{"id":"bb222fae-4665-437b-a0c8-057631edebcb","arxiv_id":"2412.14927","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"In bilayer superconductors, the lower Josephson plasmon is a c-axis-polarized, counterflowing current mode whose spectral weight vanishes at zero out-of-plane momentum, which is why it appears as a ghost.","lead":"This paper explains why the lower Josephson plasmon in bilayer superconductors is invisible to density probes at small out-of-plane momentum: its current fluctuations are transverse and staggered, so it does not produce charge density. The explanation gives a physical picture for a puzzling ghost mode in cuprate superconductors like YBCO and helps interpret RIXS experiments.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finite-q_c spectral weights are not those of a Bloch wave: Eq. (22) sums all layer matrix elements, omitting the sublattice phase e^{-iq_c d_1} of the density probe, so the claimed non-periodicity and visibility of the lower branch at q_c=1.8π/d are artifacts; W_-(0)=0 survives.","rationale":"After reading the paper in good faith, I find the central zero at q_c=0 is well supported: the lower eigenvector is the staggered (1,-1) mode, orthogonal to the uniform density probe, and this is independent of α and of the retardation approximation. The reader's concern about dropping A is plausible but not the weakest point: the cutoff qbar quoted in Sec. III.B is orders of magnitude below RIXS/EELS momenta, and the q_c=0 orthogonality is a symmetry statement. The load-bearing weakness is instead in the step from the 2x2 response matrix Eq. (20) to the scalar response Eq. (22). A physical probe with momentum q_c has different phase at the two layers, so summing all matrix elements without the Bloch factors is not the measured density response except at q_c=0. This missing form factor is the origin of the paper's non-periodicity claim, which is inconsistent with Bloch periodicity for the true c-lattice constant d. The RIXS comparison in Sec. III.C uses q_c=1.8π/d, which is -0.2π/d modulo 2π/d; a periodic response cannot have the dramatic difference shown in Figs. 3(b) and 3(c). The proposed test would settle whether the lower branch is actually visible at any experimentally accessible momentum. If the form-factor-corrected calculation still gives a strong lower-branch signal at an inequivalent large q_c, the paper's conclusion would be restored; otherwise the central application fails while the q_c=0 ghost concept remains. The verdict therefore stays conditional, but for a more concrete and checkable reason than the retardation caveat identified by the reader.","tokens_in":22138,"tokens_out":34182,"duration_ms":317399,"concrete_test":"Recompute the physical density response from Eq. (20) as χ_c(q,ω) = -Im[ v(q)^† χ̂(q) v(q) ] with v(q)=(1,e^{-iq_c d_1}) (or v(q)=(e^{-iq_c d_1/2}, e^{iq_c d_1/2}) in the symmetric gauge), instead of -Im Σ_{αβ} [χ̂]_{αβ}. Using the Fig. 3 parameters, evaluate at q_c=0, 0.2π/d, and 1.8π/d. If the corrected χ_c is periodic and W_-(1.8π/d)=W_-(−0.2π/d)≈W_-(0.2π/d), then Eq. (22)'s weighting is wrong; report whether any lower-branch intensity remains above the damping floor at these momenta and whether the upper-branch fit in Sec. III.C is then the only viable assignment.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III.A and Eq. (22) define the physical charge response as -Im Σ_{αβ} [χ̂]_{αβ}. This corresponds to δφ_1=δφ_2, i.e. a probe that is uniform across the two layers of the unit cell. For a periodic bilayer with c-lattice constant d and layer positions 0 and d_1, an external scalar potential with wavevector q_c couples to the two layers with amplitudes 1 and e^{-iq_c d_1} (up to an irrelevant global phase). The physical density response is e(q)^T χ̂(q) e(q) with e(q)=(1,e^{-iq_c d_1}); the unweighted sum is correct only at q_c=0. This immediately explains the 'non-periodicity' asserted after Eq. (24): under q_c→q_c+2π/d the matrix χ̂ undergoes a gauge transformation, and e(q) transforms contravariantly, so the physical response is periodic; the bare sum is not. Consequently W_-(q_c=0)=0 and the counterflow/transverse-polarization picture at q_c=0 are robust, but the central finite-q_c prediction—that W_- grows and the lower branch becomes visible at q_c≈1.8π/d—is not established. Indeed q_c=1.8π/d is the same Bloch momentum as -0.2π/d, so the lower-branch intensity should equal the (small or vanishing) intensity at q_c=0.2π/d of Fig. 3(b), not the large intensity of Fig. 3(c). The RIXS assignment in Sec. III.C and Fig. 7 therefore rests on an unphysical choice of density operator.