Intrinsic superconducting instabilities of a solvable model for an incoherent metal

Debanjan Chowdhury; Erez Berg

REVIEW 3 major objections 5 minor 38 references

Intrinsic superconducting instabilities of a solvable model for an incoherent metal

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In a translationally invariant large-N lattice model, repulsive random local interactions alone can make an incoherent metal superconducting—at the interaction scale, not via Fermi-surface Cooper logs.

desk verdict Model A gives a controlled large-N example of non-BCS pairing in an incoherent metal; Model B's headline Tc ~ W* is an admitted extrapolation, not a derivation. read the letter →

arxiv 1908.02757 v1 pith:HZNVYVUJ submitted 2019-08-07 cond-mat.str-el cond-mat.supr-con

Debanjan Chowdhury , Erez Berg This is my paper · ORCID

classification cond-mat.str-elcond-mat.supr-con PACS 71.27.+a74.20.-z74.20.Mn71.10.-w

keywords non-FermiliquidSachdev-Ye-KitaevmodelsuperconductivityKohn-Luttingermechanismlarge-NexpansionstrangemetalpairingwithoutCooperlogarithmtranslationallyinvariantSYKlattice

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 claims that repulsive (sign-random), local interactions of modified Sachdev-Ye-Kitaev form can by themselves make a translationally invariant large-$N$ lattice metal superconducting, without attractive bare interactions, without a Cooper logarithm, and without Fermi-surface nesting. In Model-A, where the random couplings satisfy $J_{ijkl}=J_{ikjl}$, the incoherent non-Fermi liquid at strong coupling develops an on-site, spin-singlet pairing instability at a critical temperature of order the interaction scale, $T_c\sim J$, even though there are no long-lived quasiparticles. In Model-B, where the sign is reversed, the pairing equation has no single-site solution; instead, at strong coupling the paper finds an instability at a temperature of order the renormalized bandwidth, $T_c\sim W^*$, in the crossover region where an incipient heavy Fermi liquid forms. If correct, these are among the first exact large-$N$ examples in which superconductivity is generated intrinsically by the same interactions that destroy quasiparticles, rather than by the conventional BCS mechanism.

What carries the argument

The load-bearing object is the disorder contraction in the pairing channel, $J_{ijkl}J_{ikjl}=\pm J^2$, which enters the Bethe-Salpeter equation as an effective interaction $J^2\Pi(k-q,i\omega-i\Omega)$; the sign is fixed by the permutation symmetry of the random interaction, and the polarization bubble $\Pi$ supplies the only momentum dependence. In Model-A the plus sign makes the single-site (momentum-independent) kernel self-amplify in the incoherent metal, producing a divergent eigenvalue $\sim\log^2(1/T)$ and hence $T_c\sim J$ with no Fermi surface at all. In Model-B the minus sign forbids on-site pairing, so the mechanism shifts to the momentum dependence of $\Pi$, which is the same physics as the weak-coupling Kohn-Luttinger effect; the strong-coupling statement $T_c\sim W^*$ comes from rescaling the Eliashberg equation by $W^*$ and using $Z\nu_0 J\sim 1$ to make the dimensionless pairing eigenvalue order one.

What would settle it

Solve the full large-$N$ Bethe-Salpeter equation for Model-B in the strong-coupling limit at temperatures approaching $W^*$ without the sharp-Fermi-surface approximation; if the largest eigenvalue at $T\sim W^*$ is of order one and crosses unity, the claim $T_c\sim W^*$ is confirmed, whereas if it stays below unity until temperatures are exponentially small in the inverse coupling, the claimed scale collapses and the instability is a conventional Kohn-Luttinger one.

