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arxiv: 2303.17656 · v1 · pith:EHKO6FGOnew · submitted 2023-03-30 · ❄️ cond-mat.supr-con · cond-mat.dis-nn

Incommensurability-Induced Enhancement of Superconductivity in One Dimensional Critical Systems

Ricardo Oliveira , Miguel Gon\c{c}alves , Pedro Ribeiro , Eduardo V. Castro , Bruno Amorim This is my paper

Pith reviewed 2026-05-24 09:36 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.dis-nn
keywords superconductivityquasiperiodic systemsAubry-André modelone-dimensional systemsincommensurabilitycritical phases-wave pairingcritical temperature scaling
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The pith

Incommensurate modulations raise the superconducting critical temperature in one-dimensional critical phases and change its scaling from exponential to algebraic.

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

The paper examines a generalized Aubry-André chain that carries both potential and hopping modulations. Without interactions this model hosts extended, critical, and localized phases when the modulation is incommensurate. Once s-wave pairing is added, the critical temperature inside the parent critical region rises markedly above the values found in the extended phase or in the uniform (unmodulated) limit. The same incommensurate case produces an algebraic rise of Tc with interaction strength, whereas a nearby commensurate modulation recovers the usual exponentially small BCS scaling. These two scalings together produce a clear enhancement of superconductivity at weak and intermediate coupling.

Core claim

In the generalized Aubry-André model with incommensurate modulations, a substantial region of the parent critical phase exhibits a significantly higher superconducting critical temperature than either the extended phase or the uniform limit; the critical temperature scales algebraically with interaction strength for incommensurate modulations and exponentially for commensurate ones, resulting in an incommensurability-induced enhancement accompanied by a larger zero-temperature order parameter.

What carries the argument

The generalized Aubry-André Hamiltonian with simultaneous quasiperiodic potential and hopping terms, whose incommensurate versus commensurate character controls the algebraic versus exponential scaling of Tc.

If this is right

  • Inside the critical phase the superconducting order parameter at zero temperature is larger than in the extended phase or uniform chain.
  • The algebraic scaling of Tc with interaction strength persists into the weak-coupling regime for incommensurate modulations.
  • Commensurate modulations of similar period recover the standard exponentially small BCS result once the system is large enough.
  • The enhancement is strongest in the weak-to-intermediate coupling window where the algebraic and exponential curves diverge most.

Where Pith is reading between the lines

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

  • If the algebraic scaling survives in higher dimensions or with longer-range interactions, incommensurate modulations could offer a route to higher-Tc superconductivity without increasing the bare pairing strength.
  • The same mechanism might appear in cold-atom realizations of quasiperiodic lattices where the modulation ratio can be tuned continuously across commensurate and incommensurate values.
  • Experimental searches could focus on transport or spectroscopy signatures that distinguish algebraic from exponential Tc scaling in quasiperiodic wires.

Load-bearing premise

The non-interacting extended, critical, and localized phases stay well-defined and continue to control the interacting problem, and the numerical extraction of Tc reaches the thermodynamic limit without finite-size effects that would erase the algebraic-exponential distinction.

What would settle it

A direct comparison, on sufficiently large systems, of the functional dependence of Tc on interaction strength for a fixed incommensurate modulation versus a nearby commensurate modulation with the same average density.

Figures

Figures reproduced from arXiv: 2303.17656 by Bruno Amorim, Eduardo V. Castro, Miguel Gon\c{c}alves, Pedro Ribeiro, Ricardo Oliveira.

Figure 1
Figure 1. Figure 1: (a) α BdG n ≡ − log [MIPR] / log [Fn] for the BdG eigenstates in the (V1, V2) parameter space at zero tempera￾ture, with g = 1 for a system of size N = Fn = 610. α BdG n converges to 1 for extended states, 0 for localized states and to non-integer in-between values for critical states as n → ∞. The transition lines AB, BC and BD of the localization dia￾gram for the free system are plotted as dashed black l… view at source ↗
Figure 2
Figure 2. Figure 2: Momentum-space distribution of the BdG wave [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Finite-size analysis of MIPR for the BdG [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

We show that incommensurability can enhance superconductivity in one dimensional quasiperiodic systems with s-wave pairing. As a parent model, we use a generalized Aubry-Andr\'e model that includes quasiperiodic modulations both in the potential and in the hoppings. In the absence of interactions, the model contains extended, critical and localized phases for incommensurate modulations. Our results reveal that in a substantial region inside the parent critical phase, there is a significant increase of the superconducting critical temperature compared to the extended phase and the uniform limit without quasiperiodic modulations. We also analyse the results for commensurate modulations with period close to the selected incommensurate one. We find that while in the commensurate case, the scaling of the critical temperature with interaction strength follows the exponentially small weak-coupling BCS prediction for a large enough system size, it scales algebraically in the incommensurate case within the critical and localized parent phases. These qualitatively distinct behaviors lead to a significant incommensurability-induced enhancement of the critical temperature in the weak and intermediate coupling regimes, accompanied by an increase in the superconducting order parameter at zero temperature.

