{"id":"fd0047c9-e0f5-405e-ab3c-33b61028aed4","arxiv_id":"2412.14534","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Quantum-critical intra-unit-cell loop currents are poor pairing glues, electric monopole-dipole interactions in quasi-2D Dirac systems can drive odd-parity superconductivity, and Sr2RuO4 pairing must include s, d_{x^2-y^2}, or d_{xz}+i d_{yz} admixtures.","lead":"This doctoral thesis tests whether loop-current fluctuations can drive superconductivity, proposes a new pairing mechanism based on electric dipole interactions in Dirac metals, and narrows down the pairing symmetry of Sr2RuO4. It concludes that intra-unit-cell loop currents are ineffective or pair-breaking for cuprates, and that two-component Sr2RuO4 pairing needs severe fine-tuning.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The odd-parity loop-current 'strong pair-breaker' verdict is an extrapolation of the static weak-coupling eigenvalue to the QCP; Landau damping and self-energy are uncontrolled there, and — unlike the nematic/spin positive-control cases — no complementary retarded or numerical check exists for the…","rationale":"The reader's ACCEPT/MODERATE is defensible: the thesis is careful and self-contained; the Chapter 1 symmetry analysis (forward-scattering suppression for PΘ-odd orbital order, Perron-Frobenius results, repulsiveness of odd-parity LC channels) is rigorous within the model. I checked the odd-parity LC repulsiveness claim: although the triplet kernel W_ii(p,k) is positive at backward scattering (p≈−k), its action on odd-parity gap functions is repulsive in both the forward and backward regions, so the conclusion that all pairing channels are suppressed is internally consistent. The even-parity LC non-enhancement is also robust because F(k,k)=0 is a coupling-vertex property, independent of boson dynamics. The load-bearing weak point is the extrapolation from the static weak-coupling eigenvalue to the QCP for the odd-parity branch. The static/instantaneous replacement (Eq. 1.22) is acknowledged to fail at the QCP; the thesis's criterion (weak-coupling breakdown indicates strong pairing) is supported by complementary methods only for the positive cases (nematic/FM/AFM QCPs), and the cited references do not establish the converse for the negative LC cases. No retarded/Eliashberg, RG, or numerical check of the LC channels exists; the one prior calculation (ASV 2010) is contested by argument in Sec. 2.5.7, not by an independent computation. The a posteriori consistency note in Sec. 1.3.3.3 does not determine QCP behavior. The proposed test — a Landau-damped, frequency-dependent Eliashberg solution of the Sec. 1.3.1 model at the LC QCP, for the odd- and even-parity LC vertices — is feasible, standard, and decisive. If the odd-parity LC channel develops a positive pairing eigenvalue at r=0, the pair-breaker claim is refuted; if it remains non-positive and the even-parity eigenvalue stays finite, the extrapolation is validated. I therefore recommend CONDITIONAL rather than ACCEPT: the scoped claims and the symmetry constraints stand, but the headline negative verdict on loop-current pairing at the QCP should be confirmed by the retarded calculation, or the claim should be explicitly scoped to the weak-coupling regime away from the QCP.","tokens_in":61343,"tokens_out":38392,"duration_ms":327011,"concrete_test":"Perform a frequency-dependent (Eliashberg) solution of the linearized gap equation for the Sec. 1.3.1 Yukawa model at the loop-current QCP, replacing the static propagator with the Landau-damped retarded one, χ(q,iΩ) = 1/(q² + ξ⁻² + γ|Ω|/|q|) with γ from the fermion bubble, including the fermionic self-energy, for the odd-parity (px,py) LC vertex (Sec. 2.5.4) and, as a control, the even-parity dx²−y² vertex. Decisive check: if the leading eigenvalue in the odd-parity LC channel is positive (attractive) at r = 0, the 'parametrically strong pair-breaker' claim of Fig. 1.4 is refuted; if it remains non-positive while the even-parity eigenvalue stays finite as r→0, the static extrapolation is validated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (Fig. 1.4) has two branches. The even-parity LC branch — pairing not enhanced near the QCP — rests on the PΘ-odd orbital symmetry forcing the forward-scattering form factor to vanish, F(k,k)=0 (Eq. 1.124). That vertex suppression is a property of the electron-boson coupling, independent of boson dynamics, so this branch is robust to retardation. The odd-parity LC