{"id":"a0010fd5-f886-4044-8b25-9a9f3004c93c","arxiv_id":"2502.03779","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"RIXS measurements and RPA calculations show the dominant low-energy plasmon in trilayer Bi2223 is a gapped, weakly qz-dependent out-of-phase mode (omega-), unlike the acoustic plasmons of single-layer cuprates.","lead":"Researchers used resonant inelastic x-ray scattering to find a gapped, nearly two-dimensional plasmon in the three-layer cuprate superconductor Bi2223, and they interpret it as an out-of-phase oscillation of the outer copper-oxide sheets. The measurement fills a missing experimental point in how collective charge modes change as copper-oxide layers are added, which matters for theories that tie these modes to high-temperature superconductivity.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The omega- assignment rests on the unconstrained interlayer hopping tz/tz0 and Coulomb scale Vc; without an independent constraint, the computed ~300 meV gap is a fit rather than a prediction.","rationale":"The paper presents a credible experimental observation: a nearly qz-independent, gapped (~300 meV) plasmon branch in trilayer Bi2223, with the gap directly visible in the RIXS maps and a small qz variation consistent with the inter-trilayer periodicity. The RPA interpretation with three branches is a natural framework, and the qualitative features (weak qz dependence, dominant low-energy weight) do point to an out-of-phase mode rather than the strongly qz-dependent omega+ mode. However, the quantitative assignment depends on a parameter point that is not independently determined. The reader's weakest assumption correctly identifies this: Vc, tz, and tz0 are not constrained by data outside the plasmon measurement, so the predicted gap and the branch labeling are partially fitted. My concern sharpens this by noting the specific mechanism: the omega- gap at q=0 vanishes when tz and tz0 are absent, so the 300 meV gap is essentially a function of the chosen interlayer hopping and Coulomb scale. A sensitivity analysis and quantitative fit are needed to know whether the omega- assignment is robust or merely one possible parameterization. The manuscript's own acknowledgment that Gamma is chosen phenomenologically (SM V) further underscores that the model parameters are not fixed by independent data. Overall the evidence supports a conditional acceptance: the experiment and the broad mode classification are convincing, but the 'first identification' of omega- should be treated as model-dependent until the parameter space is explored.","tokens_in":15086,"tokens_out":6429,"duration_ms":67170,"concrete_test":"Fix in-plane tight-binding parameters to those independently determined by ARPES fits (e.g., Refs. 24,25) and treat tz, tz0, and Vc as free parameters in the RPA calculation. Fit the experimental peak positions in Figs. 3(a)-3(e) and report best-fit values with confidence intervals. Then vary tz and tz0 by +/-50% and Vc by +/-20% and check whether the observed branch remains assigned to omega-; if a model with a gapless omega3 and gapped omega+ also reproduces the data, or if the best-fit tz is inconsistent with the ARPES-determined interlayer splitting, the central claim is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that the observed gapped, nearly qz-independent plasmon is the omega- branch. In the RPA calculation, the omega- gap at (qx,qy)=(0,0) is generated by the interlayer hoppings tz and tz0 entering the bare Green's function (SM Eq. 1) together with the long-range Coulomb matrix elements V2, V3 (SM Eqs. 11-12). If tz=tz0=0, the out-of-phase modes become acoustic at q=0 in this layered model; the finite gap is thus directly controlled by the product of tb and Vc. The parameters (delta=0.18, t'=-0.26, t''=0.13, tz=0.07, tz0=0.1, Vc=280) are stated in SM IV, but no sensitivity analysis is given. The cited ARPES Fermi-surface agreement is qualitative and cannot independently fix tz and tz0, which are small (7-10 meV) compared with the 300 meV gap; the gap is amplified by the Coulomb interaction, so different (tz,Vc) pairs can produce the same gap. The experimental data alone rule out the strongly qz-dependent omega+ branch and the gapless omega3 branch, but they do not uniquely select omega- without specifying the model parameters. The paper's phrase 'first identification of the omega- plasmon' is therefore stronger than the evidence supports.