Disorder and the Robustness of Superconductivity on the Flat Band

Beno\^it Gr\'emaud; G. George Batrouni; Si Min Chan

REVIEW 4 major objections 4 minor 77 references

Disorder and the Robustness of Superconductivity on the Flat Band

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read On the flat-band Creutz ladder, superconductivity survives disorder up to a finite critical strength that scales with the clean superfluid weight.

desk verdict A credible numerical case that flat-band superconductivity survives finite disorder, with Wc ∝ Ds; the main caveat is that Wc rests on a three-size BKT collapse without error bars. read the letter →

arxiv 2506.07095 v2 pith:GJHZ7YAA submitted 2025-06-08 cond-mat.supr-con cond-mat.dis-nncond-mat.str-el

Si Min Chan , Benoît Grémaud , G. George Batrouni This is my paper

classification cond-mat.supr-concond-mat.dis-nncond-mat.str-el

keywords flat-bandsuperconductivityCreutzlatticeattractiveHubbardmodeldisordersuperfluidweightBerezinskii-Kosterlitz-Thoulesstransitionpairlocalizationdensitymatrixrenormalizationgroup

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 superconductivity on a flat band survives when the lattice is dirtied. On the Creutz ladder—a two-chain lattice whose noninteracting bands are flat—attractive interactions bind fermions into pairs that move and carry supercurrent. Using density-matrix renormalization group calculations on chains of up to 200 unit cells, the authors argue that a finite disorder strength $W_c$ is needed to localize those pairs and destroy superconductivity for every interaction strength studied, $0

What carries the argument

The central object is the superconducting (superfluid) weight $D_s$, defined as the curvature of the ground-state energy with respect to an applied phase twist, $D_s=\pi L\,d^2E_{\rm GS}(\Phi)/d\Phi^2|_{\Phi=0}$, which vanishes when the system can no longer carry supercurrent. The argument measures $D_s$ as a function of disorder $W$ at several system sizes, collapses those curves using the BKT correlation length $\xi(W)=a\exp(b/\sqrt{W-W_c})$, and compares the extracted $W_c(U)$ with the clean-system $D_s(U)$. Pair correlations supply the supporting scales: they decay as $r^{-1/(2K)}$ in the clean system and as $e^{-r/\xi_{\rm disorder}}$ in the localized phase, giving both the Luttinger parameter $K$ and the disorder localization length. The large-$U$ limit is checked against an effective hard-core-boson Hamiltonian, and the finite-size collapse uses a cost-function minimization over the BKT parameters.

What would settle it

At quarter filling and $U=8$, measure $\xi_{\rm disorder}$ directly from pair correlations on chains up to $L=400$ and fit $\ln\xi$ against both $\ln(W-W_c)$ and $1/\sqrt{W-W_c}$ over a decade of $W-W_c$; if the power-law form fits better or the directly measured lengths disagree with the BKT form fitted from $D_s$ collapses, the claimed BKT transition and finite $W_c$ are falsified. Extending the $D_s$ collapse to $L=24$ and $L=28$ would also reveal whether $W_c$ drifts toward zero with system size.

Watch

Extended reading notes

Core claim

On the attractive Hubbard model on the Creutz lattice, the paper finds $W_c\neq 0$ at both quarter and eighth filling for the full range $0<U\le 25$, including interaction strengths where the Luttinger parameter $K$ lies below the usual critical value $K_c=3/2$ for spinful attractive fermions; the argument is that in the large-$U$ limit the relevant critical value is the hard-core-boson one, $K_c^{\rm HCB}=3/4$, above which the system remains. The localization transition is argued to have Berezinskii–Kosterlitz–Thouless form, supported by direct measurement of the disorder localization length from pair correlations on chains up to $L=200$ at $U=1$ and by scaling collapse of $D_s(W)$ at $L=12,16,20$. The controlling relation is $W_c(U)\approx 0.084\,D_s(U,W=0)$ at quarter filling. Even at weak attraction the localization is of pairs, not of single fermions, and mean-field theory, despite predicting a second-order rather than BKT transition, locates $W_c$ accurately when scaled with the appropriate power-law function.

Load-bearing premise

The load-bearing assumption is that the BKT collapse of the superfluid weight at only three system sizes ($L=12,16,20$) gives unbiased critical disorder values; the direct long-chain check of the BKT form was performed at one interaction strength only, and the extracted $W_c$ values carry no reported error bars.

