{"id":"5907a13c-29c7-40fb-97dc-70beb3dd4e87","arxiv_id":"2607.10464","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Fluctuating Cooper pairs above Tc produce closed-form nonreciprocal dichroism and gyrotropic birefringence that diverge as 1/(T−Tc) and require particle-hole asymmetry plus a cubic Lifshitz invariant.","lead":"Superconducting fluctuations just above Tc generate a nonreciprocal, wavevector-odd conductivity in noncentrosymmetric 2D materials. The effect is critically enhanced near Tc and is in principle readable by NV-center noise magnetometry on gated MoS2-like films.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the strongest claim (closed-form AL gyrotropy controlled by γ2 and the cubic LI) and the weakest premise (unquantified γ2 for MoS2). That premise is real but is already standard in the fluctuation-Hall literature and is stated explicitly by the authors; it does not render the derivation inconsistent. The appendices close the two natural loopholes (kinetic LI and normal-state background). Because the math is clean, the selection rules transparent, and no additional soft spot appears under scrutiny, the ACCEPT verdict stands without adjustment.","tokens_in":16242,"tokens_out":442,"duration_ms":9574,"concrete_test":"Independently recompute the frequency integral that produces δK (Eq. 16) from the product of two Lorentzian propagators (9) expanded to linear order in γ2 and in δε = k·v, then insert the cubic w(q) of Eq. (4) and perform the 2D momentum integrals that yield W(ϖ) of Eq. (19); if the closed form and the peak location ϖ≃2.1 are recovered, the analytic core is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper’s central claim is a controlled Gaussian AL calculation of the odd-in-k conductivity from the TDGL free energy with cubic LI (4) and complex γ. The derivation of the kernel expansion (16), the closed-form W(ϖ) and V(ϖ) (19,23), the 3:(-1) channel ratio, and the 1/ϵ scaling of the static birefringence are internally consistent under the stated assumptions. Appendix B correctly shows that a kinetic LI is sub-leading by one power of ϵ near Tc, and Appendix A supplies an independent normal-state baseline of the same tensor structure. The necessity of nonzero γ2 is already flagged by the reader and is the same premise that underlies the fluctuation Hall effect; it is not a hidden inconsistency. No further load-bearing technical flaw is present that would overturn the claim as formulated.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper computes the Aslamazov–Larkin contribution to the spatially dispersive conductivity of a 2D noncentrosymmetric superconductor above Tc within Gaussian TDGL theory that includes particle-hole asymmetry (complex γ) and the cubic Lifshitz invariant of the D3h point group. It shows that the conductivity acquires a nonreciprocal, odd-in-k component: the dissipative part is odd in frequency, vanishes at dc, and is controlled by the closed-form function W(ϖ) that peaks near ωτ_GL ≃ 2, while the reactive (Kramers–Kronig) partner V(ϖ) remains finite at ω → 0 and scales as 1/ϵ. Both effects require broken inversion and time-reversal symmetries and are proportional to γ2, in direct analogy with the fluctuation Hall effect. The same band model is used to evaluate the normal-state gyrotropy as a smooth baseline, and the critical enhancement is framed for gated MoS2 and NV-center noise spectroscopy.","tokens_in":16415,"tokens_out":820,"duration_ms":9812,"significance":"If correct, the work supplies a concrete, closed-form linear-response counterpart to the fluctuation-enhanced magnetochiral anisotropy and superconducting diode effect already observed near Tc in noncentrosymmetric 2D materials. The analytic frequency functions W and V, the 3:(−1) channel ratio, the 1/ϵ scaling of the static birefringence, and the explicit normal-state baseline are parameter-free once γ2, η, and τ_GL are regarded as material inputs; they therefore constitute falsifiable predictions for optical dichroism/birefringence and for direction-odd magnetic noise. The calculation cleanly places fluctuation gyrotropy in the same universality class as the fluctuation Hall effect and strengthens the case that Lifshitz invariants play the role of band geometry in the fluctuation regime.","major_comments":[],"minor_comments":[{"comment":"The magnitude of γ2/γ1 is left as a free material parameter (Eq. 7) and is not estimated microscopically for the MoS2 band