{"id":"9290c1ce-84a3-45c6-8019-c36993b6ab92","arxiv_id":"1908.02757","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A solvable lattice model shows superconductivity emerging from an incoherent, non-Fermi-liquid metal, with Tc set by the interaction scale or the renormalized bandwidth, not by Cooper nesting.","lead":"This paper analyzes large-N lattice models with SYK-type interactions and finds superconductivity can arise directly from an incoherent non-Fermi liquid, with transition temperatures set by the interaction strength or the renormalized bandwidth rather than by Fermi surface nesting. It supplies concrete solvable examples for mechanisms often invoked in high-temperature superconductivity.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Model-B's Tc ~ W* is an extrapolation: the strong-coupling pairing eigenvalue is computed only in the low-T FL and high-T IM limits, not at T ~ W*, where Section IV admits the assumptions break down, so the abstract's second example remains unproven.","rationale":"I read the paper in good faith: Model-A is a genuine exact large-N example of pairing from incoherent local excitations, with a dimensionless Bethe-Salpeter kernel and a diverging eigenvalue; no issue there. The central claim (ii) in the abstract depends on Model-B at strong coupling. The reader's verdict identified the same soft spot: the pairing eigenvalue is computed in the low-T FL regime and the high-T IM regime, and both are extrapolated to T ~ W*, where the authors explicitly disclaim the validity of their assumptions. This is not an internal inconsistency; the controlled parts of the paper are correct, but it means the headline claim overreaches. A numerical solution of the exact rescaled equations would settle it. Since the conditional verdict already reflects this uncertainty, no change is needed.","tokens_in":14749,"tokens_out":28010,"duration_ms":311983,"concrete_test":"Numerically solve the strong-coupling large-N saddle-point and Bethe-Salpeter equations (Eqs. 12 and 14) on a finite lattice, e.g., a 32x32 square lattice, without assuming the FL form of G: self-consistently determine G(k,iω) and Π(q,iΩ) at each T, then compute the largest pairing eigenvalue in the d-wave (or, for the polarized example, p-wave) channel as a function of T/W*. If λ_max(T/W*) crosses 1 at T_c/W* in the range 0.1-1, the Model-B claim is established; if the crossing occurs only at T_c/W* < 10^-2, the strong-coupling claim should be downgraded to an exponentially small Kohn-Luttinger scale.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Model-A is a controlled large-N result: the local Bethe-Salpeter kernel is dimensionless, its largest eigenvalue diverges as T -> 0, so a crossing at Tc ~ J follows. The load-bearing weakness is Model-B's strong-coupling claim Tc ~ W*. The proof of an instability uses the low-T Fermi-liquid limit of Eq. (14), where G ~ 1/(iω - ε), to establish a logarithmic divergence of the pairing eigenvalue; the scale of the crossing is then asserted to be W* because the dimensionless coupling is O(1) (Zν0J ~ 1). But the authors never solve Eq. (14) at T ~ W*. Section IV states: 'at these scales that are comparable to the renormalized bandwidth, many of our underlying assumptions are not strictly applicable... the scattering rate is large... it is not sufficient to focus only on the low-energy states near the Fermi surface... This is beyond the scope of this work.' The high-T incoherent-metal calculation gives only δχpair ~ Tcoh/T, which becomes O(1) at T ~ Tcoh but does not by itself locate a pairing eigenvalue crossing. The explicit polarized example yields |λ1| = Z^2ν0^2J^2 α(1-α); with Zν0J ~ 1 this is at most 1/4, so a BCS estimate gives Tc/W* ≥ e^{-4} ≈ 0.018, and incoherent spectral-weight corrections could push it much lower. A finite Tc is guaranteed by the Cooper log (footnote 5); what is unproven is its magnitude. Thus the abstract's 'comparable to its renormalized bandwidth' is an extrapolation, not a derivation, and if the true eigenvalue at T ~ W* is ≪ 1, Model-B reduces to a Kohn-Luttinger instability with exponentially small Tc.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15058,"tokens_out":3947,"duration_ms":49806,"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":[{"comment":"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":"Section IV.A, Eq. (14)"},{"comment":"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":"Section IV.B, Eqs. (22)-(25)"},{"comment":"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.","section":"Section V"}],"minor_comments":[{"comment":"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.","section":"Fig. 3"},{"comment":"The displayed trial-function ratio contains an unbalanced parenthesis in the numerator; this is a typographical error that should be fixed.","section":"Eq. (10)"},{"comment":"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":"Section IV.A, Eqs. (12)-(13)"},{"comment":"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.","section":"Section III"},{"comment":"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.","section":"Footnote 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about the extrapolation in Model-B, but the abstract and Section V present the strong-coupling Tc~W* as an established result. I would ask for either a numerical solution of the dimensionless Bethe-Salpeter equation at T~W* using the full self-energy, or a clear rephrasing of the second example as a suggestive extrapolation with a guaranteed but not quantitatively determined Tc. The Model-A part is convincing and should be credited in the final version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nModel 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.