Editorial extensions

If this is right

Disorder must exceed a finite $W_c$ before pair transport is destroyed; weak on-site disorder leaves the superconducting state intact on the Creutz lattice.
Because $W_c$ is proportional to the clean superconducting weight $D_s$, parameter choices that raise $D_s$—such as lower $U$ in the regime where $D_s=\pi U\rho(1-\rho)$—also raise the disorder threshold.
At strong coupling the pairs behave as hard-core bosons with critical Luttinger parameter $K_c^{\rm HCB}=3/4$, and since $K$ stays above that value, the large-$U$ system remains robust against disorder.
Mean-field theory, although it gets the universality class wrong, can still be used to locate the critical disorder if the correct power-law scaling function is applied.
The quadratic-in-$\Phi$ approximation for the phase-twist energy, valid in the clean limit, fails once disorder is present, and its use can produce a spurious $W_c=0$.

Reading between the lines

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

If the proportionality $W_c\propto D_s$ is generic, then flat-band lattices engineered to maximize $D_s$—for instance by closing the gap to the next dispersive band—would also be the most disorder-tolerant; the same density-matrix renormalization group collapse on the sawtooth or Lieb geometries could test this.
The reported constant $0.084$ is established at quarter filling; repeating the analysis at eighth filling, where the pair-correlation exponent saturates at a different value, would show whether $W_c/D_s$ is universal or density-dependent.
Because the disorder localization length can exceed experimentally accessible system sizes, finite samples may show superconducting response even when $W>W_c$; experiments would need to separate such finite-size vestiges from true thermodynamic superconductivity.
Direct large-$L$ measurement of $\xi_{\rm disorder}$ at $U>1$ would independently check whether the BKT form assumed in the small-size collapse is correct outside the single interaction strength at which it was tested directly.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a solid DMRG study of the attractive Hubbard model on a disordered Creutz ladder, and its headline claim—finite Wc at every U, with Wc roughly linear in the clean-system superfluid weight Ds—is a genuine new result that directly contradicts the Wc=0 finding of Liang et al. [53]. The paper deserves a serious referee, and I would not desk-reject it.

What is actually new: the finite-Wc result across 0

The soft spots are real but not fatal. Every Wc reported in Figs. 5–8 is obtained by collapsing Ds(W) at only L=12,16,20 with a cost-function fit (Appendix B). There are no error bars on Wc, no explicit test that the cost-function minimum is sharp, and no direct comparison to a Wc=0 fit at the same system sizes. The direct ξdisorder measurement that would independently fix the BKT form is shown only for U=1; for the intermediate and large-U cases the form is assumed rather than verified. The disagreement with Ref. [53] is explained by a plausible critique of their Ds approximation, but the authors do not benchmark that approximation by running their own code with the [53] method. If the fitting procedure is mildly flexible, the linear Wc-Ds relation could absorb some systematic bias.

Still, I do not think the central claim is broken. The evidence is consistent, the large-U HCB check is independent of the BKT collapse, and the U=1 direct correlation-length data validate the scaling form where it is actually checked. What is needed is more: L=24,28, more disorder realizations, bootstrapped error bars on Wc, and a head-to-head with the [53] approximation. That is exactly what peer review should ask for.

For whom: researchers in flat-band superconductivity, disordered ladders, and DMRG studies of BKT transitions. I would cite the finite-Wc result with a caveat on the uncertainty.

Referee Report

4 major / 4 minor

Summary. The manuscript studies the attractive Hubbard model on the quasi-one-dimensional Creutz ladder with on-site disorder, at quarter and eighth filling, using DMRG and MPS methods. It reports three main claims: (i) the critical disorder strength Wc required to destroy superconductivity is finite for all interaction strengths studied (0 < U ≤ 25); (ii) the delocalized-to-localized transition has Berezinskii–Kosterlitz–Thouless (BKT) scaling form, with the disorder correlation length diverging as ξ ∼ exp(b/√(W−Wc)); and (iii) Wc is proportional to the superfluid weight Ds of the clean system, with proportionality constant approximately 0.084 at quarter filling. The large-U regime is analyzed through an effective hard-core-boson (HCB) model, whose Ds collapse gives Wc_HCB = 1.5, and mean-field theory is shown to reproduce Wc values despite predicting the wrong universality class.

Significance. If the central claims hold, the paper provides a useful quantitative statement about the robustness of flat-band superconductivity: a finite disorder threshold exists at all U, and that threshold is controlled by the clean system's superfluid weight. The explicit HCB large-U confirmation is a genuine strength, as is the direct pair-correlation evidence at U=1 and, to a lesser extent, at U=8. However, the main quantitative claim—the linear Wc ∝ Ds relation—rests on a finite-size collapse procedure applied to only three small system sizes, with no reported error bars and no systematic exclusion of Wc=0 or power-law alternatives for most values of U. The unresolved disagreement with Ref. [53] makes the fitting methodology a load-bearing element that needs to be made more rigorous.

major comments (4)