model. A short order-of-magnitude estimate (or a pointer to existing microscopic calculations of ∂lnTc/∂lnμ) would help experimental readers judge observability.","section":null},{"comment":"Figure 1 is clear, but the caption and main text could explicitly mark the location of the maximum of W (ϖ ≃ 2.07) and the value Wmax ≃ 0.886 so that the peak frequency can be read off without re-deriving the function.","section":null},{"comment":"In Sec. V the thin-film transmission formulae for ΔA and Δφ are given under |2πσ/c| ≪ 1; a brief remark on the range of validity for typical gated-TMD sheet conductivities near Tc would be useful.","section":null},{"comment":"Appendix A recovers the same 3:(−1) ratio as the AL calculation; a one-sentence cross-reference in the main text would make this structural parallel more visible.","section":null},{"comment":"A few minor notational points: the definition of τ_GL = γ1/(2a) appears after its first use in the discussion of W; the wave-vector convention for the normal-state warping parameter Λ could be stated once in the main text for readers who skip the appendix.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is technically solid and self-contained. The self-citations to the author’s related MCA/SDE works are appropriate for context and do not inflate novelty. Fit for a specialized condensed-matter journal is excellent; the closed-form results and clear experimental fingerprints make it suitable for a high-visibility venue as well."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is the closed-form Aslamazov–Larkin gyrotropic conductivity for a 2D noncentrosymmetric superconductor above Tc. With the cubic Lifshitz invariant and complex TDGL relaxation constant, Levchenko gets the odd-in-k dissipative dichroism Re σ_odd ∝ η Bz kx W(ω τ_GL) (W odd, peaks near ω τ_GL ≃ 2) and its Kramers–Kronig partner V that stays finite at ω\to0 and scales as 1/ϵ. That is the linear-response sibling of the already-observed giant MCA and SDE, plus a concrete NV-noise observable.\n\nWhat works: the derivation is controlled and transparent. Kernel expansion to linear order in γ2 and k, frequency integrals done in closed form, single analytic G(ϖ) that yields both W and V, explicit 3:(-1) channel ratio, and the same tensor structure recovered for the normal-state D3h baseline in Appendix A. Appendix B correctly shows the kinetic LI is sub-leading by one power of ϵ near Tc. Selection rules are clean: both inversion and time-reversal must be broken, and the effect vanishes without particle-hole asymmetry, exactly as the fluctuation Hall effect does. Citations sit in the right place; the self-cites supply the cubic LI and MCA context without circularity. No invented entities, low free-parameter burden once γ2/γ1 and η are treated as material inputs.\n\nSoft spot (proportionate): the whole nonreciprocal piece is linear in γ2, and the paper does not compute its microscopic value for the MoS2 band model. That is the same premise that has always limited fluctuation Hall predictions; it is flagged honestly and does not break the calculation under the stated assumptions. Everything else holds.\n\nThis is for people already working on nonreciprocal superconductivity, fluctuation transport, or NV magnetometry of critical dynamics. The math is solid enough that a serious editor should send it out. I would read it carefully, cite the W/V forms and the 1/ϵ birefringence when I need them, and bring it to reading group.","headline":"Clean closed-form AL calculation of fluctuation gyrotropy (W and V) that supplies the linear-response counterpart to giant MCA/SDE near Tc; soft only on the usual unquantified γ2 premise.","tokens_in":17070,"tokens_out":614,"would_cite":true,"duration_ms":8633,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Above Tc, fluctuating Cooper pairs give a 2D noncentrosymmetric superconductor a nonreciprocal, wavevector-odd conductivity that peaks at the pair decay rate and grows as 1/(T-Tc).","keywords":["superconducting fluctuations","nonreciprocal dichroism","gyrotropy","Aslamazov-Larkin","Lifshitz invariant","particle-hole asymmetry","MoS2","TDGL"],"falsifier":"In a gated MoS2 film near Tc, measure the direction-odd absorption Delta A or phase shift Delta phi of counter-propagating waves (or the field-reversed magnetic noise of an NV center) and check whether a critical 1/(T-Tc) upturn appears with the predicted nonmonotonic frequency profile peaking near the GL