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a phase-only action description of collective plasma modes in bilayer superconductors and derives the density-density response matrix. Its central claim is that the lower Josephson plasmon has zero spectral weight in the density response at out-of-plane momentum q_c=0 because it is a staggered, c-axis-polarized counterflow mode that is virtually transverse at small q_c. The authors further claim that this mode becomes visible at large q_c, that the spectral weights are not periodic in q_c, and that the RIXS branch observed in Ca-YBCO at q_c=1.8π/d should be assigned to this lower plasmon. The derivation reproduces earlier RPA results and offers a physically appealing backfolding picture.","tokens_in":22550,"tokens_out":14981,"duration_ms":141198,"significance":"If the ghost mechanism is correct, the paper provides a valuable analytical and physical explanation for the invisibility of the lower Josephson plasmon in bilayer cuprates: the mode's polarization is transverse to the density probe at small q_c, unlike the in-phase upper mode. The closed forms for the spectral weights in the α→0 limit, the explicit connection to current polarizations via the gauge-invariant ψ fields, and the cross-checks against independent RPA calculations (refs. 22, 46, 47) are useful contributions. The analogy with the Pines' demon is stimulating. However, the finite-q_c visibility and the RIXS assignment rest on a questionable definition of the physical density operator, which is a load-bearing issue for the paper's experimental claims.","major_comments":[{"comment":"The charge response is defined as -Im Σ_{αβ}[χ̂_{ρρ}]_{αβ}. For a periodic bilayer with layer positions r_1=R and r_2=R+d_1, a physical plane-wave scalar potential δφ(r)=δφ_q e^{iq·r} couples to the two sublattices with amplitudes (1, e^{-iq_c d_1}) (up to a global phase). The physical density response is therefore e(q)^T χ̂(q) e(q), not the unweighted sum over all matrix elements. The unweighted sum corresponds to a probe that is identical on both layers of every unit cell, i.e., a staggered potential rather than a plane wave. Consequently, the finite-q_c spectral weights in Eqs. (23)-(24), the claimed non-periodicity, and the visibility of the lower branch at q_c=1.8π/d are not established. The vanishing at q_c=0 survives because the phase factor reduces to unity there, but the crucial finite-q_c prediction is an artifact of the probe definition.","section":"Sec. III.A, Eq. (22)"},{"comment":"The statement that the spectral weights of the density-density response are not 2π/d periodic is unphysical for a periodic crystal: any observable response function must be periodic under q_c→q_c+2π/d. The non-periodicity of W_±(q) follows directly from the use of the unweighted matrix sum. With the correct sublattice-coherent probe, the response at q_c=1.8π/d is equal to that at q_c=-0.2π/d (and hence, by inversion symmetry, to that at q_c=0.2π/d). The authors should either justify an alternative Fourier convention that makes their sum the physical response or remove the non-periodicity claim and recompute the spectral weights.","section":"Sec. III.A, after Eq. (24)"},{"comment":"The RIXS comparison relies on a large lower-branch spectral weight at q_c=1.8π/d. Since the physical spectral weight at this momentum is the same as at q_c=-0.2π/d, where the lower branch has small weight according to the authors' own Fig. 3(b), the assignment of the measured mode to the lower plasmon is not supported unless additional q-dependent matrix elements (e.g., form factors or the RIXS scattering cross-section) are explicitly modeled. The authors should re-evaluate the comparison with Ref. [22] and discuss which physical ingredients could lead to a difference between q_c=1.8π/d and q_c=0.2π/d in the measured intensity.","section":"Sec. III.C and Fig. 7"}],"minor_comments":[{"comment":"The