[Section III.C and Appendix B, Eq. (B1)] For all U values except U=1, every Wc quoted in Fig. 8(b) is extracted solely from a cost-function collapse of Ds(W) computed at L=12, 16, and 20, using the BKT form Eq. (8) with Wc and b as free parameters. The paper reports no error bars on Wc, no bootstrap over disorder realizations, no stability check with additional system sizes (e.g., L=24 or L=28), and no quantitative comparison with a power-law collapse or with a Wc=0 null hypothesis for U≠1. Because the cost function Eq. (B1) minimizes scatter by construction, a finite Wc is an output of the fitting procedure rather than an independently tested prediction. Please add confidence intervals on Wc, show the sharpness of the cost-function minimum, and perform a null test (e.g., fix Wc=0 and compare the resulting cost) to demonstrate that the finite critical disorder is required by the data.
[Section III.C, Fig. 8(a)] The central proportionality Wc = 0.084 Ds is obtained by fitting Wc values that all come from the same collapse procedure. If that procedure is biased at any U, the bias is inherited by every point in Fig. 8(a), and the apparent linear relation could be an artifact of the fitting method rather than a physical statement. Independent determinations of Wc are needed at least at one or two intermediate U values—for example, by direct measurement of ξdisorder(W) at U=8 or U=15, or by repeating the Ds collapse with more system sizes—and confidence intervals on the slope of Wc versus Ds should be reported.
[Section III.C, Ref. [53] discussion] The paper attributes the disagreement with Ref. [53], which found Wc=0 at U=8 and ρ=0.25, to Ref. [53]'s approximation Ds ≈ (8L/π)(EGS(π/2) − EGS(0)) becoming invalid in the presence of disorder (Fig. 11). However, it does not demonstrate that this approximation actually drives Wc to zero. Please compute Ds with both the full quadratic fit used here and the approximation used in Ref. [53] for the same parameters, and show that the approximation yields Wc=0 (or otherwise identify the precise source of the discrepancy). Without this, the disagreement leaves room for doubt about the finite-Wc conclusion at exactly the parameter point where the two papers conflict.
[Section III.B, Fig. 4] The BKT scaling form is directly verified from the divergence of ξdisorder only at U=1, ρ=0.5. At U=8, Fig. 3 shows power-law decay at W=0.1 and exponential decay at W=0.5, which is evidence for a finite Wc but is not a measurement of the divergence law. For all other U values, the BKT form is assumed in the Ds collapse rather than tested. Since the universality class is one of the main claims, please provide either direct ξdisorder(W) data at one or two additional parameter points or a quantitative comparison of BKT versus power-law collapses of Ds using the same system sizes.

minor comments (4)

[Throughout] There are several typos and grammatical slips: “inavlid” (Section III.C), “disagress” (Section III.C), “illsutrates” (Section III.C), “T op” (Fig. 1 caption), and “1 ≥ U ≤ 25” (Fig. 17 caption), which should be “1 ≤ U ≤ 25”.
[Eq. (7)] The effective HCB Hamiltonian contains the term n_j^α n_{j+1}^β + n_{j+1}^β n_j^α; since these operators commute for different sites, this appears to be just 2 n_j^α n_{j+1}^β. Please clarify whether the doubled notation is intentional or a typographical artifact.
[Section III.C] The sentence “Using MPS optimization, we showed that the scaling form and prefactor a cannot simply be extracted from minimizing the cost function for Ds at various system sizes” is confusing, because the same cost-function minimization is then used to obtain Wc. Please clarify the role of the prefactor a and state explicitly why it does not affect the Ds collapse.
[Appendix B] The cost function in Eq. (B1) is introduced without a brief explanation of why the normalized total variation measures collapse quality or how the minimum is located in the (b, Wc) plane. A few sentences describing the minimization grid and the reported uniqueness of the minimum would make the procedure reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Wc is fit to disordered-system data while Ds of the clean system is an independent input.

full rationale

The central claim, Wc finite and Wc ∝ Ds(U, W=0), is not circular by construction. Wc is obtained by minimizing the cost function (Appendix B, Eq. B1) for the BKT collapse of Ds(W) curves at L = 12, 16, 20 (Figs. 5-7). The clean-system Ds is computed separately via the phase-twist formula, Eq. (4), at W = 0; it is not an input to the cost-function minimization. Thus Wc ∝ Ds is an empirical correlation between two independently computed U-dependent quantities, not an identity. The BKT scaling form is justified by direct measurement of the disorder correlation length from pair correlations on chains up to L = 200 (Fig. 4), at least at U = 1; using the same form at other U is an extrapolation, which is a correctness/robustness concern rather than circularity. Some supporting clean-system results (K values, weak-U Ds formula) are cited from the same group's Refs. [11, 12, 56], but the paper recomputes the Luttinger parameters in Fig. 2, and the central Wc data are new DMRG results; those self-citations are not load-bearing. The absence of error bars and the small number of system sizes affect statistical confidence but do not make the derivation circular.

