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Luciano Zinni, [Hiroyuki Yamase](https://orcid.org/0000-0003-0328-5657), Matthias Hepting, Matías Bejas, Andrés Greco

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[Strongly correlated model of acousticlike plasmons persisting across the phase diagram of cuprate superconductors](https://mdr.nims.go.jp/datasets/e81aa123-530f-49c3-9ace-065ceb35ff3f)

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Strongly correlated model of acousticlike plasmons persisting across the phasediagram of cuprate superconductorsLuciano Zinni,1 Hiroyuki Yamase,2, ∗ Matthias Hepting,3, † Mat́ıas Bejas,4 and Andrés Greco4, ‡1Facultad de Ciencias Exactas, Ingenieŕıa y Agrimensura (UNR-CONICET),Avenida Pellegrini 250, 2000 Rosario, Argentina2Research Center of Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan3Max-Planck-Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany4Facultad de Ciencias Exactas, Ingenieŕıa y Agrimensura and Instituto de F́ısica Rosario (UNR-CONICET),Avenida Pellegrini 250, 2000 Rosario, Argentina(Dated: June 8, 2026)Layered two-dimensional electron systems exhibit both optical and acousticlike plasmons aroundthe Brillouin-zone center. In the layered cuprate La2−xSrxCuO4, resonant inelastic x-ray scattering(RIXS) has detected corresponding acousticlike plasmons in a low-energy regime comparable to thatof other collective excitations associated with distinct regions of the cuprate phase diagram. Thisoverlap in energy scale raises the question of whether the acousticlike plasmons are significantlyinfluenced by phase-specific electronic phenomena, including the pseudogap, charge and spin order,superconductivity, and strange-metal behavior. Here we show that a single parameter set of thelayered t-J-V model, which incorporates strong correlations and the long-range Coulomb interactionV , consistently describes the acousticlike plasmon dispersion across all currently available RIXS datafrom the underdoped to the heavily overdoped regime. This transferability of a single parameter setexceeds that of earlier theoretical descriptions and supports a picture in which strong correlationspersist into the heavily overdoped regime, while the collective plasmon mode exhibits only limitedsensitivity to the phase-specific electronic phenomena that distinguish different regions of the phasediagram.Introduction. High-temperature cuprate superconduc-tors exhibit a rich phase diagram as a function of holedoping and temperature, including antiferromagnetism,the pseudogap, spin and charge order, d-wave supercon-ductivity, the strange-metal regime, and Fermi-liquid-liketransport behavior [1]. Characterizing how the associ-ated low-energy excitations evolve across this phase di-agram has been a major objective of both theory andexperiment, since these excitations encode informationabout the underlying electronic states and the extentto which different ordering tendencies compete or coex-ist [2–10].A powerful experimental technique to probe the low-energy spin and charge dynamics in cuprates across awide energy and momentum range is resonant inelasticx-ray scattering (RIXS) [11–13]. Specifically, RIXS atthe Cu L3-edge can map out dispersive spin excitations,commonly referred to as paramagnons in doped metal-lic cuprates [14–22], which were first found by inelasticneutron scattering to emanate from the Brillouin-zone(BZ) corner (π, π) [23]. Notably, the paramagnon moderetains significant spectral weight deep into the over-doped regime [24], although a crossover from collectivespin dynamics to a regime more closely resembling in-dividual particle-hole excitations occurs around optimaldoping [20–22]. Charge order with an incommensuratewavevector has been observed by RIXS in parts of theunderdoped phase diagram, whereas charge-density fluc-tuations may extend over a broader doping range [25–27]. RIXS results further indicate that these dynamiccharge correlations disappear progressively upon over-doping [28]. A recent resonant x-ray diffraction studyalso proposed the emergence of distinct charge orderwith a different wavevector and out-of-plane correlationin the heavily overdoped regime [29], although other ex-periments suggest that a similar diffraction signal couldinstead