\n\nOtherwise 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.\n\nWho 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.","headline":"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.","tokens_in":15650,"tokens_out":1988,"would_cite":true,"duration_ms":21673,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["71.27.+a","74.20.-z","74.20.Mn","71.10.-w"],"model":"deepseek-v4-flash","headline":"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.","keywords":["non-Fermi liquid","Sachdev-Ye-Kitaev model","superconductivity","Kohn-Luttinger mechanism","large-N expansion","strange metal","pairing without Cooper logarithm","translationally invariant SYK lattice model"],"falsifier":"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.","tokens_in":14495,"feed_emoji":"⚡","tokens_out":13414,"duration_ms":121818,"temperature":0.7,"pith_summary":"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.","feed_headline":"Superconductivity can emerge from an incoherent metal without BCS logs","feed_subtitle":"A large-N lattice model pairs incoherent electrons at a scale set by the interaction itself, not by nesting.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original single-site Sachdev-Ye-Kitaev interaction structure that the lattice model uses as its local building block.","marker":"[12]"},{"why":"Supplies the translationally invariant lattice construction and the large-$N$ saddle-point Green's function whose low/high-temperature crossover is used throughout.","marker":"[25]"},{"why":"Defines the Kohn-Luttinger weak-coupling mechanism that Model-B generalizes in strong coupling.","marker":"[27]"},{"why":"Supplies the two-dimensional p-wave Kohn-Luttinger result for the parabolic-band example.","marker":"[28]"},{"why":"Provides the d-wave pairing solution near a spin-density-wave instability that the low-temperature Model-B equation reduces to.","marker":"[29]"},{"why":"Gives the asymptotically exact weak-coupling d-wave eigenvalue on the square lattice used to argue Model-B has a non-trivial solution.","marker":"[30]"},{"why":"Supplies the eigenvalue calculation for a partially spin-polarized Fermi sea whose strong-coupling version yields $T_c\\sim W^*$.","marker":"[31]"}],"fun_headline_variants":["Solvable model shows superconductivity from incoherent metal","Non-BCS superconductivity arises in solvable incoherent model","Large-N model reveals intrinsic pairing without nesting","Superconducting instability from incoherence, no BCS needed","Incoherent metal pairs electrons via random interactions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Solvable model shows superconductivity from incoherent metal","Non-BCS superconductivity arises in solvable incoherent model","Large-N model reveals intrinsic pairing without nesting","Superconducting instability from incoherence, no BCS needed","Incoherent metal pairs electrons via random interactions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000221,"raw_usage":{"total_tokens":1465,"prompt_tokens":972,"completion_tokens":493,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":416}},"tokens_in":588,"tokens_out":493,"duration_ms":4837,"temperature":1.0,"reasoning_tokens":416,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:35:41.246656+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Superconductivity in bad metals,","cited_arxiv_id":null,"evidence_quote":"Supplies the original single-site Sachdev-Ye-Kitaev interaction structure that the lattice model uses as its local building block."},{"cited_title":"Quantum criticality and duality in the Sachdev-Ye-Kitaev/ads2 chain,","cited_arxiv_id":null,"evidence_quote":"Supplies the translationally invariant lattice construction and the large-$N$ saddle-point Green's function whose low/high-temperature crossover is used throughout."},{"cited_title":"Coherent superconductivity with a large gap ratio from incoherent metals,","cited_arxiv_id":null,"evidence_quote":"Defines the Kohn-Luttinger weak-coupling mechanism that Model-B generalizes in strong coupling."},{"cited_title":"New mechanism for su- perconductivity,","cited_arxiv_id":null,"evidence_quote":"Supplies the two-dimensional p-wave Kohn-Luttinger result for the parabolic-band example."},{"cited_title":"Kohn-luttinger effect and the instabil- ity of a two-dimensional repulsive fermi liquid at t=0,","cited_arxiv_id":null,"evidence_quote":"Provides the d-wave pairing solution near a spin-density-wave instability that the low-temperature Model-B equation reduces to."},{"cited_title":"d-wave pair- ing near a spin-density-wave instability,","cited_arxiv_id":null,"evidence_quote":"Gives the asymptotically exact weak-coupling d-wave eigenvalue on the square lattice used to argue Model-B has a non-trivial solution."},{"cited_title":"Super- conductivity in the repulsive hubbard model: An asymp- totically exact weak-coupling solution,","cited_arxiv_id":null,"evidence_quote":"Supplies the eigenvalue calculation for a partially spin-polarized Fermi sea whose strong-coupling version yields $T_c\\sim W^*$."}],"review_version":1}