branch — a 'parametrically strong pair-breaker' whose attractive eigenvalue vanishes at the QCP — is read off from the static, linearized BCS analysis of Secs. 1.3.2–1.3.3, in which the retarded propagator is replaced by χ(q,0) (Eq. 1.22), an approximation the thesis itself says is valid only away from the QCP. The load-bearing step is the inference from the weak-coupling eigenvalue behavior to the physics at the QCP. The thesis justifies this inference by citing complementary analytical and numerical methods (Refs. 125, 149–156, 171–174) that find strong quantum-critical pairing when the weak-coupling λ diverges. Those methods validate the positive direction for the nematic/FM/AFM cases; they do not establish the two negative-direction claims needed here: that a non-divergent λ implies no strong pairing, and that a diverging repulsive eigenvalue implies pair-breaking at the QCP. No retarded/Eliashberg, RG, or numerical check exists for the LC channels. At the QCP the static χ(q,0)=1/(q²+ξ⁻²) is replaced by the Landau-damped χ(q,iΩ)=1/(q²+ξ⁻²+γ|Ω|/|q|), which alters the Cooper-channel integral over (q,Ω) and adds a self-energy not present in the BCS treatment; whether the repulsion in the odd-parity LC channel still drives the attractive eigenvalue to zero at r=0 is a quantitative question the static calculation does not settle. The a posteriori consistency remark at the end of Sec. 1.3.3.3 confirms only that the weak-coupling regime is self-consistent, not the QCP behavior.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This dissertation examines pairing mechanisms and pairing symmetries in three classes of unconventional superconductors. Chapter 1 develops a general symmetry-based weak-coupling analysis of pairing mediated by intra-unit-cell (IUC) order-parameter fluctuations near their quantum-critical points (QCPs). The central result, summarized in Fig. 1.4, is that even-parity IUC loop currents do not produce enhanced pairing near the QCP because parity-times-time-reversal symmetry forces the forward-scattering form factor to vanish [Eq. (1.124)], while odd-parity IUC loop currents act as parametrically strong pair-breakers because their time-reversal-odd repulsive interaction is peaked at q = 0. Chapter 2 applies this framework to the three-orbital Emery model of the cuprates, classifies all IUC particle-hole bilinears, and concludes that neither of the two proposed loop-current orders gives d_{x^2-y^2} pairing and that odd-parity loop-current fluctuations suppress superconductivity. Chapter 3 proposes a pairing mechanism based on electric monopole-dipole interactions in parity-mixed, spin-orbit-coupled quasi-2D Dirac metals, predicting odd-parity pseudoscalar pairing and an out-of-plane optical-conductivity signature.","tokens_in":61748,"tokens_out":8202,"duration_ms":71179,"significance":"If the central loop-current results are correct, they overturn a prominent proposal that quantum-critical IUC loop-current fluctuations are the pairing glue of the cuprates; the classification in Tab. 2.5 and the Sr_2RuO_4 symmetry analysis are also likely to become useful reference results. The thesis has real strengths: a generalized Bloch-Kirchhoff theorem, symmetry arguments that do not rely on uncontrolled numerics, extensive analytic and numerical solutions of the linearized gap equation, a systematic classification of bilinears, and falsifiable predictions such as the optical-conductivity signature of dipole pairing and the absence of a cusp or transition splitting under [110] stress. The main correctness risk is not in the symmetry algebra but in the extrapolation of static weak-coupling eigenvalues to the QCP for the odd-parity loop-current channel; the paper explicitly acknowledges that the static BCS treatment is valid only away from the QCP, and the negative-direction claims lack the retarded/Eliashberg or numerical cross-checks that the paper cites for the positive-direction cases.","major_comments":[{"comment":"The odd-parity branch of the central claim, namely that IUC loop-current fluctuations are parametrically strong pair-breakers at their QCP, is inferred from the static linearized BCS eigenvalue obtained with χ(q,0) [Eqs. (1.22) and (1.92)]. As the thesis itself states in Sec. 1.3.1, this instantaneous approximation is controlled only away from the QCP; at the QCP the frequency dependence of the boson propagator precludes a BCS treatment of the low-frequency sector. The complementary analytical and numerical