symbol d_2 is used throughout but never defined; please define d_2 = d - d_1 explicitly.","section":"Sec. III.A, Fig. 2"},{"comment":"The delta-function spectral weights in Eqs. (12) and (22) are stated without the phenomenological damping γ; the figures use Lorentzian broadening, so the relationship between the analytic expressions and the plotted intensities should be clarified.","section":"Secs. II.B and III.A, Eqs. (12) and (22)"},{"comment":"The layer-resolved responses χ_c^(1) and χ_c^(2) are introduced only in the text near Fig. 6; please give explicit definitions in the main text and specify the probe convention used for these quantities.","section":"Sec. III.B and Fig. 6"},{"comment":"The phrase 'two layers per unit cells' contains a minor grammatical error; it should read 'two layers per unit cell'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The probe-phase issue is serious and appears to invalidate the finite-q_c visibility and the RIXS assignment, which are highlighted selling points of the paper. The ghost mechanism at q_c=0, the analytical derivation, and the polarization picture are sound and could form the basis of a revised paper. If the corrected spectral weight shows that the lower branch remains essentially invisible at all experimentally relevant q_c, the paper would lose its RIXS application but would still offer a useful theoretical explanation of the ghost mode. I recommend major revision rather than rejection because the core mechanism is defensible and the probe issue is localizable to the definition of the physical density operator."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea here is good, and at q_c=0 it holds up: the lower Josephson plasmon is a counterflowing, c-axis-polarized mode whose density spectral weight vanishes because it is effectively transverse and staggered. That result follows from the matrix structure, reproduces earlier RPA calculations, and the backfolding picture plus gauge-invariant current analysis are genuinely new. Worth a serious look.\n\nBut the stress-test note is correct, and it matters. The response matrix in Eq. (20) is built in the two-layer basis, and a physical density probe at finite q_c couples to the layers with phase factors 1 and e^{-iq_c d1}. The paper instead sums all matrix elements of the response, which is only valid at q_c=0. This is not a harmless approximation: it is exactly what generates the claimed 2π/d non-periodicity of the spectral weights. A physical response in a periodic crystal must be periodic. Once the sublattice phases are included, the lower-branch weight at q_c=1.8π/d equals the weight at q_c=-0.2π/d (equivalently 0.2π/d), which is small. So the central prediction—that the lower branch becomes visible at the RIXS momentum—and the fit to the Ca-YBCO data in Sec. III.C rest on an unphysical density operator. The q_c=0 ghost survives; the finite-q_c story does not.\n\nOther caveats are minor by comparison: the alpha=0 closed forms and neglect of retardation are acknowledged and do not affect the q_c=0 result. The paper is internally consistent and honest about its own limitations, which is why I would send it to a referee: the flaw is specific and fixable in principle, and the q_c=0 mechanism could still be a useful contribution if the finite-q_c analysis is redone with the correct probe coupling.\n\nIn current form, I would not cite it for the experimental claim, and I would not trust the visibility criterion at large q_c. But the physical picture of the ghost at q_c=0 is a real step forward.","headline":"The q_c=0 ghost mechanism is real, but the finite-q_c predictions and the RIXS assignment rest on a density probe that omits the sublattice phase.","tokens_in":23197,"tokens_out":3954,"would_cite":false,"duration_ms":37433,"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":"The lower Josephson plasmon in a bilayer superconductor carries zero density-response weight at $q_c=0$ because it is a staggered, $c$-axis-polarized mode that stays transverse until the out-of-plane momentum grows.","keywords":["Josephson plasmon","bilayer superconductor","ghost mode","density-density response","phase-only action","plasmon polarization","RIXS","cuprate"],"falsifier":"A momentum-resolved density probe on a