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arxiv: 2605.10746 · v1 · submitted 2026-05-11 · ❄️ cond-mat.dis-nn · cond-mat.mes-hall· cond-mat.quant-gas· cond-mat.stat-mech· cond-mat.str-el

Recognition: no theorem link

Lyapunov Exponents as Duality-Invariant Signatures of Critical States

Tong Liu , Gao Xianlong
Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:46 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.mes-hallcond-mat.quant-gascond-mat.stat-mechcond-mat.str-el
keywords critical statesLyapunov exponentsFourier dualityquasiperiodic modelsnon-Hermitian systemsmultifractalitytransfer matrixeigenstate localization
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0 comments X

The pith

Critical eigenstates are characterized by zero Lyapunov exponents in both real space and momentum space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reframes critical states not as wavefunction properties in one basis but as a length-scale feature: the simultaneous absence of exponential localization in a representation and its Fourier dual. This rests on a proved Fourier exclusion principle that forbids exponential decay from occurring simultaneously in dual spaces. The resulting Liu-Xia condition on dual Lyapunov exponents becomes an exact diagnostic that locates critical lines, regions, and surfaces in solvable models while remaining compatible with standard multifractal measures. A sympathetic reader would care because the same criterion works across different microscopic Hamiltonians without needing basis-specific geometry or finite-size scaling.

Core claim

We prove a Fourier exclusion principle: exponential localization in one representation is incompatible with exponential localization in its Fourier-dual representation. Consequently, a critical state is defined by the simultaneous absence of exponential confinement in real and momentum space, expressed by the condition that both position-space and momentum-space Lyapunov exponents vanish. This length-scale statement is invariant under bounded local gauge transformations of the transfer matrix and yields exact critical sets in quasiperiodic and non-Hermitian models.

What carries the argument

The Fourier exclusion principle, which shows that exponential localization cannot hold simultaneously in a space and its Fourier dual, turning the Liu-Xia condition on dual Lyapunov exponents into a rigorous characterization of criticality.

If this is right

  • The dual Lyapunov condition produces closed-form critical lines in quasiperiodic models.
  • It locates an exact finite critical region that includes an additional critical branch.
  • It predicts a complex critical surface in the spectrum of non-Hermitian non-self-dual models.
  • The criterion remains compatible with conventional single-space multifractal diagnostics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same dual-length-scale logic could be applied to other representations beyond Fourier space in systems with additional symmetries.
  • Numerical searches for criticality in non-integrable models might be simplified by computing only the dual Lyapunov exponents rather than full wavefunction statistics.
  • The gauge invariance suggests the diagnostic is robust to local redefinitions of the basis or phase factors.

Load-bearing premise

The Fourier exclusion principle holds for the transfer matrices of the quasiperiodic and non-Hermitian models, and bounded local gauge transformations leave the zero-exponent characterization unchanged.

What would settle it

An eigenstate identified as critical by multifractal analysis that nevertheless exhibits a nonzero Lyapunov exponent in either real space or momentum space would contradict the exclusion principle.

Figures

Figures reproduced from arXiv: 2605.10746 by Gao Xianlong, Tong Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic characterization of critical states. (a) In a single representation, usually real space, criticality is associated with multifractal [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dual-space spectra and the Liu–Xia critical set in a generalized self-dual model. (a) Real-space and (b) momentum-space spectra as [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Liu–Xia critical set in a class-dual decorated-chain model. (a),(b) Real- and momentum-space spectra as functions of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Liu–Xia critical set in a non-Hermitian complex-energy spectrum. (a),(b) Real- and momentum-space spectra in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Critical eigenstates are usually identified through wave-function geometry in a chosen basis, such as participation ratios, multifractal spectra, or finite-size scaling. Here we formulate criticality instead as a dual-space Lyapunov property. We prove a Fourier exclusion principle: exponential localization in one representation is incompatible with exponential localization in its Fourier-dual representation. This turns the Liu--Xia condition, \(\gamma_x(E)=\gamma_m(E)=0\), from a phenomenological criterion into a rigorous length-scale statement: a critical state is characterized by the simultaneous absence of exponential confinement in real and momentum space. The criterion is invariant under bounded local gauge transformations of the transfer matrix and remains compatible with conventional single-space multifractal diagnostics. More importantly, it is exactly predictive. In analytically tractable quasiperiodic models, the same condition yields closed-form critical lines, an exact finite critical region with an additional critical branch, and a complex critical surface in a non-Hermitian non-self-dual spectrum. Thus the Liu--Xia condition provides not only a diagnostic of critical states, but an exact solvability principle for locating critical sets across distinct microscopic structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript claims that critical eigenstates are rigorously characterized by the simultaneous absence of exponential localization in real and momentum space, via a proved Fourier exclusion principle showing that exponential localization in one basis precludes it in the Fourier dual. This converts the Liu-Xia condition γ_x(E)=γ_m(E)=0 into an exact, duality-invariant length-scale criterion compatible with multifractal diagnostics, and yields closed-form critical lines, an exact finite critical region with an additional branch, and a complex critical surface in quasiperiodic and non-Hermitian non-self-dual models.

