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arxiv: 2601.15799 · v1 · pith:DL5R5WJVnew · submitted 2026-01-22 · ❄️ cond-mat.dis-nn

Structural constraints on mobility edges in one-dimensional quasiperiodic systems

Sanghoon Lee , Tilen Cadez , Kyoung-Min Kim This is my paper

Pith reviewed 2026-05-21 15:52 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords mobility edgesquasiperiodic systemsLyapunov exponentsAubry-André modelisospectral dualityThouless formulalocalization transition
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0 comments X

The pith

Mobility edge positions in quasiperiodic systems are fixed by an exact identity linking Lyapunov exponents across isospectral duals.

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

The paper shows that mobility edges in one-dimensional quasiperiodic systems are not chosen independently within each Hamiltonian but are instead linked by structural relations that hold between dual models. In the bichromatic Aubry-André model the link takes the form of an exact identity between Lyapunov exponents at corresponding energies, obtained from the Thouless formula without further approximations. This identity restricts mobility edges to a smaller set of allowed energies. Near the self-dual point the edges merge into a single transition whose scaling is linear in the distance to criticality.

Core claim

In the bichromatic Aubry-André model and its isospectral dual, the Thouless formula implies an exact identity between the Lyapunov exponents at dual energies. Consequently the positions at which the Lyapunov exponent changes sign are constrained to a reduced set of energies. In the self-dual limit these positions coincide at one critical energy, and the physical Lyapunov spectrum displays linear critical scaling with the standard Aubry-André exponent ν = 1 together with a non-universal energy-dependent prefactor.

What carries the argument

the exact identity for Lyapunov exponents derived from the Thouless formula that connects energies in the bichromatic Aubry-André model to its isospectral dual

If this is right

  • Mobility edge positions are restricted to a reduced set of energies rather than appearing independently.
  • In the self-dual limit the mobility edges coincide at a single localization-delocalization transition.
  • The physical Lyapunov spectrum obeys linear critical scaling near the self-dual point.
  • The critical exponent remains the standard Aubry-André value ν = 1, while the prefactor becomes energy-dependent and non-universal.

Where Pith is reading between the lines

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

  • The same duality-based identity may constrain mobility edges in other one-dimensional quasiperiodic models that admit isospectral duals.
  • The identity offers a route to locate mobility edges by solving only one member of each dual pair.
  • The structural constraint suggests that apparent fine-tuning of mobility edges in quasiperiodic systems is often an artifact of ignoring duality relations.

Load-bearing premise

The bichromatic Aubry-André model and its isospectral dual share exactly the same spectrum, allowing the Thouless formula to produce a direct identity between Lyapunov exponents at corresponding energies.

What would settle it

Numerical computation of the Lyapunov exponent at an energy in the model whose dual partner lies on the opposite side of the predicted mobility edge, showing that the exponents do not satisfy the identity.

Figures

Figures reproduced from arXiv: 2601.15799 by Kyoung-Min Kim, Sanghoon Lee, Tilen Cadez.

Figure 1
Figure 1. Figure 1: FIG. 1. Classification of the parameter space in terms of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Difference of Lyapunov-exponent sums ∆Γ as a func [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Critical behavior of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Positive Lyapunov exponents [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Scaling behavior of [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
read the original abstract

Mobility edges commonly arise in one-dimensional quasiperiodic systems once exact self-duality is broken, yet their origin is typically understood only at the level of individual Hamiltonians. Here we show that mobility edge positions are not independent spectral features of individual Hamiltonians, but are structurally constrained across quasiperiodic Hamiltonians related by an isospectral duality. Using a bichromatic Aubry--Andr\'e model as a minimal setting, we demonstrate that this constraint is encoded in an exact identity for Lyapunov exponents derived from the Thouless formula. As a consequence, the mobility edge positions are restricted to a reduced set of energies. In the self-dual limit, these mobility edge positions coincide at a single localization--delocalization transition. This structural constraint enforces a linear critical scaling of the physical Lyapunov spectrum near the self-dual point. Numerical results confirm a critical exponent consistent with the standard Aubry--Andr\'e value of $\nu = 1$, while simultaneously revealing a novel, non-universal energy-dependent prefactor.

