{"id":"2d7f3da8-fba1-40f3-84d9-1cc5a00b7252","arxiv_id":"1908.11385","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Odd-frequency pairing in Dirac semimetals is symmetry-allowed via chirality and can be supported by a repulsive frequency-dependent interaction, with cusps in the density of states as a proposed signature.","lead":"This paper works out how odd-frequency, Berezinskii superconductivity could arise in Dirac semimetals, using chirality as the extra degree of freedom that makes the pairing symmetry allowed. It shows such pairing can in principle be produced by a repulsive, frequency-dependent interaction, and gives a density-of-states shape to look for in experiments.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The pairing potential is solved from an approximate differential gap equation and is never checked against the exact integral equation; the central self-consistency claim is therefore unverified.","rationale":"In good faith, the paper's symmetry classification and the clean separation of even- and odd-frequency gap equations are valuable, and the authors are candid about unresolved issues. The concern raised here is internal consistency of the inverse-problem step, which is more fundamental than the missing microscopic mechanism: if V does not solve the exact gap equation, no microscopic realization can rescue the central claim. The exact check is missing, and the approximation in SM Eq. (S35) has no error estimate. The paper itself contains the limitation statements: the Conclusions admit that 'the physical nature of repulsive frequency-dependent potential should be also clarified', and SM Sec. V admits that the DOS cusps are tied to the finite momentum cutoff, with a single peak at omega -> 0 when Lambda_k -> infinity. These admissions support keeping the verdict conditional. The reader's weakest assumption, absence of a physical interaction, is the next step after this exact self-consistency check, so agreement is partial rather than full.","tokens_in":20399,"tokens_out":7823,"duration_ms":82924,"concrete_test":"Numerically reconstruct the potential V(omega) from Eq. (11) with Delta_odd = alpha Lambda_k / omega and the exact f_odd of SM Eq. (S34). Then evaluate the right-hand side of the exact gap equation (SM Eq. (S33)) on a frequency grid, e.g. omega/Lambda_k in [0.01, 1], and compute the relative error |Delta_exact(omega) - alpha Lambda_k / omega| / |alpha Lambda_k / omega|. If the error exceeds a few percent anywhere, the constructed potential is not a self-consistent solution and the inverse-problem demonstration is invalid. A stronger version is to solve the full nonlinear integral equation (S33) directly for V in the repulsive, frequency-dependent class and check whether a solution exists and, if so, whether its gap resembles alpha Lambda_k / omega.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central demonstration is an inverse construction: for the ansatz Delta_odd(omega) = alpha Lambda_k / omega, the pairing potential is solved from Eq. (11), which is the differential form of Eq. (10) (SM Eq. (S36)). But Eq. (10) is obtained from the exact gap equation (SM Eq. (S33)) by the uncontrolled replacement V(omega-omega') - V(omega+omega') approximately -2[theta(omega-omega') omega' V'(omega) + theta(omega'-omega) omega V'(omega')] (SM Eq. (S35)). The computed V is never substituted back into the exact integral equation (S33), and no residual is reported. Since the constructed V diverges at omega -> 0 and decays only as 1/omega^2, the first-order Taylor-type replacement is not obviously accurate. If the exact residual is not small, the conclusion that a repulsive frequency-dependent potential supports Delta_odd is an artifact of the approximation, and the paper's central self-consistency check fails. This concern is distinct from, and prior to, the acknowledged question of which microscopic interaction produces V.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper formulates a mean-field effective-action description of even- and odd-frequency superconducting pairing in three-dimensional (and two-dimensional) Dirac semimetals. The authors derive an integral gap equation, approximate it by differential gap equations that involve only the frequency derivative of the pairing potential for the odd-frequency case, and then solve these equations in inverse form: for a chosen gap ansatz, they reconstruct the pairing potential. For the odd-frequency ansatz Δ_odd(ω)=αΛ_k/ω, the reconstructed potential is found to be repulsive and to decay approximately as 1/ω² at large frequency. The paper further shows that both even- and odd-frequency pairings require a critical coupling at charge neutrality, and it computes the density of states, identifying cusp-like features as a proposed experimental signature of odd-frequency pairing. The central conclusion is that repulsive, strongly frequency-dependent interactions can generate Berezinskii pairing in Dirac semimetals.","tokens_in":20663,"tokens_out":3940,"duration_ms":40803,"significance":"If the central claim is correct, the paper offers a conceptually new route to odd-frequency superconductivity: repulsive interactions, rather than the attractive BCS interaction, could stabilize the Berezinskii state in Dirac and Weyl semimetals. The symmetry classification using the chirality degree of freedom (SP*χT* = −1) is a clean and useful extension of the standard SP*OT* rule, and the effective-action framework for frequency-dependent pairing is a