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Extended reading notes

Core claim

The central claim is that sign-correlated random interactions generate an effective pairing attraction at leading order in $1/N$ while the same interactions destroy coherent quasiparticles. For Model-A, the linearized Bethe-Salpeter kernel in the spin-singlet particle-particle channel contains the contraction $J_{ijkl}J_{ikjl}=+J^2$, and in the single-site limit its largest eigenvalue diverges like $\sim\log^2(1/T)$ as $T\to0$; since all eigenvalues vanish as $T\to\infty$, the eigenvalue crosses unity at a finite $T_c$ of order $J$. Long-range order is then established by inter-site Josephson coupling of order $N T_{\rm coh}$. For Model-B, the contraction has the opposite sign, so there is no single-site instability; after rescaling all energies by the renormalized bandwidth $W^*\sim W^2/J$, the strong-coupling Eliashberg equation becomes dimensionless and at low temperature reduces to a conventional Fermi-liquid gap equation. Because the quasiparticle residue is small, $Z\sim W/J$, the effective pairing eigenvalue is of order unity, which the paper takes to imply $T_c\sim W^*$; it explicitly notes that at this scale the Fermi surface is no longer sharp and the scattering rate is of order $W^*$, leaving the extrapolation as an assumption beyond the paper's strict control.

Load-bearing premise

Model-B's strongest claim rests on treating a pairing eigenvalue computed in the low-temperature Fermi-liquid regime, where quasiparticles are ordinary, as if it stayed of order unity when extrapolated up to temperatures of order the renormalized bandwidth $W^*$, where the Fermi surface is not sharp, the scattering rate is of order $W^*$, and the quasiparticle residue is no longer frequency independent; the paper explicitly says this step is beyond its scope.

Editorial extensions

If this is right

Superconductivity can appear in a parent state with no quasiparticles: Model-A orders at $T_c\sim J$ from a locally critical metal whose single-electron spectral function has no coherent peak.
In Model-B, the superconducting instability occurs at the same scale as the renormalized bandwidth, so it preempts the formation of the Fermi liquid rather than developing inside it.
Neither example relies on nesting or on time-reversed Cooper pairs at $\pm k$; the pairing scale is set by the local interaction, so the mechanism is genuinely non-BCS.
Pair-hopping repulsion $U$ suppresses Model-A's on-site pairing for $U\gtrsim J$; increasing $U$ moves the system toward the Model-B behavior.
In strong coupling, density-wave order with $T_c^{\rm DW}\sim W^*$ may coexist or compete with superconductivity, with the precise ratio set by microscopic details.

Reading between the lines

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

If Model-B's extrapolation to $T\sim W^*$ fails, the strong-coupling example would collapse to a weak-coupling Kohn-Luttinger instability with exponentially small $T_c$; the sign structure guarantees that no single-site pairing can rescue it.
A controlled numerical solution of the full large-$N$ Bethe-Salpeter eigenvalue for Model-B at $T\sim W^*$, keeping the full frequency-dependent self-energy and incoherent spectral weight, would settle whether $T_c\sim W^*$ or only an exponentially small scale exists.
The same sign-correlation construction should extend to other solvable building blocks; the appendix already shows $T_c\sim J$ for SYK$_q$ with $q>4$, so the mechanism is presumably insensitive to the specific operator content.

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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

The useful part of this paper is Model A. In a translationally invariant large-N SYK lattice model with spin, they show that the incoherent non-Fermi liquid is intrinsically unstable to on-site singlet pairing at Tc ~ J. The Bethe-Salpeter kernel is dimensionless, the trial function gives a log^2 divergence, and the numerical eigenvalue plot supports the crossing. That is a controlled large-N result and a genuine non-BCS route to pairing: no Fermi surface nesting, no Cooper log. This alone is worth the paper.

Model B is where the abstract overreaches. The weak-coupling limit is standard Kohn-Luttinger, fine. The strong-coupling claim, Tc comparable to the renormalized bandwidth W*, is an extrapolation, not a derivation. The computation is done in the low-T Fermi liquid, where the pairing eigenvalue is O(1) because Zν0J ~ 1, and then the crossing is placed at T ~ W*. At that scale the authors admit their own assumptions break down: scattering rate ~ W*, no sharp Fermi surface, frequency-dependent residue, incoherent spectral weight. The stress-test note lands: the explicit polarized example only gives |λ1| ≤ 1/4 in the p-wave channel, so a BCS estimate puts Tc/W* ≥ e^{-4} ~ 0.02, and incoherent corrections could push it much lower. A finite Tc is guaranteed by the Cooper log; the W* scale is not. The paper says this almost explicitly in Section IV, then in the abstract and discussion states it as a result. That mismatch should be fixed.