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports O K-edge resonant inelastic x-ray scattering (RIXS) measurements on optimally doped trilayer Bi2223, revealing a plasmon branch that disperses weakly with the out-of-plane momentum L and exhibits a gap of approximately 300 meV at the two-dimensional zone center. The authors model the charge susceptibility in RPA using a three-layer tight-binding model with long-range Coulomb interactions, obtaining three plasmon branches. They assign the observed branch primarily to the omega- mode, in which the outer CuO2 sheets oscillate out of phase while the inner sheet is unaltered at qz=0, and argue that this represents a qualitative change in the dominant low-energy charge mode as the number of CuO2 layers increases.","tokens_in":15381,"tokens_out":3917,"duration_ms":37302,"significance":"If the mode assignment is correct, the result is significant because it provides the first experimental indication that the dominant low-energy plasmon in a trilayer cuprate is the out-of-phase omega- mode, in contrast to the omega+ mode identified in single- and infinite-layer cuprates. The experimental observation of a nearly qz-independent, gapped plasmon branch is new and directly visible in the data, and the theoretical framework is clearly described in the main text and Supplemental Material. The main weakness is that the quantitative branch assignment depends on the choice of interlayer hopping parameters and the Coulomb scale in the RPA model, which are not independently constrained by data outside the plasmon measurement; the reported gap and mode identity are therefore partly emergent from parameter choices rather than robust predictions.","major_comments":[{"comment":"The omega- gap at (qx,qy)=(0,0) is directly controlled by the interlayer hopping parameters tz and tz0 entering the bare Green's function and by the long-range Coulomb matrix elements V2 and V3. With tz=tz0=0, the out-of-phase modes become acoustic at q=0 in this layered model, so the finite gap is a direct consequence of the stated parameters (delta=0.18, t'=-0.26, t''=0.13, tz=0.07, tz0=0.1, Vc=280). The cited Fermi-surface agreement with ARPES is qualitative and cannot fix tz and tz0 at the few-meV level, and different (tz,Vc) pairs can produce the same gap. The paper should provide a systematic sensitivity study over a plausible parameter range, and ideally demonstrate that the omega- assignment and the ~300 meV gap persist for all parameter sets consistent with independent constraints.","section":"Supplemental Material IV, Eq. (1) and Eqs. (10)-(12)"},{"comment":"The separation of the omega- and omega3 branches into distinct features is obtained with a small broadening Gamma=0.1. When the broadening is increased to Gamma=1.0 to reproduce the experimental linewidths (SM Fig. S3), the two branches merge into a single broad feature. Thus the RIXS intensity cannot directly distinguish the omega- and omega3 contributions at the experimental resolution, and the statement that the observed branch is 'primarily' the omega- mode is a theoretical inference rather than a direct experimental identification. The claim of the 'first identification of the omega- plasmon' in the concluding paragraph is therefore stronger than the data and present analysis support.","section":"Fig. 4 and SM Fig. S3"},{"comment":"The experimental plasmon peak positions are presented without error bars, and the quantitative comparison with the RPA dispersions relies on these positions. In particular, the ~300 meV gap at zone center and the weak qz-dispersion of only ~0.02 eV between L=-2 and L=-3 are reported as definitive numbers without uncertainty estimates. A fitting procedure or at least error bars on the peak positions are needed to assess whether the agreement with the omega- branch is statistically meaningful and whether the small qz dispersion is within the experimental uncertainty.","section":"Figs. 3 and 4"}],"minor_comments":[{"comment":"The conclusion that the qz dispersion is much smaller than in single- and infinite-layer cuprates is based on the L range from -2 to -3 only; the text should state the accessible L range and how the c'-periodicity argument extrapolates to other L values.","section":"Fig. 3 and main text"},{"comment":"The notation omega3 for the acoustic branch is used without a local definition; please define the subscript numbering explicitly (e.g., in analogy with Ref. [18]) when the branches are first introduced.","section":"Main text, after Eq. (13) area"},{"comment":"The statement that the model Fermi surfaces are in 'good agreement' with ARPES is qualitative and not shown quantitatively; please provide a reference to a figure or a numerical measure of the