valuable contribution. The authors are also commendably explicit that their construction is an inverse problem: the pairing potential is derived from a chosen gap ansatz, and the physical origin of the repulsive frequency-dependent interaction is left open. However, the load-bearing results—the repulsive character of the potential, the critical-coupling statement, and the DOS signature—are all obtained from the same ansatz through an approximate differential gap equation, and their robustness is not established. The significance is therefore conditional on additional consistency checks.","major_comments":[{"comment":"The central reduction from the exact integral gap equation, SM Eq. (S33), to the differential form used throughout, SM Eq. (S36) and main-text Eq. (10), relies on the approximation V(ω−ω′)−V(ω+ω′) ≈ −2[θ(ω−ω′)ω′V′(ω)+θ(ω′−ω)ωV′(ω′)]. This is not an exact identity for a general potential, and no validity condition or error estimate is given. Because the reconstructed potential V(ω) for the ansatz Δ_odd=αΛ_k/ω diverges at small ω and decays only as 1/ω², the derivative approximation is not obviously accurate in the region that matters. The authors never substitute the derived V back into the exact integral equation (S33) and never report a residual. Without such a check, the conclusion that a repulsive frequency-dependent potential supports the assumed odd-frequency gap could be an artifact of the approximation. I ask the authors either to justify Eq. (S35) quantitatively for the reconstructed potential or to verify the solution directly against Eq. (S33).","section":"SM Eq. (S35) and main-text Eq. (10)"},{"comment":"The claim that odd-frequency pairing 'can naturally appear' in Dirac semimetals is built on an inverse construction: the potential V(ω) is solved from the chosen ansatz Δ_odd=αΛ_k/ω, and the paper explicitly states in the Conclusions that 'the physical nature of repulsive frequency-dependent potential should be also clarified.' This is an acknowledged limitation, but it is load-bearing for the main message. The paper demonstrates that a particular gap ansatz corresponds, within the approximate equation, to a repulsive potential with a certain derivative profile; it does not show that any known or plausible microscopic interaction produces such a potential. The authors should either identify a microscopic mechanism that yields the required V(ω) or restrict the claims accordingly, for example by presenting the result as a conditional possibility rather than as a natural realization.","section":"Main-text Eq. (18) and Conclusions, fourth paragraph"},{"comment":"The proposed experimental signature—cusp-like features in the density of states—is computed from the same ansatz Δ_odd=αΛ_k/ω, and the SM itself notes that the appearance of the cusps is directly related to the finite momentum cutoff Λ_k and that for Λ_k→∞ the odd-frequency DOS shows only a single peak at ω→0. This makes the cusp signature a cutoff-dependent feature of the ansatz rather than a robust prediction of the pairing state. The authors also show that other odd-frequency gap ansätze give different spectral details. To support the experimental claim, the DOS should be computed either from a self-consistent gap or at least with an explicit demonstration that the cusp structure survives for physical, finite screening scales and is not an artifact of the ansatz.","section":"SM Sec. V and main-text Fig. 3"}],"minor_comments":[{"comment":"The title contains an obvious typo: 'sem imetals' should be 'semimetals'.","section":"Title"},{"comment":"The sentence 'the OF gap is determined by the derivative of the with respect to frequency' is missing a noun; it should read 'the derivative of the potential with respect to frequency.'","section":"Introduction, third paragraph"},{"comment":"The notation V′_odd(ω) in the boundary condition for the odd-frequency potential is confusing because the potential itself is not odd in frequency; it should be simply V′(ω).","section":"SM Eq. (S43)"},{"comment":"The even-frequency ansatz is written as Δ_even(ω)=α, but the following discussion would benefit from explicitly stating that α is a constant amplitude; this is clear from context but is not stated before the equation.","section":"Main-text Eq. (17)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central result is an inverse construction, and the authors are honest about its limitations, but the uncontrolled approximation leading to the differential gap equation is the main technical risk. If the authors can verify the reconstructed potential against the exact integral equation and either identify a microscopic origin for the repulsive frequency-dependent interaction or explicitly moderate the 'can naturally appear' claim, the paper could be acceptable. The DOS signature also needs to be shown to be robust beyond the specific cutoff and ansatz. This is not a rejection of the idea; it is a request to establish the self-consistency of the central mechanism."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth reading but not worth taking as proof. The authors derive a clean framework for odd- and even-frequency pairing in Dirac semimetals, show that an odd-frequency gap depends only on the frequency-derivative of the pairing potential, and use an inverse construction to argue that a repulsive 1/omega^2 potential can support a Berezinskii gap. That is a genuine contribution. They also give a nice chirality-based extension of the SP*OT* = -1 rule and show that at charge neutrality both even- and odd-frequency pairings require supercritical coupling. The writing is honest: they explicitly say the microscopic origin of the repulsive potential remains open.