references cited in Sec. 1.3.3.3 validate the positive-direction inference, that a diverging attractive λ implies strong pairing near the QCP, but they do not establish the two negative-direction inferences needed here: that a non-divergent attractive λ implies no strong pairing, and that a diverging repulsive eigenvalue implies pair-breaking at the QCP. For the even-parity loop-current branch, Eq. (1.124) is a property of the electron-boson vertex that is independent of boson dynamics, so that branch is robust to retardation. For the odd-parity branch, I ask for a concrete check: solve the linearized Eliashberg equation for the odd-parity loop-current vertex with a Landau-damped propagator χ(q,iΩ) = 1/(q^2 + ξ^{-2} + γ|Ω|/|q|), including the frequency-dependent self-energy, and report whether the leading attractive eigenvalue remains suppressed at the QCP. Absent such a check, the claim should be stated as a weak-coupling result away from the QCP rather than as a verdict on the QCP itself.","section":"Sec. 1.3.3.3, Fig. 1.4, Eqs. (1.22) and (1.92)"},{"comment":"The closing paragraph states that 'the absence of a strong attractive pairing interaction at the QCP justifies a posteriori the weak-coupling analysis employed in our analysis.' This justification is not independent: in the odd-parity loop-current case the weak-coupling eigenvalue does diverge as r → 0, albeit with a negative sign, so the weak-coupling framework breaks down there by the paper's own criterion. The asymmetry between treating a diverging attractive eigenvalue as evidence for a superconducting dome and treating a diverging repulsive eigenvalue as evidence for pair-breaking needs explicit support, for example from a calculation of the pairing susceptibility or of T_c at the QCP in the odd-parity loop-current channel. Without such support, the 'a posteriori justification' does not add evidential weight beyond the static BCS calculation.","section":"Sec. 1.3.3.3, final paragraph"},{"comment":"The uniqueness claim, that odd-parity loop currents are 'unique among all orders' as parametrically strong pair-breakers, is established in the strict zero-spin-orbit-coupling limit, where the derivation of Eqs. (1.122)-(1.124) assumes purely orbital Γ matrices of the form γ ⊗ σ_0. The text asserts in one sentence that strong pairing and strong pair-breaking 'continue to be so with SOC,' but no calculation or estimate of the crossover scale in spin-orbit-coupling strength is provided. Since the cuprate application in Chapter 2 explicitly neglects spin-orbit coupling, the cuprate-specific conclusion may be safe; however, the general statement about all systems would be strengthened by estimating the SOC scale at which the forward-scattering suppression is lifted and the odd-parity repulsive kernel is weakened.","section":"Sec. 1.3.3.3, Table 1.2"}],"minor_comments":[{"comment":"There are several typographical errors: 'bare in mind' should be 'bear in mind' (Sec. 2.1.2), 'purpler region' should be 'purple region' (caption context near Fig. 2.2), and the title of Sec. 4.3 reads 'Constrains' instead of 'Constraints.'","section":"Secs. 2.1.2 and 4.3"},{"comment":"The proportionality F_{BA}(p_n,k_n)|_{p→k} ∝ (p-k)^2 is stated as a generic result, but the proportionality constant could in principle vanish on special Fermi-surface points; a brief comment on the generic condition under which the leading quadratic coefficient is nonzero would make the statement more precise and easier to check.","section":"Eq. (1.124) and surrounding text"},{"comment":"The notation Λ^{ζ p_Θ}_{n,a} is dense and the table is long; a short 'how to read this table' paragraph, explaining which index labels the irrep, the time-reversal sign, the copy number, and the component of a multidimensional irrep, would significantly improve accessibility.","section":"Sec. 2.4, Table 2.5"},{"comment":"The literature discussion would benefit from a clear statement of which microscopic studies found IUC loop currents to be competitive versus which did not; the current text lists the results but does not summarize the disagreement in one sentence, making it harder for a reader to assess the weight of the experimental and numerical evidence.","section":"Sec. 2.2.