agreement, especially for the kz-dependent parts that constrain tz and tz0.","section":"Fig. 4 and SM IV"},{"comment":"The paper notes that recent analyses suggest the bilayer YBCO dispersion may be explained by the omega- mode rather than the omega+ mode (Refs. [30,31]); this ambiguity is relevant to the claim of a qualitative change with layer number and should be addressed more explicitly in the discussion.","section":"Main text, discussion of bilayer cuprates"}],"recommendation":"major_revision","confidential_remarks":"The paper reports a new experimental result that is likely of interest to the cuprate and RIXS communities, and the near-qz-independent gapped plasmon is a convincing observation. However, the central claim of omega- mode identification hinges on a parameter-dependent RPA calculation with no sensitivity analysis; without additional constraints on tz/tz0 and Vc, the mode assignment remains one of several possibilities. The presentation would also benefit from error bars on peak positions and a more careful wording of the 'first identification' claim. The manuscript is well within the scope of the journal, and the issues are addressable through additional analysis and toned-down claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the useful core: this is the first RIXS look at plasmons in a trilayer cuprate, and the central observation—a gapped branch near 300 meV with almost no dispersion between L=-2 and L=-3—is directly visible in the spectra. That is a real datum and a step forward for the n-dependence story. The theoretical part extends the Griffin-Pindor RPA to a three-layer model with finite intra-trilayer hopping, and the mode decomposition into omega+, omega-, omega3 is a sensible framework.\n\nWhere it gets soft is the branch assignment. The omega- identification is produced by the parameter set in SM IV (delta=0.18, t'=-0.26, t''=0.13, tz=0.07, tz0=0.1, Vc=280, Gamma=0.1). The gap at q=0 is controlled by the product of tz/tz0 and the Coulomb scale Vc; with no sensitivity analysis, and only qualitative ARPES agreement to fix the hoppings, the ~300 meV gap is as much fit as prediction. The data alone rule out the strongly qz-dependent omega+ branch and the gapless omega3, but they don't uniquely select omega- once the model parameters are allowed to vary. So \"first identification of the omega- plasmon\" is stronger than the evidence supports; \"first evidence consistent with omega-\" would be honest.\n\nMinor but worth saying: peak positions are given without error bars, and the qz comparison covers only half a Brillouin zone. The linewidth discussion is a bit hand-wavy, though the large-Gamma supplement helps.\n\nNone of this is fatal. The experiment is real, the qualitative n-dependence point is well taken, and the paper is clearly written. It deserves peer review; the referee should push for a sensitivity analysis of tz/Vc and a softened claim. My verdict: conditional acceptance, not in its current form.","headline":"First RIXS plasmon data on a trilayer cuprate with a plausible but under-constrained omega- assignment; the data are solid, the branch identification needs sensitivity analysis.","tokens_in":16024,"tokens_out":2220,"would_cite":true,"duration_ms":21305,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["71.45.Gm","74.25.Gz","78.70.Ck"],"model":"deepseek-v4-flash","headline":"O K-edge RIXS in optimally doped trilayer Bi2223 finds a nearly two-dimensional plasmon branch with a 300 meV gap, which the paper assigns to the out-of-phase $\\omega_-$ charge mode.","keywords":["plasmon","cuprate superconductors","trilayer cuprate","Bi2223","resonant inelastic x-ray scattering","charge susceptibility","collective charge excitations","out-of-phase mode"],"falsifier":"Remeasure the plasmon branch along the $(H,H,L)$ path with higher momentum resolution to check whether the 300 meV gap and the near-$q_z$-independence survive, and independently fix $t_z$, $t_{z0}$, and $V_c$ from Fermi-surface and optical measurements; if the lowest gapped branch then moves far from 300 meV or its dominant spectral weight falls on $\\omega_+$ instead of $\\omega_-$, the central claim fails.","tokens_in":14870,"feed_emoji":"🔬","tokens_out":12613,"duration_ms":109647,"temperature":0.7,"pith_summary":"Using oxygen K-edge resonant inelastic x-ray scattering on optimally doped trilayer Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\\delta}$, this paper identifies a plasmon branch that is almost independent of the out-of-plane momentum and opens a gap of roughly 300 meV at the in-plane zone center. A random-phase-approximation