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

The central W_c-D_s law depends on fitted scaling parameters (W_c, b), on the BKT ansatz, and on effective descriptions (bosonization and hard-core bosons) imported from prior work. No genuinely new entities are postulated.

free parameters (4)

W_c(U) = 0.061 (U=1, rho=0.5); 0.272 (U=8, rho=0.5); 0.131 (U=8, rho=0.25); 0.2 (U=15, rho=0.5); 0.119 (U=25, rho=0.5)
Critical disorder extracted by minimizing the cost function for BKT collapse of D_s versus L/xi at three system sizes; no uncertainty reported.
b(U) = 1.66 at U=25; about 1.2 at U=1
Parameter of the BKT form xi = a exp(b/sqrt(W-Wc)) tuned with W_c in the cost function; not measured directly for most U.
c (proportionality constant in W_c = c D_s) = 0.084
Slope of the linear fit W_c = 0.084 D_s in Fig. 8(a), reported without uncertainty.
n_MFT (mean-field power-law exponent) = not quoted in text
Power-law collapse exponent used to extract W_c from mean-field D_s curves; MFT has a second-order rather than BKT transition, so this exponent is a fitting device.

assumptions (4)

domain assumption Attractive Hubbard model on the Creutz ladder with uniform on-site disorder is the model of interest.
Defined in Eqs. (1)-(2); no microscopic derivation from a material model is attempted.
domain assumption Bosonization critical Luttinger parameters Kc=3/2 for spinful attractive fermions and K_HCB=3/4 for hard-core bosons on coupled ladders apply to this flat-band system.
Imported from Refs. [44-47] in Sec. III A to interpret the DMRG W_c; not derived in this paper.
domain assumption The BKT scaling form xi = a exp(b/sqrt(W-Wc)) is the correct functional form for the localization transition.
Chosen after comparing BKT and power-law fits at U=1 (Fig. 4), then assumed for all subsequent W_c extractions.
domain assumption The effective hard-core-boson model Eq. (7) is accurate for large U.
Standard strong-coupling expansion; used to simulate the large-U limit directly and to derive the 2 W_HCB/U asymptote.

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Cite this review

Pith. "Pith review of Disorder and the Robustness of Superconductivity on the Flat Band." pith.science (2026). https://pith.science/paper/GJHZ7YAA

@misc{pith2026250607095,
  author       = {Pith},
  title        = {Pith review of: Disorder and the Robustness of Superconductivity on the Flat Band},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GJHZ7YAA}},
  note         = {Machine review of arXiv:2506.07095}
}

abstract

We study the interplay between on-site disorder and fermion pairing on the quasi one-dimensional flat band Creutz lattice. Both disorder and flat bands localize particles, but an attractive interaction results in pair formation and delocalization giving rise to superconductivity. In this work, we examine the attractive Hubbard model on the Creutz lattice to study the competition between these two effects and elucidate the properties of the superconducting phase and the localization quantum phase transition as the disorder strength is increased. Our main result is that flat band superconductivity is robust against disorder: The critical disorder strength, $W_c$, required to localize the fermion pairs and destroy superconductivity, is finite at any interaction strength, $U$, and is proportional to the superconducting weight, $D_s$, of the clean system. Using large scale density matrix renormalization group computations, we show that this transition is of the BKT form. In addition, even at very small interaction strength, the localization is not due to single fermion localization but to pair localization. For completeness, we briefly study this disorder-induced localization with mean field theory and show that $W_c$ can be accurately determined by using an appropriate scaling function.

Figures

Figures reproduced from arXiv: 2506.07095 by the authors.

**Figure 2.** FIG. 2. (Color online) (a) The pair correlation, Eq.( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (Color online) Pair correlation functions at [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: FIG. 4. (Color online) Creutz lattice, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. (Color online) (a) Superfluid weight as a function [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 7.** Figure 7: FIG. 7. (Color online) (a) Superfluid weight as a function [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 9.** Figure 9: FIG. 9. (Color online) The correlation functions in the effec [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10. (Color online) Data collapse for the HCB effec [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: FIG. 11. An example of the quadratic fit used to calculate [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

**Figure 13.** Figure 13: FIG. 13. (Color online) [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]

**Figure 12.** Figure 12: FIG. 12. (Color online) The site-dependent density and pair [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

**Figure 15.** Figure 15: FIG. 15. (Color online) [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]

**Figure 16.** Figure 16: FIG. 16. (Color online) With [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]

**Figure 17.** Figure 17: FIG. 17. (Color online) [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]

**Figure 18.** Figure 18: FIG. 18. (Color online) [PITH_FULL_IMAGE:figures/full_fig_p013_18.png]

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