clean bilayer superconductor, sweeping $q_c$ through zero at small fixed $q_a$, would falsify the claim if it resolved a lower-branch peak at $q_c=0$ or if the spectral weight $W_-(q)$ computed from the microscopic parameters stayed finite as $q_c\\to 0$.","tokens_in":21812,"feed_emoji":"👻","tokens_out":7209,"duration_ms":54294,"temperature":0.7,"pith_summary":"This paper explains why one of the two Josephson plasmons in a bilayer superconductor is a ghost: it has zero weight in the density response at zero out-of-plane momentum $q_c=0$, even though the mode genuinely exists. Working with a phase-only action for the superconducting order parameter, the authors derive the full density-density response and show that the lower mode's spectral weight vanishes at $q_c=0$ for every in-plane momentum. The reason is that the lower mode is built from counterflowing currents polarized along the $c$-axis, so at small $q_c$ it is virtually transverse and cannot be excited by a longitudinal density probe. As $q_c$ grows, the mode acquires a longitudinal projection and becomes visible, which the authors use to argue that a RIXS branch in a bilayer cuprate is this lower plasmon.","feed_headline":"Lower Josephson plasmon is a ghost at zero c-axis momentum","feed_subtitle":"The mode only appears in density probes once out-of-plane momentum grows enough to give it a longitudinal component.","key_machinery":"The machinery is a Gaussian, phase-only action for superconducting phase fluctuations, promoted to a $2\\times2$ matrix form for the two layers per unit cell and dressed with the Coulomb interaction through the scalar potential, Eqs. (15)-(16). The density-density response, Eq. (20), is rearranged so the two Josephson-mode poles $\\omega_\\pm(q)$ appear explicitly, giving the spectral weights $W_\\pm(q)$ in Eqs. (23)-(24). To expose the physical mechanism, the authors introduce gauge-invariant current fields $\\psi_a,\\psi_c$ and the normalized longitudinal projection $\\psi^\\pm_L(q)$; this is the object that shows the upper mode is longitudinal for all momenta while the lower mode is transverse at small $q_c$ and becomes longitudinal only at large $q_c$.","core_discovery":"The central claim is that in a bilayer superconductor with two inequivalent interlayer Josephson couplings, the lower Josephson plasmon is invisible to density probes at $q_c=0$ because it is a staggered, $c$-axis-polarized mode. When the two layers per unit cell break the translational symmetry along $c$, the single-layer plasmon dispersion backfolds from the zone boundary to $q_c=0$; the folded branch keeps the polarization it had at the boundary, namely currents perpendicular to the planes that counterflow between the intrabilayer and interbilayer spacings. Such a mode is transverse at small $q_c$, so its spectral weight $W_-(q)$ in Eq. (24) vanishes identically at $q_c=0$ for all $q_a$, while the upper mode remains fully longitudinal. For larger $q_c$ the lower mode develops a longitudinal projection and reappears in the density response, with opposite-sign density fluctuations in the two layers.","pith_inferences":["An extension of the mechanism suggests that any multicomponent superconductor whose unit cell contains several layers should show similar ghost branches whenever a dispersion folded from the zone boundary is polarized transverse to the density-probe direction; artificial bilayer and superlattice systems could test this directly.","The distinction between a ghost mode and a truly neutral mode matters experimentally: a transverse mode should still appear in optical conductivity or transverse current probes at small $q_c$, offering a separation between the two explanations that the paper does not work out.","The analogy with the acoustic demon mode raises a broader question the paper leaves open: whether out-of-phase density oscillations generically produce acoustic dispersions in multicomponent metals and superconductors, independent of the folding mechanism that creates them.","Because the spectral weight is not periodic in $q_c$, analyses that fold experimental momenta into the first Brillouin zone should assign branch intensities carefully, since the same physical mode can appear bright or dark depending on which zone image is