relaxation rate.","tokens_in":17097,"feed_emoji":"⚡","tokens_out":805,"duration_ms":19264,"temperature":0.7,"pith_summary":"The paper shows that, just above the superconducting transition, thermally fluctuating Cooper pairs produce a linear-response nonreciprocal conductivity that is odd in wavevector. Using a time-dependent Ginzburg-Landau description that includes particle-hole asymmetry and the cubic Lifshitz invariant allowed by trigonal symmetry, it obtains the Aslamazov-Larkin contribution in closed form for all frequencies. The dissipative piece is directional dichroism: odd in frequency, vanishing at dc, and peaking when the probe frequency matches the Cooper-pair decay rate. Its Kramers-Kronig partner is gyrotropic birefringence that remains finite at zero frequency and diverges as the temperature approaches Tc. Both effects require broken inversion and time-reversal symmetry, depend on the same particle-hole asymmetry that controls the fluctuation Hall effect, and share the microscopic origin of the superconducting diode effect and giant magnetochiral anisotropy. The critical upturn is predicted to dominate the smooth normal-state background, making it a measurable signature in gated MoS2 films via optics or nitrogen-vacancy noise magnetometry.","feed_headline":"Fluctuating pairs give superconductors a nonreciprocal conductivity","feed_subtitle":"Dichroism peaks at the pair decay rate; birefringence grows as 1/(T-Tc) above the transition","key_machinery":"The Aslamazov-Larkin current-current loop built from the statistical pair propagator of TDGL theory with complex relaxation constant gamma = gamma1 + i gamma2 and cubic Lifshitz invariant w(q). Expanding the frequency-integrated kernel to linear order in gamma2 and in the inversion-odd velocity content of k yields the compact odd response that is then integrated over momentum to produce the closed-form functions W(omega) and V(omega).","core_discovery":"Fluctuation-induced Cooper pairs above Tc generate a nonreciprocal, odd-in-wavevector component of the spatially dispersive conductivity. Its dissipative part is proportional to eta Bz kx times an odd frequency function W that peaks near omega tau_GL ~ 2; its reactive part remains finite at omega = 0 and scales as 1/epsilon, where epsilon = (T - Tc)/Tc. The response is controlled by the product of particle-hole asymmetry and the cubic Lifshitz invariant and has the same trigonal tensor structure as the normal-state gyrotropy of the same band model.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Fluctuating Cooper pairs yield nonreciprocal dichroism above Tc","Pair fluctuations create odd-wavevector conductivity near transition","Superconducting fluctuations drive gyrotropic birefringence above Tc","Fluctuation pairs give nonreciprocal conductivity from broken symmetries","Cooper fluctuations enhance nonreciprocal dichroism peaking at decay rate"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The entire nonreciprocal signal is proportional to the imaginary part of the TDGL relaxation constant (particle-hole asymmetry); if that asymmetry vanishes, both dichroism and birefringence disappear.","fun_headline_variants_meta":{"raw":{"variants":["Fluctuating Cooper pairs yield nonreciprocal dichroism above Tc","Pair fluctuations create odd-wavevector conductivity near transition","Superconducting fluctuations drive gyrotropic birefringence above Tc","Fluctuation pairs give nonreciprocal conductivity from broken symmetries","Cooper fluctuations enhance nonreciprocal dichroism peaking at decay rate"]},"model":"grok-4.5","effort":"low","cost_usd":0.008548,"raw_usage":{"total_tokens":2084,"prompt_tokens":893,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":85480000,"prompt_tokens_details":{"text_tokens":893,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1101,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":893,"tokens_out":90,"duration_ms":13191,"temperature":1.0,"reasoning_tokens":1101,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T11:30:30.108090+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"In a gated MoS2 film near Tc, measure the direction-odd absorption Delta A or phase shift Delta phi of counter-propagating waves (or the field-reversed magnetic noise of an NV center) and check whether a critical 1/(T-Tc) upturn appears with the predicted nonmonotonic frequency profile peaking near the GL relaxation rate.","supporting_citations":[],"review_version":1}