Otherwise the paper is honest. It cites its own earlier work for the normal-state saddle point; those Green's functions are published, and the analytic specification is complete enough to reproduce. No fitted parameters, no invented entities. The note about concurrent independent work is appropriate.

Who is this for? People working on SYK models, strange metals, and non-BCS pairing will get real value from Model A and from the general framework. It deserves a serious referee. My recommendation: send it out, but require the authors to either compute or clearly label the Model B strong-coupling claim as a conjecture, and soften the abstract accordingly.

Referee Report

3 major / 5 minor

Summary. The paper studies a translationally invariant lattice generalization of the SYK model with N orbitals per site and a spin-1/2 label, focusing on pairing instabilities of the strongly coupled incoherent metal. In the large-N limit the authors obtain self-consistent saddle-point equations for the Green's function and linearized Bethe-Salpeter equations in the spin-singlet (and, in a polarized example, spin-triplet) pairing channel. They distinguish two models by the permutation symmetry of the random interaction tensor: Model-A, with Jijkl=Jikjl, is claimed to have an intrinsic superconducting instability of the incoherent non-Fermi liquid at Tc~J, driven not by a Cooper logarithm but by the local SYK quantum-critical dynamics; Model-B, with Jijkl=-Jikjl, is claimed to show a Kohn-Luttinger-type instability at weak coupling with exponentially small Tc, and at strong coupling a superconducting transition at Tc~W*~Tcoh, the renormalized bandwidth. A q>4 generalization is treated in an appendix.

Significance. If the Model-A result is correct, it is a rare exact large-N example in which pairing arises from an incoherent, locally critical metal without any Fermi-surface nesting or quasiparticle Cooper instability; the dimensionless Bethe-Salpeter kernel, the trial-function proof, and the numerical eigenvalue plot make this part of the paper credible and essentially parameter-free. The Model-B strong-coupling claim, if correct, would show that an incipient heavy Fermi liquid can become superconducting at a scale comparable to its renormalized bandwidth, which would be of considerable conceptual interest. However, as discussed below, that second claim is currently an extrapolation rather than a derivation, and the paper itself states that the assumptions break down at the relevant scale. The manuscript is clearly written and openly flags its main limitation, which is a strength, but the abstract and Section V state the strong-coupling Model-B result more definitively than the body of the paper supports.

major comments (3)

[Section IV.A, Eq. (14)] The claim that Model-B has Tc~W* at strong coupling is not established by the arguments given. The proof of a divergent pairing eigenvalue uses the low-temperature Fermi-liquid form Gi~1/(iω−εk), which is valid only for ω,T<<W*. It shows that a pairing instability exists at some finite Tc, as already stated in footnote 5, but it does not locate the crossing scale. Extrapolating the logarithmic growth all the way to T~W* is precisely the regime where Section IV states that 'the scattering rate is large... it is not sufficient to focus only on the low-energy states near the Fermi surface' and that the calculation is 'beyond the scope of this work.' The high-temperature estimate δχpair~Tcoh/T becomes O(1) at T~Tcoh, but that does not by itself imply that the Bethe-Salpeter eigenvalue reaches unity at that scale. The abstract's claim that Tc is 'comparable to its renormalized bandwidth' therefore remains an extrapolation.
[Section IV.B, Eqs. (22)-(25)] The explicit polarized example uses a weak-coupling expression for the pairing eigenvalue, |λ1,↑|=Z^2ν0^2J^2 α(1−α), and then promotes it to strong coupling by setting Zν0J~1. This promotion is not justified within the calculation: the static-polarization approximation and the frequency-independent quasiparticle residue Z are controlled only when the dimensionless coupling is small. At strong coupling, Zν0J~1 means the ladder kernel is not small, the frequency dependence of Π and the incoherent part of the spectral function can substantially modify the eigenvalue, and even a conservative BCS-style estimate with |λ1|≤1/4 gives Tc/W*~e^(−1/|λ1|)≤e^(−4), which is not O(1). To support the strong-coupling claim the authors would need to solve the scaled Bethe-Salpeter equation (14) at T~W* using the full self-consistent Green's function, or explicitly revise the abstract and Section V to present Tc~W* as a conjecture.
[Section V] The discussion repeats the strong-coupling claim as a result: 'we therefore find that Tc is of the order of W*.' This overstates what Section IV.A has shown, which is that W* is the only scale in the dimensionless Eliashberg equation and that a Cooper-log instability exists. Dimensional analysis alone does not determine the prefactor of Tc, and the admitted breakdown of the Fermi-liquid assumptions at T~W* leaves the magnitude of Tc unproven. The wording of the abstract should be aligned with the actual evidence.