\n\nThe soft spot is the one the stress-test flags. The whole inverse construction goes through Eq. (S35), a piecewise first-order replacement for V(omega-omega') - V(omega+omega'). For the V they end up with (diverging at omega -> 0, decaying as 1/omega^2), that replacement has no obvious justification, and they never substitute the computed V back into the exact integral gap equation to check the residual. Without that check, the central existence claim is unverified, not just microphysically incomplete. This is the difference between a consistency check and a demonstration, and the paper sometimes blurs it.\n\nSecond, the DOS double-cusp signature is computed from the same ansatz and is tied to the finite momentum cutoff; the SM admits that for Lambda_k -> infinity the DOS shows a single peak at omega -> 0. Calling the double cusps 'universal' overstates it. Third, the inverse construction is by design circular: fixing Delta and solving for V tells you what potential would produce that gap, not that such a potential exists. The authors are upfront about the inverse nature, but the abstract and conclusions lean on 'can naturally appear' language that goes beyond what is shown. The citation pattern is otherwise unremarkable, with relevant self-citations rather than padding.\n\nAt the end of the day: the symmetry framework and the differential-gap-equation machinery are useful for people working on odd-frequency pairing and Dirac materials. The central physical claim needs an exact numerical self-consistency check. This paper deserves a serious referee, but the referee should insist on that check before publication. I would not cite the repulsive-potential conclusion as established.","headline":"A useful inverse-gap framework for odd-frequency pairing in Dirac semimetals, undermined by an unchecked Taylor approximation; referee it, but don't take the existence claim as established.","tokens_in":21151,"tokens_out":3589,"would_cite":false,"duration_ms":35771,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Repulsive interactions can drive odd-frequency superconductivity in Dirac semimetals.","keywords":["odd-frequency superconductivity","Berezinskii pairing","Dirac semimetals","gap equation","effective action","density of states","repulsive interaction","chirality"],"falsifier":"A first-principles calculation of the effective frequency-dependent electron-electron interaction in a candidate Dirac semimetal would settle the mechanism: if the resulting $V(\\omega)$ is not repulsive with the derivative profile required by the differential gap equation, then odd-frequency pairing cannot naturally appear by this route.","tokens_in":20200,"feed_emoji":"⚛️","tokens_out":6678,"duration_ms":60904,"temperature":0.7,"pith_summary":"This paper argues that odd-frequency (Berezinskii) superconductivity — pairing in which the Cooper-pair amplitude changes sign when the two electrons' times are exchanged — can arise naturally in Dirac semimetals, materials whose low-energy electrons behave like massless relativistic fermions with a chirality label. The central assertion is that a repulsive, strongly frequency-dependent electron-electron interaction can generate an odd-frequency gap, even though repulsive interactions cannot generate the conventional BCS gap. The reason is structural: the odd-frequency gap equation is sensitive only to the derivative of the interaction potential with respect to frequency, so a potential that is repulsive overall can still act attractively in the odd-frequency channel. The paper also claims that at charge neutrality both even- and odd-frequency pairing require a critical coupling strength, and that cusp-like features in the density of states provide a measurable fingerprint of the odd-frequency state.","feed_headline":"Repulsive forces can still make Dirac semimetals superconduct","feed_subtitle":"In these materials, chirality opens a pairing channel that needs no attractive interaction.","key_machinery":"The load-bearing object is the frequency-resolved mean-field gap equation, converted from an integral equation into a differential equation with boundary conditions. For the odd-frequency gap $\\Delta_{\\rm odd}(\\omega)$, the equation takes the form $\\omega V''(\\omega)-V'(\\omega) = [\\omega \\Delta'_{\\rm odd}(\\omega)-\\Delta_{\\rm odd}(\\omega)]/(-2\\int_0^\\omega d\\omega' \\omega' f_{\\rm odd}(\\omega'))$, so only the derivative $V'(\\omega)$ of the pairing potential enters; together with the boundary condition $V(\\omega)\\to 0$ at large $\\omega$, this determines the potential from a chosen gap ansatz. For the ansatz $\\Delta_{\\rm odd}=\\alpha\\Lambda_k/\\omega$, the recovered potential is repulsive and roughly $1/\\omega^2$, which is the concrete mechanism claimed to favor Berezinskii pairing. The chirality label $\\chi$ plays the role that orbital parity plays in multiorbital systems, making the odd-frequency spin-singlet s-wave gap symmetry-allowed.","core_discovery":"In a Dirac semimetal at zero chemical potential, with a single Dirac node and spin-singlet s-wave pairing, the paper solves the mean-field gap equation for two gap symmetries: the conventional even-frequency gap $\\Delta_{\\rm even}(\\omega)=\\alpha$ and the odd-frequency Berezinskii gap $\\Delta_{\\rm odd}(\\omega)=\\alpha\\Lambda_k/\\omega$. For