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a doctoral thesis and is long by journal standards; if it is being considered as a single submission, the editor may wish to indicate whether the four chapters will be judged as one body of work or as separable papers. The main substantive concern is the static-to-QCP extrapolation in the odd-parity loop-current channel; this is fixable either by adding a retarded Eliashberg or numerical check or by explicitly limiting the claim to the weak-coupling regime away from the QCP. The text-recycling from the author's own published papers is transparently disclosed and should not be treated as a novelty concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, this is a real thesis, not a padded one. It has three independent parts, each with a sharp claim. The loop-current chapter is the one you'll care about: it argues that intra-unit-cell even-parity loop currents cannot provide quantum-critical pairing enhancement because the forward-scattering vertex vanishes by symmetry, and that odd-parity loop currents actually suppress pairing near their QCP. If the latter claim holds, it removes a leading explanation for cuprate superconductivity and the pseudogap. The other two parts are a proposal for odd-parity pairing from electric dipole fluctuations in doped Bi2Se3/SnTe with a concrete optical conductivity prediction, and a careful reanalysis of Sr2RuO4 uniaxial stress data that allows only s, d_{x^2-y^2}, or body-centered d_{xz}+i d_{yz} admixtures and exposes the fine-tuning problem for bulk two-component order.\n\nBest things: the symmetry machinery is thorough and honest. The generalized Bloch-Kirchhoff theorem is clean; the extended-basis classification of all particle-hole bilinears in the three-orbital Emery model is a genuine contribution that will be used by others. The proof that PΘ-odd orbital order suppresses the forward-scattering form factor by two powers of momentum (Eq. 1.124) is the backbone of the even-parity LC result, and that part is robust—it does not depend on boson dynamics. The thesis also flags its own limitations (weak-coupling, instantaneous interaction, no Coulomb repulsion) and the text recycling from the author's own papers is disclosed, not hidden. The comparison to Aji-Shekhter-Varma is valuable and points out genuinely different assumptions.\n\nWhere it's softer: the odd-parity LC 'parametrically strong pair-breaker' conclusion is read off from the static linearized gap equation, which replaces the boson propagator by χ(q,0). The thesis says this is valid away from the QCP, then extrapolates to the QCP. The even-parity branch survives that worry because the suppression is a vertex property, not a dynamics property. But the odd-parity branch depends on the repulsion surviving when you include Landau damping and self-energy. I'd call that a real caveat, not an error: the thesis's own a posteriori consistency check only establishes that the weak-coupling calculation is self-consistent, not that the QCP physics is settled. A retarded/Eliashberg calculation in that channel would be the natural follow-up. The dipole Tc estimate is modest and screening-dependent, but the optical conductivity prediction is a clean falsifiable target. The Sr2RuO4 analysis is data-driven and careful; it is probably the least controversial part.\n\nWho is this for: condensed matter theorists working on unconventional pairing, cuprates, topological superconductivity, and Sr2RuO4. It deserves a serious referee. I would send it to review rather than desk reject, but I'd ask the referee to focus on Chapter 1 and to demand an explicit statement about the extrapolation.","headline":"A serious three-part theory thesis: the even-parity loop-current result is solid and important, the odd-parity pair-breaking claim is plausible but needs a retarded check, and the dipole mechanism and Sr2RuO4 analysis are worth engaging.","tokens_in":62339,"tokens_out":4265,"would_cite":true,"duration_ms":33278,"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":"Loop currents within a unit cell cannot supply the pairing glue for superconductivity: even-parity currents stay weak and odd-parity ones break pairs.","keywords":["unconventional superconductivity","loop currents","quantum-critical pairing","cuprate pseudogap","electric dipole pairing","spin-orbit coupling","Sr2RuO4 pairing symmetry","three-orbital model"],"falsifier":"Solve the retarded Eliashberg equations at the loop-current quantum-critical point with the same order-parameter couplings and including boson damping; if the odd-parity loop-current channel yields an attractive pairing eigenvalue that diverges as $r\\to 0$, the central claim fails. A simpler empirical check would be to tune