minor comments (5)

[Fig. 3] The vertical axis is labeled as the eigenvalue of M^(1/2), while the text and Eq. (10) refer to the largest eigenvalue of M itself; the notation should be made consistent.
[Eq. (10)] The displayed trial-function ratio contains an unbalanced parenthesis in the numerator; this is a typographical error that should be fixed.
[Section IV.A, Eqs. (12)-(13)] The rescaled Green's function in Eq. (13) omits the bare iω term, but Eq. (4) includes a quasiparticle pole with residue Z; the relation between these two forms in the strong-coupling scaling limit should be stated explicitly to avoid confusion.
[Section III] The statement that 'all the eigenvalues of M go to zero in the opposite limit T→∞' is plausible but not shown; a one-sentence justification would make the intermediate-value argument rigorous.
[Footnote 5] The important caveat that a Cooper logarithm guarantees a pairing instability in Model-B appears only in a footnote; it should be integrated into the main text because it materially qualifies the distinction between models A and B.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Model-A and Model-B derivations are self-contained large-N analyses whose only cited inputs are prior published results that are independent of the target pairing-instability conclusions.

full rationale

The paper's central derivations do not reduce to their own inputs. For Model-A, the Tc~J prediction follows from rescaling the local Bethe-Salpeter equation (Eq. 8) into a manifestly dimensionless form, with the existence of a crossing established by an explicit trial-function argument whose largest eigenvalue diverges as log^2(1/T) (Eq. 10). This is a direct mathematical argument, not a fitted parameter or a definitional restatement. For Model-B, the predicted scale Tc~W* follows from the same non-dimensionalization of the Eliashberg equation (Eq. 14), and the existence of a pairing instability is imported from the independent weak-coupling Hubbard-model literature (Refs. 29, 30), which is a borrowed external result rather than a self-citation. The Green's functions quoted in Eq. (4) are taken from the authors' earlier Ref. [25], but that prior work is a published, independently checkable saddle-point solution whose stated assumptions do not include the present paper's pairing instability; citing it is legitimate background support, not circularity. The paper itself flags the principal limitation in Sec. IV: 'at these scales that are comparable to the renormalized bandwidth, many of our underlying assumptions are not strictly applicable... This is beyond the scope of this work.' That caveat concerns the validity of extrapolating the low-temperature Fermi-liquid pairing eigenvalue to T~W*, which is a correctness risk, not a definitional equivalence, a fitted-input-renamed-as-prediction, or a self-citation chain. No step was found where an output equals an input by construction, and no load-bearing conclusion depends solely on a self-citation. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the large-N saddle-point expansion, the disorder-averaged treatment of translationally invariant random couplings, and standard SYK scaling forms. No parameters are fitted to data, and no new physical entities are introduced. The model's coupling constants J, U, t, W, h, N are inputs; the ratio alpha in the Zeeman example is a dimensionless model parameter, not a fitted constant.