calculation of the charge susceptibility of a three-layer tight-binding model yields three collective branches, and the measured mode is assigned to the $\\omega_-$ branch, in which the charge density on the two outer CuO$_2$ sheets oscillates out of phase while the inner sheet remains unchanged at $q_z=0$. The paper's central claim is that the dominant low-energy charge mode in a trilayer cuprate is therefore qualitatively different from the in-phase $\\omega_+$ mode seen in single-layer and infinite-layer cuprates. This matters because layer-number-dependent charge fluctuations have been proposed as a factor controlling the layer-number dependence of $T_c$ in cuprates.","feed_headline":"Trilayer cuprate's dominant plasmon is an out-of-phase mode","feed_subtitle":"O K-edge RIXS finds a nearly 2-D plasmon with a 300 meV gap, a shift from single-layer cuprates.","key_machinery":"The machinery is the RPA dynamical charge susceptibility $\\chi=(1-\\chi_0 V_q)^{-1}\\chi_0$ of a three-layer tight-binding model, where $\\chi_0$ contains the in-plane hoppings and the intra-trilayer hopping $t_z$ and $t_{z0}$, and $V_q$ is the long-range Coulomb interaction within and between trilayers. Diagonalizing the coupled charge response produces three collective modes: the in-phase $\\omega_+$ mode, the out-of-phase $\\omega_-$ mode, and the $\\omega_3$ mode; the paper identifies the measured 300 meV gapped, weakly $q_z$-dependent branch with $\\omega_-$ by comparing spectral weight and dispersion along the experimentally scanned $q$ paths. The model's Fermi surfaces are checked against ARPES data, and a finite broadening parameter $\\Gamma$ is used to mimic correlation effects on the plasmon linewidth.","core_discovery":"The measured RIXS plasmon in optimally doped Bi2223 disperses nearly identically on the $L=-2$ and $L=-2.5$ scattering planes, and its energy rises only weakly from 0.51 eV to 0.53 eV as $L$ goes from $-2$ to $-3$; the branch has a large excitation gap of about 300 meV at the two-dimensional zone center. The RPA charge susceptibility for three coupled CuO$_2$ sheets predicts three branches: the $q_z$-dependent $\\omega_+$ mode, the weakly $q_z$-dependent $\\omega_-$ mode, and the acoustic $\\omega_3$ mode. The $\\omega_-$ branch carries the dominant low-energy spectral weight and reproduces the measured nearly two-dimensional, gapped dispersion, while the $\\omega_3$ mode is weak and nearly vanishes near the zone center. The paper concludes that this is the first identification of a $\\omega_-$ plasmon in cuprates, and that the eigenmode of the dominant low-energy plasmon changes with the number of CuO$_2$ layers.","pith_inferences":["If the $\\omega_-$ branch dominates in trilayer cuprates, the recent debate over whether the bilayer plasmon is $\\omega_+$ or $\\omega_-$ is tilted toward the $\\omega_-$ assignment, since out-of-phase modes can carry dominant spectral weight once more than one CuO$_2$ layer is present.","A natural extension is to track the 300 meV gap and branch ordering with doping in Bi2223; because the gap is controlled by the intra-trilayer hopping and the Coulomb strength, it should be a sensitive probe of how interlayer coupling evolves toward overdoping.","The out-of-phase $\\omega_-$ mode has no net interlayer charge polarization at $q_z=0$, so its coupling to light and to the superconducting pairing interaction should differ sharply from the $\\omega_+$ mode; if plasmons participate in pairing, the resulting interaction would have a distinct momentum structure.","The RPA calculation with a larger broadening $\\Gamma=1.0$ merges the $\\omega_-$ and $\\omega_3$ branches into a single broad feature, so an experimental separation of the two branches at larger in-plane momentum would provide a sharper test of the model."],"forward_implications":["In the trilayer cuprate Bi2223, the dominant low-energy plasmon is the out-of-phase $\\omega_-$ mode, not the in-phase $\\omega_+$ mode that explains single- and infinite-layer data.","The nearly $q_z$-independent dispersion implies that inter-trilayer hopping is negligible and the charge dynamics are controlled by coupling within a single trilayer.","Because the 300 meV gap at the zone center is larger than the superconducting gap $2\\Delta$ on the three Fermi surfaces, the low-energy plasmon of Bi2223 sits above the pairing scale, unlike the acoustic plasmons in single-layer cuprates.","The acoustic $\\omega_3$ branch carries little spectral weight near the zone center, so RIXS at small in-plane momentum is a direct probe of the $\\omega_-$ branch in trilayer