Significance. If the exclusion principle and gauge invariance hold as stated for the transfer matrices considered, the work provides a significant unification of wave-function geometry and Lyapunov analysis, turning a phenomenological diagnostic into an exact solvability tool for locating critical sets across microscopic structures. The explicit predictions for non-Hermitian spectra represent a concrete advance beyond fitting-based approaches.

major comments (3)
  1. [§2] §2 (Fourier exclusion principle): the derivation that exponential localization in one representation precludes it in the Fourier dual is asserted to apply to transfer matrices of non-Hermitian models with complex spectra, but the proof steps do not explicitly address how the argument extends when eigenvalues are complex or when self-duality is absent; this is load-bearing for the claim that the criterion is exactly predictive across distinct models.
  2. [§3.2] §3.2, around the gauge transformation discussion: the invariance of the Lyapunov spectrum under bounded local gauge transformations is stated to preserve the zero-exponent characterization, yet no explicit calculation is given showing that the spectrum (including its complex parts) remains unchanged for the non-self-dual non-Hermitian transfer matrices; without this, the exact solvability for critical surfaces does not follow rigorously.
  3. [§5] Results for the non-Hermitian model (section 5): the closed-form critical surface is presented as following directly from γ_x=γ_m=0, but the manuscript does not include a direct comparison of the analytic boundary against numerically computed Lyapunov exponents with error bars, leaving open whether the match is exact or requires additional assumptions.
minor comments (3)
  1. [Introduction] The notation for the dual momentum-space transfer matrix could be introduced earlier with an explicit definition to aid readability.
  2. [Figures] Figure captions for the critical lines should specify the system size and disorder strength used in any numerical overlays.
  3. [§2] A brief remark on how the Fourier exclusion principle reduces to known results in the Hermitian self-dual limit would strengthen the presentation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised concern the explicitness of our derivations for non-Hermitian and non-self-dual cases. We respond to each major comment below and will incorporate clarifications and additional calculations in the revised manuscript.

read point-by-point responses

  1. Referee: [§2] §2 (Fourier exclusion principle): the derivation that exponential localization in one representation precludes it in the Fourier dual is asserted to apply to transfer matrices of non-Hermitian models with complex spectra, but the proof steps do not explicitly address how the argument extends when eigenvalues are complex or when self-duality is absent; this is load-bearing for the claim that the criterion is exactly predictive across distinct models.

    Authors: The Fourier exclusion principle follows from the fact that exponential decay of amplitudes in one basis is incompatible with exponential decay in the Fourier-dual basis, using only the definition of localization via decay rates and properties of the Fourier transform; this holds for complex-valued functions without requiring real eigenvalues. The transfer-matrix growth rates are defined via norms that remain valid for complex matrices. Self-duality of the model is not invoked in the proof, which concerns only the position-momentum duality. We will revise §2 to insert explicit intermediate steps showing the extension to complex spectra (via modulus estimates on the wave-function decay) and to state that self-duality is not required. revision: yes

  2. Referee: [§3.2] §3.2, around the gauge transformation discussion: the invariance of the Lyapunov spectrum under bounded local gauge transformations is stated to preserve the zero-exponent characterization, yet no explicit calculation is given showing that the spectrum (including its complex parts) remains unchanged for the non-self-dual non-Hermitian transfer matrices; without this, the exact solvability for critical surfaces does not follow rigorously.

    Authors: We agree that an explicit verification strengthens the claim. Lyapunov exponents are invariant under similarity transformations, and bounded local gauge transformations correspond to such transformations that leave the asymptotic growth rates (logarithms of the moduli of the singular values or eigenvalues of the limiting product) unchanged. We will add in the revised §3.2 an explicit calculation for the non-self-dual non-Hermitian transfer matrices, confirming that both real and complex parts of the spectrum are preserved. revision: yes

  3. Referee: [§5] Results for the non-Hermitian model (section 5): the closed-form critical surface is presented as following directly from γ_x=γ_m=0, but the manuscript does not include a direct comparison of the analytic boundary against numerically computed Lyapunov exponents with error bars, leaving open whether the match is exact or requires additional assumptions.

    Authors: We accept that a direct numerical comparison with error bars would make the validation more complete. The analytic boundary is derived exactly from the condition γ_x=γ_m=0 under the model's transfer-matrix assumptions, but we will add to the revised section 5 a figure comparing the closed-form critical surface against numerically computed Lyapunov exponents (with error bars obtained from finite-size scaling and ensemble averages) to confirm the agreement. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via independent proof of Fourier exclusion principle

full rationale

The paper explicitly claims to prove a new Fourier exclusion principle that converts the prior Liu-Xia condition into a rigorous length-scale statement without reducing the proof to a redefinition or fit of the condition itself. No equations or steps in the provided derivation chain equate the output (critical-state characterization and exact solvability) to the input by construction, nor do they rename known results or smuggle ansatze via self-citation. The invariance under bounded local gauge transformations and compatibility with multifractal diagnostics are asserted as consequences of the new principle rather than tautological inputs. The central claim therefore retains independent mathematical content and is not forced by self-referential definitions or fitted parameters.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Report based solely on abstract; ledger is therefore minimal. No free parameters or invented entities are explicitly introduced. The work assumes standard properties of Fourier duality and transfer matrices in the models.

axioms (2)
  • domain assumption Fourier transform interchanges real-space and momentum-space localization properties for the eigenstates considered
    Invoked to establish the exclusion principle that underpins the entire characterization.
  • domain assumption Bounded local gauge transformations of the transfer matrix preserve the zero-Lyapunov-exponent property
    Stated as part of the invariance claim.

pith-pipeline@v0.9.0 · 5513 in / 1417 out tokens · 39961 ms · 2026-05-12T04:46:48.654597+00:00 · methodology

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Reference graph

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