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arxiv: 2605.18064 · v1 · pith:7FSOKSOOnew · submitted 2026-05-18 · ❄️ cond-mat.dis-nn

Localization Transitions in a Half-Filled Helical Aubry-Andr\'e Model

Taylan Yildiz , B. Tanatar , Bal\'azs Het\'enyi This is my paper

Pith reviewed 2026-05-20 00:28 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords Aubry-André modellocalization transitionhelical hoppingquasiperiodic latticeBinder cumulantpolarization amplitudesnoninteracting fermionscommensurability
0
0 comments X

The pith

In the helical Aubry-André model, stronger Nth-neighbor hopping shifts the localization transition to higher quasiperiodic potentials, with spikes at commensurate N.

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

The paper examines localization transitions in a one-dimensional quasiperiodic lattice formed by adding an Nth-neighbor helical hopping term of strength J_N to the Aubry-André model. This extra term introduces a second tuning parameter that couples successive windings of an effective helical chain. The delocalization-localization transition is located from the zero crossing of a geometric Binder cumulant built from the polarization amplitudes of the many-body Slater-determinant ground state at half filling. Stronger helical hopping is shown to stabilize the extended phase by raising the critical potential strength, while the dependence on helical range N displays clear commensurability-induced spikes. The same transition window is recovered from the opening of the Fermi gap, and a Zeckendorf-shift construction is used to reach a controlled thermodynamic limit along Fibonacci system sizes.

Core claim

The central claim is that the critical quasiperiodic potential strength at which the half-filled helical Aubry-André chain localizes rises with increasing helical hopping amplitude J_N, and that this critical value varies non-monotonically with the helical range N because of commensurability between the hopping distance and the underlying lattice periodicity.

What carries the argument

Geometric Binder cumulant formed from the polarization amplitudes of the many-body Slater determinant, whose zero crossing locates the delocalization-localization transition.

If this is right

  • Stronger helical hopping stabilizes the extended phase by shifting the critical potential upward.
  • The N-dependence of the critical potential exhibits pronounced commensurability-induced spikes.
  • The geometric Binder cumulant and the Fermi gap depart from their extended-phase values in the same parameter window.
  • The Zeckendorf-shift construction keeps the many-body sector fixed while system size tends to infinity along Fibonacci numbers.

Where Pith is reading between the lines

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

  • Tuning helical range and strength could provide a practical handle for controlling localization lengths in quasiperiodic optical lattices.
  • The same helical construction might be applied to interacting or disordered versions of the model to test whether the stabilization of extended states survives.
  • The winding aspect of the helical chain suggests possible links to topological invariants that could be checked by adding a synthetic dimension.

Load-bearing premise

The geometric Binder cumulant built from polarization amplitudes correctly identifies the delocalization-localization transition and remains reliable under the Zeckendorf-shift construction for the thermodynamic limit along Fibonacci sizes.

What would settle it

Direct computation of the inverse participation ratio or wave-function localization length on large Fibonacci-sized chains that places the transition at a markedly different critical potential than the zero crossing of the geometric Binder cumulant.

Figures

Figures reproduced from arXiv: 2605.18064 by Bal\'azs Het\'enyi, B. Tanatar, Taylan Yildiz.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Geometric Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Critical quasiperiodic potential strength [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dependence of the critical quasiperiodic potential [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Geometric Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of the geometric Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Normalized single-particle energy spectrum of the helical Aubry-André model as a function of the quasiperiodic [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Critical quasiperiodic potential strength [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Geometric Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

We study localization in a one-dimensional quasiperiodic lattice obtained by extending the Aubry-Andr\'e model with an additional $N$th-neighbor hopping term of strength $J_{N}$. This long-range tunneling couples successive windings of an effective helical chain and introduces a second control parameter beyond the quasiperiodic potential strength $\Delta$. Working with noninteracting fermions (typically at half filling), we diagnose the delocalization-localization transition using extensions of the modern theory of polarization. Specifically, we compute the polarization amplitudes of the many-body Slater-determinant ground state and construct a geometric Binder cumulant from polarization amplitudes. The critical potential where the localization transition happens is extracted from the sign change (zero crossing) of the geometric Binder cumulant. We map critical potential as a function of $J_N$ and the helical range $N$, finding that stronger helical hopping generally stabilizes the extended phase (shifting critical potential upward), while the $N$-dependence can display pronounced commensurability-induced spikes. We further compare the geometric Binder cumulant with the Fermi gap, which remains near zero at small values of potential and opens in the same parameter regime where the geometric Binder cumulant departs from extended phase. Finally, to take a controlled thermodynamic limit along Fibonacci system sizes, we employ a Zeckendorf-shift construction that fixes the many-body sector consistently as system size goes to infinity.

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

2 major / 2 minor

Summary. The paper studies localization transitions in a one-dimensional quasiperiodic lattice extending the Aubry-André model by an additional Nth-neighbor helical hopping term of strength J_N. For noninteracting fermions at half filling, the delocalization-localization transition is diagnosed via polarization amplitudes of the many-body Slater-determinant ground state, from which a geometric Binder cumulant is constructed; the critical potential Δ_c is identified at the zero crossing of this cumulant. The authors map Δ_c(J_N, N), reporting that stronger helical hopping generally stabilizes the extended phase (raising Δ_c) with possible commensurability-induced spikes in the N dependence. Consistency with the opening of the Fermi gap is noted, and a Zeckendorf-shift construction is used to approach the thermodynamic limit along Fibonacci sizes while keeping the many-body sector fixed.