the odd-frequency case the integral gap equation can be rewritten so that only $V'(\\omega)$, the frequency derivative of the pairing potential, appears; solving the inverse problem for the potential yields a repulsive potential, $V(\\omega)>0$, falling approximately as $1/\\omega^2$ at large frequency. The paper concludes that odd-frequency pairing is not only possible but natural in Dirac semimetals because the chirality degree of freedom makes the inter-chirality, spin-singlet s-wave channel odd under time exchange, satisfying a generalized $SP^*\\chi T^*=-1$ rule. Both gaps need coupling above a critical value because the density of states vanishes at the Dirac point, and the calculated density of states for the odd-frequency gap shows double-cusp features at low frequency, a signature absent for the even-frequency gap.","pith_inferences":["If confirmed, the mechanism predicts that tuning the frequency dependence of the interaction, for example by coupling to a soft bosonic mode that changes the sign of $V'(\\omega)$, could switch between BCS and Berezinskii states, since the two channels couple to different functionals of $V$.","The cusp-like DOS signature could be tested in existing Dirac semimetal candidates with scanning tunneling spectroscopy; absence of the low-frequency double-cusp at charge neutrality would count against the specific odd-frequency ansatz.","The paper's inverse-problem method, recovering the potential from an assumed gap, could be applied to other odd-frequency proposals to check whether their assumed gaps imply physically reasonable repulsive potentials.","Because the required potential is repulsive, competing charge or insulating instabilities may preempt the superconducting state; the paper notes this possibility but does not analyze it."],"forward_implications":["In compensated Dirac and Weyl semimetals at charge neutrality, both even- and odd-frequency gaps require coupling strengths above a critical value, so the usual BCS no-threshold intuition does not apply.","A repulsive frequency-dependent interaction can be a source of superconductivity, giving experimental search a new target: materials with strong repulsive frequency-dependent scattering, not just attractive BCS channels.","The density of states of a Dirac semimetal with an odd-frequency gap exhibits cusp-like features at low frequencies and generically no coherence peaks even away from charge neutrality, providing a spectroscopic fingerprint.","The same qualitative behavior holds in two-dimensional Dirac semimetals, so graphene-like systems are also candidate platforms.","Chirality provides a symmetry-allowed odd-frequency channel in spin-singlet s-wave pairing, extending Berezinskii pairing beyond systems that require orbital multiplicity."],"supporting_citations":[{"why":"Introduces odd-frequency superconductivity as a possible pairing state, the phenomenon this paper seeks to realize in Dirac semimetals.","marker":"[1]"},{"why":"States the SP*OT* = -1 symmetry rule that the paper generalizes to chirality as SP*chi T* = -1.","marker":"[3]"},{"why":"Together these supply the effective-action mean-field formulation for frequency-dependent pairing on which the gap-equation derivation is built.","marker":"[18, 19]"},{"why":"Shows that a critical coupling is needed to generate an odd-frequency gap; the paper extends this result to Dirac semimetals and uses it as a comparison.","marker":"[38]"}],"fun_headline_variants":["Odd-frequency pairing turns repulsion into superconductivity","Repulsive forces spark odd-frequency superconductivity in Dirac semimetals","Chirality enables odd-frequency superconductivity without attraction","From repulsion to superconductivity via odd-frequency pairing in Dirac semimetals","No attraction needed for odd-frequency superconductivity in Dirac semimetals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central result depends on real Dirac semimetals actually having a repulsive interaction that changes strongly with frequency in the required way; the paper derives what that interaction must look like but does not prove one exists.","fun_headline_variants_meta":{"raw":{"variants":["Odd-frequency pairing turns repulsion into superconductivity","Repulsive forces spark odd-frequency superconductivity in Dirac semimetals","Chirality enables odd-frequency superconductivity without attraction","From repulsion to superconductivity via odd-frequency pairing in Dirac semimetals","No attraction needed for odd-frequency superconductivity in Dirac semimetals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001118,"raw_usage":{"total_tokens":4644,"prompt_tokens":923,"completion_tokens":3721,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":3635}},"tokens_in":539,"tokens_out":3721,"duration_ms":24678,"temperature":1.0,"reasoning_tokens":3635,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:16:11.409561+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A first-principles calculation of the effective frequency-dependent electron-electron interaction in a candidate Dirac semimetal would settle the mechanism: if the resulting $V(\\omega)$ is not repulsive with the derivative profile required by the differential gap equation, then odd-frequency pairing cannot naturally appear by this route.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces odd-frequency superconductivity as a possible pairing state, the phenomenon this paper seeks to realize in Dirac semimetals."}],"review_version":1}