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REVIEW 3 major objections 5 minor 42 references

The nature of the "pseudogap" in the insulating phase of highly disordered superconductors

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The single-particle gap survives across the superconductor–insulator transition and grows with disorder to more than twice its superconducting-side value, while filling rather than closing as temperature rises.

desk verdict First tunneling spectroscopy deep in the insulating side of the SIT; the gap-growth trend is credible, but the Cooper-pair reading rests on an untested 9 T normalization and a pairing-presupposing fit. read the letter →

arxiv 2608.12508 v1 pith:EK5VRRNL submitted 2026-08-12 cond-mat.supr-con cond-mat.dis-nncond-mat.str-el

Mark Nikolaevsky , Abhisek Samanta , Nandini Trivedi , Aviad Frydman This is my paper · ORCID
classification cond-mat.supr-concond-mat.dis-nncond-mat.str-el
keywords pseudogapsuperconductor-insulatortransitionCooper-pairinsulatorlocalizedCooperpairstunnelingspectroscopyamorphousindiumoxideattractiveHubbardmodelquantumMonteCarlo
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether a gap in the single-particle spectrum can arise purely from pairing without global phase coherence. It answers yes: tunneling measurements on amorphous indium oxide films, taken deep in the insulating regime, show a single-particle gap that survives the loss of superconductivity and grows with disorder, reaching more than twice the superconducting-side value. Quantum Monte Carlo simulations of the disordered attractive Hubbard model reproduce the same growth. The paper interprets this as spectroscopic evidence for a Cooper-pair insulator, where electrons remain paired in the valleys of the disorder potential but the phase of the order parameter is not coherent across the film. This unifies the pseudogap above $T_c$ and the insulating gap as two manifestations of pairing without global phase coherence.

What carries the argument

The load-bearing object is the Cooper-pair insulating state: electrons bind into local pairs in the minima of the random disorder potential, forming small superconducting islands whose phases are not coherent with one another. Experimentally, the enabling tool is ultra-high-resistance planar tunnel junctions, around 100 MΩ, with picoampere current sensitivity, which permit tunneling spectroscopy on films too resistive for scanning tunneling microscopy. The gap is extracted from the measured spectra using a model that accounts for spatial and temporal fluctuations of the superconducting order parameter. Theoretically, the argument is carried by determinant quantum Monte Carlo simulations of the disordered attractive Hubbard model, which include both quantum and thermal phase fluctuations and yield the density of states through maximum-entropy analytic continuation.

What would settle it

Measure the extracted gap as a function of the film's charging energy, or attempt to destroy pairing by applying a magnetic field while keeping disorder fixed: if the gap persists when pairing is suppressed, or tracks the grain charging energy $e^2/2C$, the Cooper-pair interpretation fails; if it disappears or follows the predicted localization scaling, the interpretation holds.

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Extended reading notes

Core claim

The central discovery is that the single-particle excitation gap does not vanish when superconductivity is destroyed by disorder; instead it persists and grows monotonically with disorder, by a factor of more than two. The gap has a distinctive temperature signature: as temperature rises, states fill in below the gap and coherence peaks are suppressed, but the gap edge barely moves until a temperature $T^*$ where it disappears, which is not the BCS closing-gap behavior. The same behavior appears in determinant quantum Monte Carlo simulations of the attractive Hubbard model with random on-site disorder. The paper concludes that the underlying state is a Cooper-pair insulator with localized pairs trapped in potential minima and no global phase coherence, and proposes that the pseudogap in cuprates and the insulating gap in these films share the mechanism of pairing without phase coherence.

Load-bearing premise

The interpretation rests on the fitted tunneling gap's being a pairing gap from localized Cooper pairs; the paper does not explicitly rule out single-electron Coulomb-blockade or charging effects in the strongly insulating film.

