Out-of-phase Plasmon Excitations in the Trilayer Cuprate Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$

A. Fujimori; A. Greco; D. J. Huang; D. Shiga; H. Eisaki; H. Kumigashira; H. Suzuki; H. Wadati; H. Yamase; H. Y. Huang

REVIEW 3 major objections 4 minor 41 references

Out-of-phase Plasmon Excitations in the Trilayer Cuprate Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2502.03779 v2 pith:KQHJRCX6 submitted 2025-02-06 cond-mat.str-el

S. Nakata , M. Bejas , J. Okamoto , K. Yamamoto , D. Shiga , R. Takahashi , H. Y. Huang , H. Kumigashira

show 9 more authors

H. Wadati J. Miyawaki S. Ishida H. Eisaki A. Fujimori A. Greco H. Yamase D. J. Huang H. Suzuki

This is my paper · ORCID

classification cond-mat.str-el PACS 71.45.Gm74.25.Gz78.70.Ck

keywords plasmoncupratesuperconductorstrilayerBi2223resonantinelasticx-rayscatteringchargesusceptibilitycollectiveexcitationsout-of-phasemode

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

Where 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.

Minor 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.

None 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Supplemental Material IV, Eq. (1) and Eqs. (10)-(12)] 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.
[Fig. 4 and SM Fig. S3] 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.
[Figs. 3 and 4] 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.

minor comments (4)

[Fig. 3 and main text] 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.
[Main text, after Eq. (13) area] 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.
[Fig. 4 and SM IV] 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.
[Main text, discussion of bilayer cuprates] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RIXS observation is independent, and the RPA branch assignment is a model comparison with stated input parameters, not a reduction of the result to its inputs.

full rationale

The measured ~300 meV gapped, nearly q_z-independent plasmon is an experimental RIXS result. The RPA susceptibility (SM Eqs. 1–13) is a standard many-body calculation with a stated parameter set (δ=0.18, t'=-0.26, t''=0.13, tz=0.07, tz0=0.1, Vc=280, Γ=0.1), and the experimental peak positions are overlaid on the computed intensity maps rather than used as fitting constraints in the equations shown. The ω− assignment follows from the model's three eigenmodes and their spectral weights; this is an interpretation of the data, not a definitional identity. The principal weakness is that tz, tz0, and Vc are not independently pinned down and no sensitivity analysis is provided, so the computed gap magnitude is contingent on those choices; that is a robustness/identifiability concern, not circularity. Self-citations (Refs. 16, 19, 30) supply background and bilayer context and are not load-bearing for the Bi2223 claim; no uniqueness theorem or ansatz is smuggled in via a self-citation, and no equation reduces to its input by construction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new microscopic particles, forces or dimensions are introduced. The omega+, omega- and omega3 branches are standard eigenmodes of the RPA multilayer model and were defined in prior work. The ledger is dominated by model parameters and domain assumptions: the quantitative gap and branch assignment depend on the untested combination of Vc=280, tz=0.07t, tz0=0.1t, doping, hopping parameters and broadening.

free parameters (6)

delta (doping) = 0.18
Carrier doping level of the tight-binding model; set to the nominal optimally doped level and not derived within the paper.
t' and t'' (in-plane hopping) = t'=-0.26t, t''=0.13t
Tight-binding hopping parameters chosen to reproduce ARPES Fermi surfaces; the mapping from ARPES data is not shown step by step.
tz (intra-trilayer hopping) = 0.07t (0.007 eV with t=0.1 eV)
Controls the gapped dispersion of the omega- branch and is central to the claimed ~300 meV gap; no independent determination is demonstrated.
tz0 (interlayer hopping term) = 0.1t
Additional interlayer hopping parameter that affects Fermi surface shape and mode energies; chosen to match ARPES qualitatively.
Vc (Coulomb coupling strength) = 280 (in units where t=0.1 eV)
Sets the overall plasmon energy scale; without a first-principles derivation, the 300 meV gap and branch energies depend strongly on this choice.
Gamma (broadening) = 0.1 (or 1.0 for broadened spectra)
Phenomenological broadening parameter; the SM states the choice can be arbitrary when mimicking correlation effects.