a material through an odd-parity intra-unit-cell loop-current quantum-critical point and measure whether $T_c$ is suppressed there rather than enhanced.","tokens_in":61150,"feed_emoji":"🌀","tokens_out":8751,"duration_ms":69019,"temperature":0.7,"pith_summary":"The thesis sets out to test whether fluctuating intra-unit-cell loop currents can act as the pairing glue of unconventional superconductors, the proposal most often invoked for the cuprates. Its central claim is that they cannot: in two-dimensional systems with weak spin-orbit coupling, even-parity loop-current fluctuations mediate pairing that does not intensify as the assumed quantum-critical point is approached, while odd-parity loop currents make the Cooper channel repulsive and actively suppress pairing. The same machinery shows that nematic, ferromagnetic, and altermagnetic fluctuations do enhance pairing near their quantum-critical points, so loop currents are uniquely ineffective among intra-unit-cell orders. The thesis also proposes that electric monopole-dipole interactions in parity-mixed, spin-orbit-coupled quasi-2D metals produce odd-parity unconventional superconductivity, and it uses new elastocaloric and $T_c$ measurements to constrain the pairing symmetry of Sr$_2$RuO$_4$ to combinations of $s$, $d_{x^2-y^2}$, and body-centered $d_{xz}+i d_{yz}$ states.","feed_headline":"Loop currents cannot glue high-temperature superconductivity","feed_subtitle":"Even-parity loop currents give no boost; odd-parity ones actively suppress Cooper pairing.","key_machinery":"The load-bearing object is the Cooper-channel pairing interaction $W_{BA}(p_m,k_n)$, built from the static boson susceptibility $\\chi(q,0)$ and a pairing form factor $F_{BA}(p_m,k_n)$ that measures how a Cooper pair at $(k,-k)$ scatters to $(p,-p)$. Near an intra-unit-cell quantum-critical point, $\\chi(q,0)$ is peaked at $q=0$ and would naturally make $\\lambda$ diverge; the argument turns on whether the form factor survives at forward scattering. Symmetry analysis shows that parity-times-time-reversal-odd purely orbital orders have $F_{BA}(k_n,k_n)=0$, suppressed as $(p-k)^2$, while odd-parity loop currents make the form factors strictly positive so the $q=0$ divergence becomes a repulsive Coulomb-like interaction. Applied through the linearized BCS gap equation, this form-factor analysis is what separates loop currents from nematic, ferromagnetic, and altermagnetic orders, and it also supplies the classification of particle-hole bilinears in the three-orbital cuprate model.","core_discovery":"The core discovery, stated for a general itinerant Fermi liquid coupled to a soft order-parameter field, is a symmetry result about the Cooper-channel interaction. For quantum-critical intra-unit-cell order in two dimensions without spin-orbit coupling, the pairing eigenvalue $\\lambda$ is enhanced as $r\\to 0$ only if the forward-scattering form factor is attractive and nonvanishing; combined parity and time-reversal symmetry forces that form factor to vanish as $(p-k)^2$ for even-parity loop currents, turning the would-be divergence of the susceptibility into a finite $\\lambda$. For odd-parity loop currents the interaction is repulsive at all transferred momenta and strongly peaked at $q=0$, so the largest attractive eigenvalue is driven to zero near the quantum-critical point, just as pairing is suppressed by unscreened Coulomb repulsion. The thesis therefore concludes that intra-unit-cell loop currents, including the odd-parity order proposed for the cuprate pseudogap, are not an effective pairing glue, while staggered finite-$q$ loop currents remain viable. In the same volume it establishes a new electronic pairing mechanism from electric dipole fluctuations in quasi-2D parity-mixed systems and narrows the pairing state of Sr$_2$RuO$_4$.","pith_inferences":["A sharper experimental test than the thesis states directly: in a material tuned through an odd-parity intra-unit-cell loop-current quantum-critical point, $T_c$ should dip or vanish as the critical point is approached, in contrast to the dome expected near nematic or magnetic quantum-critical points.","The form-factor mechanism likely generalizes to other particle-hole orders beyond those tabulated, so the classification can serve as a screening tool: any order whose forward-scattering form factor is symmetry-forbidden should be parametrically weak regardless of material details.","The electric-dipole pairing proposal implies that the search for