Editorial extensions

If this is right

  • In the paired-insulating regime, the single-particle spectrum remains gapped up to a disorder-dependent temperature $T^*$, so a film can be electrically insulating while still showing a spectroscopic pairing gap.
  • Gap filling with temperature, long considered a fingerprint of cuprates, can appear in a weak-coupling s-wave superconductor once disorder suppresses phase coherence.
  • Spectral weight removed from the gap is redistributed to energies well above the gap scale, with the high-energy scale set by the disorder landscape.
  • Ultra-high-resistance planar junctions extend tunneling spectroscopy deep into the insulating phase, a regime that was previously inaccessible to STM.
  • The agreement between experiment and quantum Monte Carlo strengthens the case that local pairing alone, without competing electronic orders, can produce pseudogap phenomenology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test of the pairing interpretation would be to measure the same films in a magnetic field strong enough to suppress pairing: a pairing gap should evolve or close at a pair-breaking field, whereas a Coulomb-blockade gap would be insensitive to such a field; the paper does not report this check.
  • The localization-based expression $\Delta \propto U/2\xi^2$ implies a quantitative relationship between the extracted gap and the disorder-driven localization length; measuring both on the same film would test whether the growth rate matches the mechanism.
  • The same ultra-high-resistance planar junction technique could be applied to other amorphous superconductors or thickness-tuned films to see whether the paired-insulator signature is a general feature of the disorder-driven superconductor–insulator transition.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper reports tunneling spectroscopy of amorphous indium oxide films driven deep into the insulating side of the disorder-driven superconductor–insulator transition (SIT), using planar junctions with resistances up to 100 MΩ and picoampere current sensitivity, combined with determinant quantum Monte Carlo (DQMC) simulations of the disordered attractive Hubbard model. The normalized conductance (dI/dV at 0 T divided by the spectrum at 9 T) shows a single-particle gap that survives across the transition and grows with disorder, reaching values more than twice those on the superconducting side; with increasing temperature the gap fills rather than closes, and spectral weight is redistributed to high energies. The DQMC results exhibit the same qualitative trends, and the paper interprets the combined observations as direct quantitative confirmation of a Cooper-pair insulating state—local pairing without global phase coherence—and as a unifying link to pseudogap physics in cuprates.

Significance. If the identification of the measured gap as a pairing gap holds, this is a significant advance: it is, to my knowledge, the first tunneling spectroscopic access well into the insulating phase of the SIT, a regime previously inaccessible because of the junction-resistance and current-sensitivity constraints that the authors have overcome. The theoretical side is genuinely independent: the DQMC parameters (U/t, V/t, μ, temperature) are inputs and are not fitted to the data, the method is sign-problem-free for the attractive Hubbard model, and the predicted gap growth with disorder agrees in trend with the experiment. The temperature-filling behavior and its similarity to cuprate pseudogap phenomenology are interesting and add breadth. These strengths are real. However, "direct quantitative experimental confirmation" is stronger than the evidence supports: the high-energy redistribution scale is admitted by the authors to disagree by nearly two orders of magnitude, and the pairing-gap identification rests on a normalization reference and a fitting model whose validity deep in the insulating regime is not established.