materials.","Temperature-dependent RIXS across $T_c$ shows no appreciable change in the plasmon lineshape at $q=(-0.1,0,-2)$, so any reconstruction of the charge response at the superconducting transition is not visible at this momentum."],"supporting_citations":[{"why":"Provides the trilayer RPA formulation and the $\\omega_+$, $\\omega_-$, $\\omega_3$ branch notation on which the calculation is built.","marker":"[18]"},{"why":"Reports the bilayer plasmon dispersion and branch assignment that the trilayer result is compared against.","marker":"[19]"},{"why":"Documents the gapped plasmon in infinite-layer cuprates whose gap mechanism (interlayer hopping) is contrasted with the intra-trilayer hopping gap here.","marker":"[14]"},{"why":"Gives the single-layer electron-doped cuprate plasmon with strong $q_z$ dispersion that the weakly $q_z$-dependent $\\omega_-$ branch is contrasted with.","marker":"[10]"},{"why":"Shows acoustic plasmons in hole-doped single-layer cuprates measured by O K-edge RIXS, establishing the measurement and the $\\omega_+$ behavior in the one-layer case.","marker":"[12]"},{"why":"Supplies the ARPES Fermi surface of optimally doped Bi2223 used to validate the tight-binding model parameters.","marker":"[24]"},{"why":"Provides the superconducting gap magnitudes on the three Fermi surfaces used to argue that the 300 meV plasmon gap exceeds $2\\Delta$.","marker":"[32]"}],"fun_headline_variants":["First omega- plasmon identified in trilayer cuprate","Out-of-phase plasmon dominates in trilayer cuprate","Trilayer cuprate's dominant plasmon is out-of-phase","RIXS reveals out-of-phase plasmon in Bi2223","Gapped out-of-phase plasmon dominates trilayer cuprate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The branch assignment rests on the specific model parameter set of Supplemental Material IV ($\\delta=0.18$, $t'=-0.26$, $t''=0.13$, $t_z=0.07$, $t_{z0}=0.1$, $V_c=280$, $\\Gamma=0.1$); if the interlayer hoppings and Coulomb strength are not fixed by data independent of the plasmon measurement, the computed 300 meV gap and the $\\omega_-$ identification could change.","fun_headline_variants_meta":{"raw":{"variants":["First omega- plasmon identified in trilayer cuprate","Out-of-phase plasmon dominates in trilayer cuprate","Trilayer cuprate's dominant plasmon is out-of-phase","RIXS reveals out-of-phase plasmon in Bi2223","Gapped out-of-phase plasmon dominates trilayer cuprate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000787,"raw_usage":{"total_tokens":3492,"prompt_tokens":986,"completion_tokens":2506,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":2422}},"tokens_in":602,"tokens_out":2506,"duration_ms":15969,"temperature":1.0,"reasoning_tokens":2422,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T00:46:51.912224+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Remeasure the plasmon branch along the $(H,H,L)$ path with higher momentum resolution to check whether the 300 meV gap and the near-$q_z$-independence survive, and independently fix $t_z$, $t_{z0}$, and $V_c$ from Fermi-surface and optical measurements; if the lowest gapped branch then moves far from 300 meV or its dominant spectral weight falls on $\\omega_+$ instead of $\\omega_-$, the central claim fails.","supporting_citations":[{"cited_title":"Greco, H","cited_arxiv_id":null,"evidence_quote":"Provides the trilayer RPA formulation and the $\\omega_+$, $\\omega_-$, $\\omega_3$ branch notation on which the calculation is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the gapped plasmon in infinite-layer cuprates whose gap mechanism (interlayer hopping) is contrasted with the intra-trilayer hopping gap here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the single-layer electron-doped cuprate plasmon with strong $q_z$ dispersion that the weakly $q_z$-dependent $\\omega_-$ branch is contrasted with."},{"cited_title":"Hepting, L","cited_arxiv_id":null,"evidence_quote":"Shows acoustic plasmons in hole-doped single-layer cuprates measured by O K-edge RIXS, establishing the measurement and the $\\omega_+$ behavior in the one-layer case."},{"cited_title":"Karppinen, S","cited_arxiv_id":null,"evidence_quote":"Supplies the ARPES Fermi surface of optimally doped Bi2223 used to validate the tight-binding model parameters."},{"cited_title":"Theory of charge dynamics in bilayer electron system with long-range Coulomb interaction","cited_arxiv_id":"2411.13650","evidence_quote":"Provides the superconducting gap magnitudes on the three Fermi surfaces used to argue that the 300 meV plasmon gap exceeds $2\\Delta$."}],"review_version":1}