unconventional superconductivity should be extended to low-carrier-density parity-mixed metals without strong correlations, using the out-of-plane optical conductivity to measure the pairing glue directly.","The Sr$_2$RuO$_4$ constraints suggest that future uniaxial-stress experiments along other directions could distinguish the three allowed states by their distinct nodal structures."],"forward_implications":["If the central claim is right, the odd-parity intra-unit-cell loop-current order proposed for the cuprate pseudogap cannot simultaneously be the source of the high-temperature superconducting dome; it would suppress rather than generate pairing near its quantum-critical point.","Even-parity loop currents may still coexist with superconductivity, but only as a subordinate order; they provide no reason for $T_c$ to peak near their quantum-critical point.","The same analysis leaves staggered (finite-$q$) loop currents, of the $d$-density-wave type, as a viable pairing glue, since no symmetry forbids the $q=Q$ scattering that would enhance pairing.","In doped Bi$_2$Se$_3$ and SnTe, the proposed dipole mechanism predicts unconventional odd-parity superconductivity that should be visible as a feature in the out-of-plane optical conductivity.","For Sr$_2$RuO$_4$, the pairing state must contain $s$, $d_{x^2-y^2}$, or body-centered $d_{xz}+i d_{yz}$ admixtures; a bulk two-component state is only consistent with ultrasound and elastocaloric data under heavy fine-tuning."],"supporting_citations":[{"why":"The proposal that quantum-critical intra-unit-cell loop-current fluctuations are the pairing glue of the cuprates, which the thesis's central claim targets.","marker":"[34–36]"},{"why":"The earlier analysis by Aji, Shekhter, and Varma that claimed robust $d_{x^2-y^2}$ pairing from loop-current fluctuations; the thesis argues it rests on incorrect assumptions.","marker":"[41]"},{"why":"Supplies the strategy of approaching the quantum-critical point from the disordered Fermi-liquid side and reading weak-coupling breakdown as an indicator of strong pairing.","marker":"[124]"},{"why":"Justifies the static, instantaneous treatment of the boson-mediated interaction away from the quantum-critical point and the interpretation of the weak-coupling breakdown.","marker":"[150]"},{"why":"The $d$-density-wave or staggered loop-current proposal for the pseudogap, which the thesis keeps viable because finite-$q$ scattering is not symmetry-forbidden.","marker":"[64–66]"}],"fun_headline_variants":["Loop currents don't glue superconductivity","Even-parity loop currents fail as pairing glue","Odd-parity loop currents suppress Cooper pairing","Intra-unit-cell loop currents: not a pairing source","Quantum-critical loop currents: no boost, only suppression"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The negative verdict on loop-current pairing rests on approaching the quantum-critical point from the disordered Fermi-liquid side and treating the boson-mediated interaction as static and instantaneous; if retardation and damping of the fluctuating order parameter are important enough near the critical point to invalidate that extrapolation, the pairing verdict could change.","fun_headline_variants_meta":{"raw":{"variants":["Loop currents don't glue superconductivity","Even-parity loop currents fail as pairing glue","Odd-parity loop currents suppress Cooper pairing","Intra-unit-cell loop currents: not a pairing source","Quantum-critical loop currents: no boost, only suppression"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000969,"raw_usage":{"total_tokens":4217,"prompt_tokens":1138,"completion_tokens":3079,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":754,"completion_tokens_details":{"reasoning_tokens":3008}},"tokens_in":754,"tokens_out":3079,"duration_ms":20422,"temperature":1.0,"reasoning_tokens":3008,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:08:36.380971+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the retarded Eliashberg equations at the loop-current quantum-critical point with the same order-parameter couplings and including boson damping; if the odd-parity loop-current channel yields an attractive pairing eigenvalue that diverges as $r\\to 0$, the central claim fails. A simpler empirical check would be to tune a material through an odd-parity intra-unit-cell loop-current quantum-critical point and measure whether $T_c$ is suppressed there rather than enhanced.","supporting_citations":[],"review_version":1}