major comments (3)
  1. [The gap at high disorder; Appendix A and Figs. 5–6] The normalized conductance is formed as dI/dV(0 T)/dI/dV(9 T), and the text refers to the 9 T curves as "normal state" spectra, but the paper nowhere demonstrates that at 9 T, deep in the insulating regime, the single-particle DOS is featureless. Appendix A and Fig. 6 track only the zero-bias conductance versus field orientation; the full field-dependent DOS and the bias-dependent shape of the 9 T reference are not shown. In the deep insulator the high-field state is itself an insulator, so a gap-like suppression in the 9 T reference (from charging effects or field-dependent localization) would directly distort the normalized spectrum and the extracted gap, and the distortion could evolve with disorder. Because the growing-gap trend in Fig. 1(b) is the central experimental result, the validity of the 9 T reference is load-bearing.
  2. [The gap at high disorder (Fig. 1b); Appendix A] The gap values in Fig. 1(b) are extracted by fitting to the Dentelski et al. fluctuating-superconductivity model (ref. 40), whose functional form presupposes a locally paired state. The manuscript provides no test of whether a single-electron charging (Coulomb-blockade) gap or another non-pairing mechanism can account for the same spectra, no control experiment or distinguishing observable, and no fit-quality or residual analysis. As a result, the central interpretation—that the measured insulating gap is a Cooper-pair gap—is built into the analysis rather than independently established, and the abstract's wording "direct quantitative experimental confirmation" overstates what the measurement alone demonstrates. The authors should compare the fits against at least one non-pairing gap model and report goodness-of-fit for both.
  3. [Redistribution of spectral weights (Fig. 3)] In the spectral-weight redistribution section the authors state that the energy scale "differs significantly" between theory and experiment—the calculated shift is of order Δ whereas the experimental redistribution "extends nearly two orders of magnitude"—and they attribute the difference to the finite numerical bandwidth. This self-stated limitation directly contradicts the abstract's claims of "striking agreement" and "quantitative" confirmation for this part of the data. Moreover, the dc bias was swept only over ±100 mV, so the "two orders of magnitude" scale may be constrained by the measurement window rather than by the underlying physics. The manuscript should either provide a quantitative metric for the agreement on the spectral-weight redistribution or restrict the quantitative-confirmation claim to the gap magnitude and its temperature filling.
minor comments (5)
  1. [The gap at high disorder; Appendix A] The main text states that the trend in Fig. 1(b) "was found for nine measured films," while Appendix A states that "Four samples were measured"; the relationship between films and samples (e.g., number of junctions per film, annealing stages) should be clarified.
  2. [Appendix A] The sentence "fitting it to a model published by David et al. in 2018" misnames the authors of ref. 40, which is the Dentelski et al. model; in addition, the main text contains a stray parenthesis in "(see supplementary material) for details)" and repeatedly references a supplementary material that is not present in the submission, although Appendices A and B partially cover the same ground.
  3. [Throughout] Several typos remain: "insulting regime" should be "insulating regime" in the gap-growth paragraph; "This poses a question as to regarding the conservation of spectral density" contains a duplicated preposition; "Fig. 2 (b) depicts ... calculations. showing that" has a lowercase word after a period; and "InO/AlO/Alpla planar tunnel junctions" appears to be missing a letter in the top electrode metal.
  4. [Fig. 1(d)] The caption states that the gap is calculated "at T=0" using quantum Monte Carlo, but DQMC is a finite-temperature method and the simulations in Appendix C are presented at low but nonzero T/t=0.1; the caption should specify the temperature or the extrapolation procedure used.
  5. [Fig. 1(b)] The gap extraction points in Fig. 1(b) are shown without error bars or any measure of fit uncertainty; since the gap values are the core experimental quantity, the authors should report the fitting uncertainty and the number of spectra averaged per disorder level.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the experimental tunneling gaps and the DQMC calculations are independent inputs, and the coauthor-involved references are used as quantitative tools rather than as forced conclusions.

full rationale

The central comparison is between measured tunneling spectra and determinant quantum Monte Carlo simulations of the disordered attractive Hubbard model. The DQMC parameters (U/t, V/t, temperature, system size) are stated inputs and are not fitted to the experimental tunneling data; the gap-versus-disorder and gap-filling-versus-temperature curves are independent computations. The experimental gap is quantified by fitting to the Dentelski et al. fluctuating-superconductivity model (ref 40), but the gap is already visible in the raw normalized dI/dV spectra, and the model is used as a known quantitative extraction framework, not as part of the prediction being tested. The theoretical prediction in ref 39 is by a coauthor, and ref 40 also involves a coauthor, but neither citation carries the argument by itself: the paper performs its own DQMC calculation, and the experimental observation is new and external to those theoretical works. The paper explicitly acknowledges a two-order-of-magnitude discrepancy in the spectral-weight redistribution energy scale and attributes it to the numerical bandwidth; this is a stated limitation, not a circular step. Possible experimental artifacts, such as a non-featureless H=9T reference spectrum or an alternative Coulomb-blockade interpretation of the insulating gap, are scientific validity risks rather than instances of the derivation reducing to its own inputs. No equation defines a predicted quantity in terms of the data used to produce it, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim relies on interpreting the measured gap as a pairing gap. The model parameters are inputs, not fitted to the experiment, which keeps the theory side independent. The main assumptions are the validity of the Hubbard model for a-InO, the 9 T normalization, and the gap extraction model.