assumptions (6)

domain assumption Large-N self-consistent 'watermelon' diagrams (Eqs. 2-3) give the exact electron Green's function at leading order in 1/N.
This is the standard SYK large-N technique. The paper assumes no other diagrams contribute at leading order and does not compute subleading corrections.
domain assumption A single realization of the translationally invariant random couplings behaves like the disorder-averaged large-N solution.
The J_ijkl are identical at every site, so this is not conventional quenched disorder. The paper relies on Ref. [25] for the validity of this averaging.
domain assumption SYK scaling forms G(i omega) ~ i sgn(omega)/sqrt(|omega|) and Pi(Omega) ~ -log(1/max(|Omega|,T)) hold in the incoherent regime for omega,T << J.
These forms enter Eq. (9) and are the basis for the proof that the Model-A pairing eigenvalue diverges as T -> 0. They are imported from SYK literature.
standard math Eq. (14) in the low-frequency FL limit is identical to the weak-coupling Hubbard model gap equation, and the known d-wave solution of Refs. [29,30] applies.
The authors identify the equivalence but do not rederive the d-wave solution for their frequency-dependent kernel; they import the result from prior literature.
domain assumption The partially polarized Fermi sea can be treated with parabolic, rotationally invariant dispersions with spin-dependent Fermi momenta kF,s.
Used for the explicit angular integrals in Eqs. (20-25); assumes no band structure effects alter the Kohn-Luttinger result.
domain assumption The linearized Bethe-Salpeter equation at zero center-of-mass momentum determines the superconducting transition temperature, and long-range order in Model-A follows from Josephson coupling between sites.
The computation identifies the on-site linear instability; the phase stiffness and nonlinear effects are treated separately and heuristically.

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Pith. "Pith review of Intrinsic superconducting instabilities of a solvable model for an incoherent metal." pith.science (2026). https://pith.science/paper/HZNVYVUJ

@misc{pith2026190802757,
  author       = {Pith},
  title        = {Pith review of: Intrinsic superconducting instabilities of a solvable model for an incoherent metal},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZNVYVUJ}},
  note         = {Machine review of arXiv:1908.02757}
}

abstract

We construct a family of translationally invariant lattice models with a large number ($N$) of orbitals at every site coupled together via single electron tunneling. By tuning the relative strength of the electronic bandwidth and on-site interactions, that have a modified Sachdev-Ye-Kitaev (SYK) form, we demonstrate a number of unusual features at strong coupling and in the large$-N$ limit. We find examples of (i) an intrinsic non-BCS superconducting instability arising out of an incoherent non-Fermi liquid metal, and, (ii) an instability of an incipient heavy Fermi liquid metal to superconductivity with transition temperatures comparable to its renormalized bandwidth. At strong-coupling, these solvable models display pairing instabilities that are not driven by any special "nesting" properties associated with an underlying Fermi surface.

Figures

Figures reproduced from arXiv: 1908.02757 by the authors.

**Figure 2.** FIG. 2. The Bethe-Salpeter equation for the pairing ver [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. The maximum eigenvalue of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Y. Wang, “A Solvable Random Model With Quantum- critical Points for Non-Fermi-liquid Pairing,” arXiv e-prints , arXiv:1904.07240 (2019), arXiv:1904.07240 [cond-mat.str-el]

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Cooper pairing of incoher- ent electrons: an electron-phonon version of the Sachdev- Ye-Kitaev model,

I. Esterlis and J. Schmalian, “Cooper pairing of incoher- ent electrons: an electron-phonon version of the Sachdev- Ye-Kitaev model,” arXiv e-prints , arXiv:1906.04747 (2019), arXiv:1906.04747 [cond-mat.str-el]

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A generalization of sachdev-ye-kitaev,

D. J. Gross and V. Rosenhaus, “A generalization of sachdev-ye-kitaev,” Journal of High Energy Physics 2017, 93 (2017). Appendix A: Generalization to translationally invariant SYK q models In this appendix, we consider a generalization of the model originally introduced in Ref....

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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pair hopping

shows similar physics, as discussed in Appendix A. IV. MODEL −B: GENERALIZED KOHN-LUTTINGER MECHANISM It is clear from the discussion in the previous section that in model-B, as a result of the sign of JijklJikjl = −J2, the corresponding Bethe-Salpeter equation in the incoherent regime (i.e. Eqn. 6b) does not have a non- trivial solution at the level of a...

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