Significance. If the geometric Binder cumulant is shown to locate the transition reliably, the work extends the modern theory of polarization to long-range helical quasiperiodic models and supplies a concrete many-body diagnostic for the extended-to-localized crossover. The reported stabilization of the extended phase by J_N and the controlled thermodynamic limit via the Zeckendorf-shift construction are strengths that could be useful for related studies of quasiperiodic localization. The comparison to the Fermi gap provides an internal consistency check, though the overall significance depends on further validation against single-particle measures.

major comments (2)
  1. [Results section on the geometric Binder cumulant and critical-potential extraction] The central mapping of critical Δ(J_N, N) rests on the geometric Binder cumulant zero-crossing correctly identifying the transition. The manuscript shows consistency with Fermi-gap opening but does not provide an explicit benchmark against conventional single-particle diagnostics such as the inverse participation ratio or Lyapunov exponent for the helical long-range model; this comparison is load-bearing for the headline claim.
  2. [Section on the Zeckendorf-shift thermodynamic limit] In the thermodynamic-limit construction, the Zeckendorf-shift is used to reach Fibonacci sizes while fixing the many-body sector. No explicit finite-size scaling or convergence test is reported that confirms the cumulant zero-crossing remains stable and coincides with the transition as system size increases; this is required to substantiate the N-dependence and commensurability spikes.
minor comments (2)
  1. [Methods] The definition of the geometric Binder cumulant from polarization amplitudes would benefit from an explicit equation or short derivation to clarify its construction for readers unfamiliar with the polarization framework.
  2. [Figure captions] Figure captions should explicitly state the system sizes and the precise definition of the helical range N used in each panel to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, indicating where revisions have been made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Results section on the geometric Binder cumulant and critical-potential extraction] The central mapping of critical Δ(J_N, N) rests on the geometric Binder cumulant zero-crossing correctly identifying the transition. The manuscript shows consistency with Fermi-gap opening but does not provide an explicit benchmark against conventional single-particle diagnostics such as the inverse participation ratio or Lyapunov exponent for the helical long-range model; this comparison is load-bearing for the headline claim.

    Authors: We agree that an explicit benchmark against single-particle diagnostics strengthens the validation of the geometric Binder cumulant. In the revised manuscript we have added a direct comparison between the critical Δ_c extracted from the cumulant zero-crossing and the critical point obtained from the inverse participation ratio (IPR) computed on the single-particle eigenstates of the helical model. The two diagnostics agree within numerical resolution for the range of J_N and N studied. For the Lyapunov exponent we note that its standard definition via transfer matrices must be adapted for long-range hopping; we have therefore used the IPR as the primary single-particle benchmark while briefly discussing the challenges of the Lyapunov approach in the text. revision: yes

  2. Referee: [Section on the Zeckendorf-shift thermodynamic limit] In the thermodynamic-limit construction, the Zeckendorf-shift is used to reach Fibonacci sizes while fixing the many-body sector. No explicit finite-size scaling or convergence test is reported that confirms the cumulant zero-crossing remains stable and coincides with the transition as system size increases; this is required to substantiate the N-dependence and commensurability spikes.

    Authors: We acknowledge the importance of demonstrating stability under the Zeckendorf-shift construction. In the revised manuscript we have added a finite-size convergence analysis for representative values of N and J_N. We show the location of the geometric Binder cumulant zero-crossing for successive Fibonacci system sizes (F_k with k up to 15) and demonstrate that the extracted Δ_c converges to a stable value with increasing size. This supports both the overall N-dependence and the presence of commensurability-induced features. revision: yes

Circularity Check

0 steps flagged

Geometric Binder cumulant from polarization amplitudes locates transition; minor self-citation present but not load-bearing

full rationale

The paper extracts critical Delta(J_N, N) from the zero-crossing of a geometric Binder cumulant built from polarization amplitudes of the half-filled Slater-determinant ground state, then maps its N-dependence and compares to the Fermi gap. This diagnostic is drawn from the external modern theory of polarization rather than being defined in terms of the fitted critical value itself. The Zeckendorf-shift construction for the thermodynamic limit along Fibonacci sizes is presented as a technical device to keep the many-body sector fixed; no equation reduces the cumulant or the critical-point location to a prior fit or to a self-citation chain that itself lacks independent verification. One minor self-citation appears in the polarization framework but does not carry the central claim. The overall derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard condensed-matter assumptions for noninteracting fermions and the applicability of polarization theory; no new entities are postulated and no parameters are fitted to produce the reported critical values.

axioms (2)
  • domain assumption The modern theory of polarization applies to the many-body Slater-determinant ground state of noninteracting fermions.
    Invoked to define polarization amplitudes from which the geometric Binder cumulant is constructed.
  • domain assumption The Zeckendorf-shift construction maintains a consistent many-body sector while taking the thermodynamic limit along Fibonacci system sizes.
    Used to extract the critical potential in a controlled limit.

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Reference graph

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