free parameters (4)
  • U/t (attractive interaction) = 1.5 and 4
    Chosen by hand to represent weak and intermediate coupling; not fitted to the experimental spectra.
  • V/t (disorder strength) = swept from 0.1 to 6
    Independent variable in the model; corresponds to different experimental disorder levels but not mapped quantitatively to sheet resistance.
  • mu (chemical potential) = not stated
    Determines electron density; likely set for a half-filled band, but the paper does not report the value.
  • Dentelski model fit parameters = gap Delta and auxiliary parameters
    Used to extract the experimental gap from each tunneling spectrum; the model itself comes from prior work with overlapping authorship.
assumptions (5)
  • domain assumption The attractive Hubbard model on a square lattice with random onsite disorder (Eq. 1) captures the essential physics of amorphous indium oxide films.
    The paper maps the experimental system to this model without quantitative justification; a-InO is amorphous, not a clean square lattice.
  • domain assumption The 9 T field fully suppresses superconductivity and provides a true normal-state DOS, so normalizing by the 9 T spectrum removes non-pairing effects.
    The entire spectral analysis divides zero-field spectra by 9 T spectra; if pairing persists or the normal state changes with field, the gap estimate is biased.
  • domain assumption The Dentelski et al. model (ref 40) correctly describes the tunneling DOS in disordered films, so the fitted gap is the true single-particle gap.
    The experimental gap values are extracted by fitting this model, which assumes fluctuating superconductivity; a different mechanism (e.g., Coulomb blockade) could produce a similar spectral gap.
  • standard math Maximum entropy analytic continuation reliably reconstructs N(omega) from imaginary-time QMC data, including gap edges and peak positions.
    MEM is approximate and can smooth or shift features; the paper does not validate against exact diagonalization or alternative methods.
  • domain assumption The localization scaling Delta proportional to U/(2 xi^2) is valid in the parameter regime.
    This scaling is invoked to explain the gap increase; it references prior theory (ref 39) but is not derived in this work.

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Pith. "Pith review of The nature of the "pseudogap" in the insulating phase of highly disordered superconductors." pith.science (2026). https://pith.science/paper/EK5VRRNL

@misc{pith2026260812508,
  author       = {Pith},
  title        = {Pith review of: The nature of the "pseudogap" in the insulating phase of highly disordered superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EK5VRRNL}},
  note         = {Machine review of arXiv:2608.12508}
}
abstract

Disordered thin films undergoing a superconductor-insulator transition provide a controlled setting for studying pseudogap physics in the absence of competing electronic orders. Although theory predicts that local Cooper pairing can survive deep into the insulating phase, direct spectroscopic confirmation has remained experimentally inaccessible because tunneling measurements in highly insulating films require ultra-high-resistance junctions and picoampere current sensitivity. Here we combine ultra-high-resistance planar tunneling spectroscopy on amorphous indium oxide films with quantum Monte Carlo simulations of the attractive Hubbard model to probe the single-particle excitation spectrum deep in the insulating regime. We find striking agreement between experiment and theory: the single-particle gap not only survives across the superconductor-insulator transition, but increases substantially with disorder, reaching values more than twice those observed on the superconducting side. With increasing temperature, the gap fills rather than closes, while coherence peaks are suppressed and spectral weight redistributes to energies far exceeding the gap scale. Our results provide direct quantitative experimental confirmation of the theoretically predicted Cooper-pair insulating state with localized Cooper pairs and establish a unified connection between the pseudogap above $T_c$ and the insulating gap as manifestations of pairing without global phase coherence.

Figures

Figures reproduced from arXiv: 2608.12508 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. , is extended in this work deep into the insulating regime of highly disordered InO. Experimentally we find ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic illustration of superconductivity in a highly [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Reference graph

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