assumptions (5)

domain assumption The RPA susceptibility chi = (1 - chi0 Vq)^-1 chi0 describes the charge dynamics of optimally doped Bi2223 at energies near 300 meV.
Invoked in SM Section IV, Eq. (13); RPA neglects vertex corrections and short-range correlations, which the authors acknowledge when discussing the broad experimental linewidth.
domain assumption The O K-edge RIXS signal at 527.9 eV is dominated by charge scattering from the itinerant holes, so the measured intensity can be compared with -Im chi(q, omega).
Used to justify the RIXS measurement as a probe of the dynamical charge susceptibility; standard but not independently verified for this compound and energy range.
domain assumption Inter-trilayer electron hopping is negligibly small because the distance c' - 2d = 11.98 angstroms is large.
Invoked in the main text discussion of the n dependence; the model therefore includes only intra-trilayer hopping.
domain assumption The long-range Coulomb interaction between CuO2 layers has the Griffin-Pindor form with the parameter Vc=280.
SM Section IV adopts the interaction matrix from Ref. [4] of the SM; this form and strength are assumed rather than derived from the material's dielectric properties.
domain assumption A single-band tight-binding model with three CuO2 layers captures the low-energy electronic structure of Bi2223.
The model omits the outer-to-outer layer hopping within a trilayer, which the paper says may matter only in the overdoped region; this is a modeling choice.

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Cite this review

Pith. "Pith review of Out-of-phase Plasmon Excitations in the Trilayer Cuprate Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$." pith.science (2026). https://pith.science/paper/KQHJRCX6

@misc{pith2026250203779,
  author       = {Pith},
  title        = {Pith review of: Out-of-phase Plasmon Excitations in the Trilayer Cuprate Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_10+\delta$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQHJRCX6}},
  note         = {Machine review of arXiv:2502.03779}
}

abstract

Within a homologous series of cuprate superconductors, variations in the stacking of CuO$_2$ layers influence the collective charge dynamics through the long-range Coulomb interactions. We use O $K$-edge resonant inelastic x-ray scattering to reveal plasmon excitations in the optimally-doped trilayer Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$. The observed plasmon exhibits nearly $q_z$-independent dispersion and a large excitation gap of approximately 300 meV. This mode is primarily ascribed to the $\omega_{-}$ mode, where the charge density on the outer CuO$_2$ sheets oscillates out of phase while the density in the inner sheet remains unaltered at $q_z=0$. The intensity of the acoustic $\omega_3$ mode is relatively weak and becomes vanishingly small near $(q_x, q_y)=(0, 0)$. This result highlights a qualitative change in the eigenmode of the dominant low-energy plasmon with the number of CuO$_2$ layers.

Figures

Figures reproduced from arXiv: 2502.03779 by the authors.

**Figure 1.** (a) shows the crystal structure of Bi2223 and the key parameters that determine the plasmon dispersion. The CuO2 trilayers (light blue planes) are sep- (c) (b) (a) FIG. 2. (a) Measurement paths in the three-dimensional q space. (b) Representative O K-edge RIXS spectrum of Bi2223 at q = (-0.05, 0, -2.0). The spectrum includes the elastic line (blue), phonon (orange), plasmon (green), bimagnon (red), and dd excitatio… view at source ↗

**Figure 3.** (h). At this in-plane momentum, the plasmon lineshape is almost independent of L, and the peak energy shows only a weak dispersion from 0.51 eV at L = −2 to 0.53 eV at L = −3. This energy difference at these equivalent q vectors shows that the periodicity of the plasmon intensity is determined by the inter-trilayer distance c ′ , instead of the crystallographic lattice constant c [10]. This qz dispersion is signi… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Dynamical charge susceptibility computed within th [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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compounds La 2−xCexCuO4 [ 10, 11], La 2−xSrxCuO4 [12, 13], and Bi 2Sr1. 6La0. 4CuO6+δ (Bi2201) [ 12] are all acoustic-like, with a small energy gap in the limit (qx, q y) → (0, 0). In contrast, the inﬁnite-layer ( n = ∞) Sr0. 9La0. 1CuO2 has a gapped plasmon branch [ 14]. This energy gap arises from interlayer electron hopping [ 16], which brings the syst...

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This is because the RPA does not incorporate spectral broadening due to electron correlations, such as short- range antiferromagnetic correlations [ 27]

On the other hand, the calculated RPA charge susceptibility predicts the three sharp plasmon branches. This is because the RPA does not incorporate spectral broadening due to electron correlations, such as short- range antiferromagnetic correlations [ 27]. In the single- layer case, this eﬀect is discussed in the strong-coupling t- J-V model [ 28]. It can...

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