• 0 citations
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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.

Journal reference
Phys. Rev. B(Letter) 113, L241108 (2026)
DOI
10.1103/wzs6-3s61
Comments
8 pages, 8 figures

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We introduce and analyze a class of model systems to study transitions in the integer quantum Hall effect (IQHE). Even without disorder our model exhibits an IQHE transition as a control parameter is varied. We find that the transition is in the two-dimensional Ising universality class and compute all associated exponents and critical transport properties. The fixed point has time-reversal, particle-hole, and parity invariance. We then consider the effect of quenched disorder on the IQHE transition and find the following. (i) Randomness in the control parameter (which breaks all the above symmetries) translates into bond randomness in the Ising model and is hence marginally irrelevant. The transition may equally well be viewed as a quantum percolation of edge states localized on equipotentials. The absence of random-phase factors for the edge states is responsible for the nongeneric (Ising) critical properties. (ii) For a random magnetic field (which preserves particle-hole symmetry in every realization) the model exhibits an exactly solvable fixed line, described in terms of a product of a Luttinger liquid and an SU(n) spin chain. While exponents vary continuously along the fixed line, the longitudinal conductivity is constant due to a general conformal sum rule for Kac-Moody algebras (derived here), and is computed exactly. We also obtain a closed expression for the extended zero-energy wave function for every realization of disorder and compute its exact multifractal spectrum f(\ensuremath{\alpha}) and the exponents of all participation ratios. One point on the fixed line corresponds to a recently proposed model by Gade and Wegner. (iii) The model in the presence of a random on-site potential scales to a strong disorder regime, which is argued to be described by a symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible global phase diagram in which all forms of disorder are simultaneously considered. In this generic case, the presence of random-phase factors in the edge-state description indicates that the transition is described by a Chalker-Coddington model, with a so far analytically inaccessible fixed point.
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Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two — a metal–insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.