measured."],"forward_implications":["At $q_c=0$ the density response contains only the upper Josephson plasmon; the lower branch cannot be detected by RIXS or EELS at that momentum, regardless of in-plane momentum.","The spectral weight of the lower branch is not periodic in $q_c$ with period $2\\pi/d$: the mode reappears when $q_c$ approaches the zone boundary, so a measurement at $q_c=1.8\\pi/d$ can see the branch that is invisible at $q_c=0.2\\pi/d$.","For the bilayer cuprate Ca-YBCO, the RIXS-measured dispersion is most plausibly the lower Josephson plasmon, with the upper branch overdamped in the quasiparticle continuum.","In the region where it becomes visible, the lower plasmon's density fluctuations in the two layers have opposite signs and its dispersion is approximately linear, resembling the acoustic demon-like mode discussed for multiband metals."],"supporting_citations":[{"why":"Supplies the bilayer phase-only action and gauge-invariant current formalism on which the paper's derivation is built.","marker":"[51]"},{"why":"Establishes that dropping the phase-vector-potential coupling (retardation effects) is quantitatively good at the momenta probed by RIXS and EELS.","marker":"[50]"},{"why":"Provides the RIXS data on Ca-YBCO that the paper compares with and whose measured branch it assigns to the lower plasmon.","marker":"[22]"},{"why":"Gives a parallel recent theory assigning the observed mode to the lower branch, which the paper corroborates and explains microscopically.","marker":"[47]"},{"why":"Reports the acoustic demon mode in a multiband metal used as the analogy for out-of-phase, acoustic collective modes.","marker":"[21]"},{"why":"Documents that RIXS probes the density-density response, defining the experimental channel in which the lower mode is a ghost.","marker":"[2]"},{"why":"Documents that EELS probes density fluctuations, the other channel in which the mode should be absent at small $q_c$.","marker":"[3]"}],"fun_headline_variants":["Ghost Josephson plasmon hides at zero c-axis momentum","Lower plasmon is invisible to density probes at q_c=0","Counterflow mode explains missing plasmon in bilayers","Plasmon ghost appears only when momentum grows","Bilayer superconductor hides one plasmon from view"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that retardation effects are negligible, so the superconducting phase couples only to the scalar potential; if the neglected phase-vector-potential coupling matters at the smallest probed $q_c$, the lower mode could acquire a small density response and would not be strictly ghost.","fun_headline_variants_meta":{"raw":{"variants":["Ghost Josephson plasmon hides at zero c-axis momentum","Lower plasmon is invisible to density probes at q_c=0","Counterflow mode explains missing plasmon in bilayers","Plasmon ghost appears only when momentum grows","Bilayer superconductor hides one plasmon from view"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1712,"prompt_tokens":902,"completion_tokens":810,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":733}},"tokens_in":518,"tokens_out":810,"duration_ms":6658,"temperature":1.0,"reasoning_tokens":733,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:47:10.369043+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A momentum-resolved density probe on a clean bilayer superconductor, sweeping $q_c$ through zero at small fixed $q_a$, would falsify the claim if it resolved a lower-branch peak at $q_c=0$ or if the spectral weight $W_-(q)$ computed from the microscopic parameters stayed finite as $q_c\\to 0$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the RIXS data on Ca-YBCO that the paper compares with and whose measured branch it assigns to the lower plasmon."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives a parallel recent theory assigning the observed mode to the lower branch, which the paper corroborates and explains microscopically."},{"cited_title":"branch separately vanishes in each layer for smallqc, due to the fact that the current fluctuations are transverse and do not induce density fluctuations","cited_arxiv_id":null,"evidence_quote":"Documents that EELS probes density fluctuations, the other channel in which the mode should be absent at small $q_c$."}],"review_version":1}