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 / 1 minor

Summary. The manuscript studies superconductivity in a one-dimensional generalized Aubry-André model with s-wave pairing and quasiperiodic modulations in both potential and hoppings. It claims that, inside the parent non-interacting critical phase, incommensurate modulations produce a substantial increase in the superconducting critical temperature Tc relative to the extended phase and the uniform limit. The scaling of Tc with interaction strength U is reported to be algebraic for incommensurate modulations (within critical and localized phases) but exponentially small (BCS-like) for commensurate modulations of similar period, yielding an incommensurability-induced enhancement at weak and intermediate coupling together with an increase in the zero-temperature order parameter.

Significance. If the algebraic-versus-exponential distinction survives the thermodynamic limit, the result would identify a concrete mechanism by which quasiperiodicity can enhance superconductivity in one dimension beyond standard weak-coupling expectations. The direct comparison between incommensurate and commensurate cases supplies a falsifiable signature that could guide future numerical and experimental work on quasiperiodic superconductors.

major comments (2)
  1. [Numerical extraction of Tc and scaling analysis] The load-bearing claim is the survival of algebraic Tc(U) scaling for incommensurate modulations versus exponential scaling for commensurate ones in the thermodynamic limit. Because incommensurate cases are realized via rational approximants whose period grows with system size, the accessible lengths remain finite; at weak U the superconducting correlation length can exceed these lengths, allowing an exponentially small Tc to appear algebraic over the simulated window. No explicit finite-size scaling collapse, extrapolation procedure, or error bars on extracted Tc values are described that would control this artifact.
  2. [Phase identification and interaction effects] The parent non-interacting phases (extended, critical, localized) are used to interpret the interacting results, yet the manuscript provides no direct evidence that these phases remain sharply defined or that the localization properties survive once interactions are introduced. This assumption is central to identifying the 'substantial region inside the parent critical phase' where enhancement occurs.
minor comments (1)
  1. [Abstract] The abstract states that the model 'contains extended, critical and localized phases for incommensurate modulations' but does not specify the numerical method (e.g., DMRG, exact diagonalization) or the interaction cutoff employed to obtain Tc.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address the major comments point by point below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Numerical extraction of Tc and scaling analysis] The load-bearing claim is the survival of algebraic Tc(U) scaling for incommensurate modulations versus exponential scaling for commensurate ones in the thermodynamic limit. Because incommensurate cases are realized via rational approximants whose period grows with system size, the accessible lengths remain finite; at weak U the superconducting correlation length can exceed these lengths, allowing an exponentially small Tc to appear algebraic over the simulated window. No explicit finite-size scaling collapse, extrapolation procedure, or error bars on extracted Tc values are described that would control this artifact.

    Authors: We agree that finite-size effects must be carefully controlled when using rational approximants, as the distinction between algebraic and exponential scaling is central to the claim. Our data were obtained from a sequence of approximants with increasing period, showing consistent algebraic behavior for incommensurate modulations and exponential decay for commensurate ones once the period is sufficiently large. To strengthen the evidence for the thermodynamic limit, the revised manuscript will include additional approximant sizes, a finite-size scaling analysis with extrapolation of Tc, and error estimates on the extracted values. revision: yes

  2. Referee: [Phase identification and interaction effects] The parent non-interacting phases (extended, critical, localized) are used to interpret the interacting results, yet the manuscript provides no direct evidence that these phases remain sharply defined or that the localization properties survive once interactions are introduced. This assumption is central to identifying the 'substantial region inside the parent critical phase' where enhancement occurs.

    Authors: The phases are defined in the non-interacting limit and used to classify the parameter regimes in which the interacting superconducting properties are studied. For the weak-to-intermediate couplings considered, we expect the qualitative distinctions to remain relevant. We acknowledge that direct probes of localization (such as interacting inverse participation ratios) are not presented. In the revised manuscript we will expand the discussion of this assumption and its limitations, and add any feasible supporting analysis. revision: partial

Circularity Check

0 steps flagged

No significant circularity; numerical results are independent of inputs

full rationale

The paper reports direct numerical computations of Tc and its scaling in a generalized Aubry-André model, distinguishing algebraic vs exponential behavior as an output of the simulations for incommensurate vs commensurate cases. No equations or claims reduce by construction to fitted parameters, self-definitions, or self-citation chains; the central enhancement result is presented as emerging from the calculations without load-bearing self-references or ansatzes smuggled in. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated beyond the standard assumption that the generalized Aubry-André Hamiltonian plus s-wave pairing term captures the relevant physics.

axioms (1)
  • domain assumption The non-interacting generalized Aubry-André model possesses well-defined extended, critical, and localized phases that remain diagnostically useful once interactions are added.
    Invoked when the authors locate the enhancement inside the parent critical phase.

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