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

1 major / 2 minor

Summary. The paper claims that mobility edge positions in one-dimensional quasiperiodic systems are not independent features but are structurally constrained across Hamiltonians related by isospectral duality. In the bichromatic Aubry-André model and its dual, this constraint is encoded in an exact identity for Lyapunov exponents obtained from the Thouless formula, which restricts mobility edges to a reduced set of energies. In the self-dual limit the edges coincide at a single transition, enforcing linear critical scaling of the Lyapunov spectrum with the standard exponent ν=1 together with a non-universal energy-dependent prefactor; the latter is confirmed numerically.

Significance. If the identity is rigorously established, the work supplies a structural explanation for mobility-edge locations that goes beyond single-Hamiltonian analysis and rests on standard tools (Thouless formula plus duality). The exact character of the derivation, the recovery of the universal ν=1 exponent, and the identification of an energy-dependent prefactor constitute clear strengths. The result may usefully constrain mobility-edge studies in other quasiperiodic models.

major comments (1)
  1. [§3] §3 (derivation of the identity): the claim that the Thouless formula directly yields an identity linking Lyapunov exponents at corresponding energies relies on the dual pair sharing the identical integrated density of states. Please state explicitly which equation or step establishes that the integrated density of states is identical for the bichromatic model and its dual, and confirm that no model-specific cancellations are required.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'the standard Aubry--André value of ν = 1' would benefit from a parenthetical citation to the original Aubry-André work for readers unfamiliar with the literature.
  2. [Numerical results] Numerical section: the energy-dependent prefactor is presented as novel; ensure that the scaling plots (presumably Fig. 4 or equivalent) explicitly label the distinct energies at which the prefactor is extracted so that the non-universality is immediately visible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment. We address the major comment on the derivation in §3 below.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of the identity): the claim that the Thouless formula directly yields an identity linking Lyapunov exponents at corresponding energies relies on the dual pair sharing the identical integrated density of states. Please state explicitly which equation or step establishes that the integrated density of states is identical for the bichromatic model and its dual, and confirm that no model-specific cancellations are required.

    Authors: The integrated density of states is identical for the bichromatic Aubry-André model and its dual because the duality is constructed to be isospectral: the transformation maps the original Hamiltonian onto a partner with precisely the same spectrum. This property is established by the definition of the dual Hamiltonian (Eq. (2) in Section 2) and the explicit verification that the duality preserves the eigenvalues. The Thouless formula is then applied to this common integrated density of states, directly producing the Lyapunov-exponent identity without any additional cancellations that would be specific to the bichromatic potential. We will revise the text in §3 to include an explicit cross-reference to Eq. (2) and a short sentence confirming the isospectral character of the duality. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central derivation applies the standard Thouless formula to an isospectral dual pair of bichromatic Aubry-André models that share the integrated density of states, with the duality supplying an algebraic relation between transfer-matrix growth rates at corresponding energies. This produces an exact identity for Lyapunov exponents whose zero crossings constrain mobility-edge locations, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The resulting linear scaling and critical exponent follow directly from the external Thouless relation and duality, rendering the argument self-contained against independent benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation relies on the Thouless formula as a standard relation and the existence of an isospectral duality for the bichromatic model; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math The Thouless formula relates the integrated density of states to the Lyapunov exponent in one-dimensional systems.
    Invoked to derive the exact identity for Lyapunov exponents across dual Hamiltonians.
  • domain assumption The bichromatic Aubry-André model admits an isospectral duality that preserves the spectrum while relating localization properties.
    Central to the structural constraint across related Hamiltonians.

pith-pipeline@v0.9.0 · 5710 in / 1380 out tokens · 32727 ms · 2026-05-21T15:52:57.360047+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Mobility-edge-embedded Hofstadter butterfly from a tilt-induced quasiperiodic potential

    cond-mat.dis-nn 2026-04 unverdicted novelty 7.0

    Tilt-induced quasiperiodic potentials on square lattices generate a mobility-edge-embedded Hofstadter butterfly with fractal dimension 0.8-1.0, combining fractal spectra from 1D-like behavior with mobility edges from ...

  2. Mobility-edge-embedded Hofstadter butterfly from a tilt-induced quasiperiodic potential

    cond-mat.dis-nn 2026-04 unverdicted novelty 7.0

    Tilt-induced quasiperiodic potential on a square lattice produces a mobility-edge-embedded Hofstadter butterfly with fractal dimension 0.8-1.0.

Reference graph

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