arise from oxygen vacancy ordering [30]. Thedoping evolution of both the paramagnon excitation andcharge order has also been extensively investigated theo-retically [31–42].A distinct type of charge excitation has been observedby Cu L3-, O K-, and Cu K-edge RIXS in the vicinity ofthe BZ center [43–54]. Although the origin was initiallycontroversial [55, 56], it is now established that these ex-citations correspond to acousticlike plasmons, character-istic of layered cuprates [46–54]. This type of plasmonsemerges in the presence of long-range Coulomb interac-tion in a layered structure, while their dispersion is fur-ther shaped by interlayer electron hopping. Specifically,the generic plasmon spectrum in a layered electron sys-tem depends not only on the in-plane momentum transferq∥, but also on the out-of-plane momentum qz [57–60].For qz = 0, one obtains the optical plasmon branch witha conventional quadratic dispersion, observed in earlyelectron-energy loss spectroscopy studies [61, 62]. In thepresence of interlayer hopping tz [50, 63], this minimumis shifted to finite energy, rendering the branches acousti-clike rather than purely acoustic. For the theoretical de-scription of plasmons in cuprates, a variety of approacheshas been employed [63–74].2RIXS measurements on the prototypical cuprate su-perconductor La2−xSrxCuO4 (LSCO) [48, 49, 51], wherethe Sr substitution x is equivalent to the hole doping δ,have reported acousticlike plasmon excitations at variouslocations in the phase diagram (Fig. 1). These measure-ments span the pseudogap, superconducting, strange-metal, and Fermi-liquid regimes, and were performed onboth thin-film and single-crystal samples at the OK- andCu K-edges. Most recently, a detailed high-resolutionRIXS data set was reported for heavily overdoped LSCOwith δ = 0.35 [75], measured at T = 40 K deep in theFermi-liquid regime, where spin fluctuations are stronglydamped and the pseudogap as well as superconductivityare absent [17–19, 24, 76–78].In this Letter, we compile the available RIXS data onLSCO between δ = 0.05 and 0.40 [48, 49, 51, 75] andmodel the experimentally reported plasmon dispersionusing the layered t-J-V model, which extends the stan-dard two-dimensional t-J model to incorporate both in-terlayer hopping and long-range Coulomb interaction Vin a form that respects the lattice structure [63]. Notably,we find that a single microscopic parameter set of the t-J-V model, previously determined for optimally dopedLSCO (δ = 0.16) [50], provides a consistent descriptionof all reported plasmon dispersions across the phase di-agram when only adjusting the hole doping δ and thetemperature T . This degree of transferability and con-sistency surpasses that of other theoretical descriptionsof the plasmon dispersion in cuprates, including the ran-dom phase approximation (RPA) models [66, 67, 75]. Itfurther suggests that strong correlations, which are in-herently included in the t-J-V model, persist in the over-doped regime and that, unlike many other physical ob-servables [79], the acousticlike plasmon remains a robustcollective mode across different hole doping and temper-ature regions, with its dispersion showing only limitedsensitivity to the electronic states in the cuprate phasediagram.Model and formalism. To describe plasmon excitationsin LSCO, we employ the layered t-J-V model on a squarelattice,H =−∑i,j,σtij c̃†iσ c̃jσ +∑⟨i,j⟩Jij(S⃗i · S⃗j −14ninj)+12∑i,ji̸=jVijninj , (1)where c̃†iσ (c̃iσ) creates (annihilates) an electron with spinσ in the Hilbert space without double occupancy. Thehopping amplitudes tij include nearest-neighbor (t) andnext-nearest-neighbor (t′) processes within each CuO2plane, as well as interlayer hopping tz. The exchange in-teraction Jij = J acts between nearest neighbors withinthe planes, and Vij denotes the long-range Coulomb in-teraction. ni =∑σ c̃†iσ c̃iσ and S⃗i are the electron den- 0 50 100 150 200 250 300 350 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4  0.45ASCPseudogapStrange metalFermi liquidSC (thin films)FRIXSRef. [51]Ref. [49]Ref. [48]Ref. [75]Temperature [K]Hole doping δFIG. 1. Schematic phase diagram of La2−xSrxCuO4 as afunction of hole doping and temperature, compiled fromRefs. [51, 80–84]. Colored regions indicate the antiferro-magnetic (AF) phase, the pseudogap regime (orange) withits crossover region (dark orange), the strange-metal regime(green) with its crossover region (dark green), and the regimeof Fermi-liquid-like transport behavior (blue). The supercon-ducting (SC) dome is shown by the solid red line for bulksamples and by the dashed red line for thin films. Symbolsmark the locations in the phase diagram of RIXS measure-ments: purple triangles correspond to O K-edge RIXS on thinfilms in Ref. [51], the yellow square to O K-edge RIXS on asingle crystal in Ref. [49], the red diamond to O K-edge RIXSon a single crystal in Ref. [48], the pink triangle to Cu K-edgeRIXS on a single crystal in Ref. [51], and the black circle toO K-edge RIXS on a thin film in Ref. [75].sity and spin operators, respectively. The non-double-occupancy constraint is treated via a path-integral for-mulation of Hubbard operators within a large-N expan-sion (technical details are given in the Supplemental Ma-terial [85]; see also Refs. [86–89] therein). The quasipar-ticle dispersion εk = ε∥k + ε⊥k is given byε∥k =− 2(tδ2+ ∆)(cos kx + cos ky)− 4t′δ2cos kx cos ky − µ , (2)ε⊥k = −2tzδ2(cos kx − cos ky)2 cos kz , (3)where δ/2 originates from strong correlations and con-trols the effective bandwidth and interlayer hopping, andthe bond field ∆ and chemical potential µ are determinedself-consistently [85]. The long-range Coulomb interac-tion consistent with a layered square lattice [90] takesthe formV (q) =VcA(qx, qy)− cos qz, (4)300.20.40.60.81.00 0.1(a) δ = 0.05<L> = 0.740 0.1(b) δ = 0.10<L> = 0.740 0.1 0.2(c) δ = 0.12L = 0.70 0.1<L> = 0.74L = 0.6L = 0.8L = 1(d) δ = 0.1600.20.40.60.81.00 0.1<L> = 0.74L = 13K = H(e) δ = 0.200 0.1(f) δ = 0.30<L> = 0.740 0.1(g) δ = 0.35<L> = 1.080 0.1H [r.l.u.]Plasmon energy [eV](h) δ = 0.40<L> = 0.74FIG. 2. In-plane plasmon dispersion. (a-h) Open symbols are plasmon energies determined in RIXS measurements on LSCOwith hole doping δ = 0.05, 0.10, 0.12, 0.16, 0.20, 0.30, 0.35, and 0.40, respectively. Measurements were performed by varying themomentum transfer along the in-plane H direction, while the out-of-plane momentum L was fixed. For experiments where theout-of-plane momentum was not strictly fixed during H variation, the label ⟨L⟩ denotes the average out-of-plane momentum.The data labeled with H = K in panel (e) were measured for in-plane momentum transfer along the diagonal direction, whereasin all other measurements K = 0 was fixed. Solid lines are plasmon dispersions calculated with the t-J-V model, using thesame set of microscopic parameters (see text) determined for LSCO with δ = 0.16 in Ref. [50], while adjusting only δ andthe temperature according to the corresponding experiment. RIXS data in panel (a) and (b) are from Ref. [51], in (c) fromRef. [49], in (d) from Refs. [48, 51], in (e) from Ref. [51], in (f) from Ref. [51], in (g) from Ref. [75], and in (h) from Ref. [51].with Vc = e2d/(2ε⊥a2) and A(qx, qy) = α(2 − cos qx −cos qy) + 1. Here α = ε̃/(a/d)2 with ε̃ = ε∥/ε⊥, whereε∥ and ε⊥ are the dielectric constants parallel and per-pendicular to the CuO2 planes, respectively. The latticeconstant within the planes is denoted by a, the interlayerspacing by d, and e is the electron charge. Within thet-J-V framework, the charge-charge correlation functionis obtained at leading order asχc(q, iωn) = N(δ2)2D11(q, iωn) , (5)whereD11 is the (1, 1) element of the 6×6 dressed bosonicpropagator obtained from the Dyson equation [85], andq and iωn are the momentum transfer and bosonic Mat-subara frequency, respectively. After analytical continu-ation iωn → ω+iΓ and setting the physical value N = 2,we compute Imχc(q, ω), which is directly compared withRIXS intensity maps.Results. We now examine whether a coherent descrip-tion of the plasmon dispersion can be achieved across theentire doping range measured with RIXS. To this end, weadopt the parameter set previously introduced for opti-mally doped LSCO (δ = 0.16) in Ref. [50]: t′/t = −0.2,tz/t = 0.01, Vc/t = 31, α = 3.5, Γ/t = 0.1, and J/t = 0.3,with t/2 = 0.35 eV. Since in the large-N formalism theoriginal Hamiltonian hopping t is scaled to t/N , andN = 2 is set at the end of the calculation [35, 63], thenatural unit of energy is t/2, consistent with the hoppingexpected for cuprates (see Supplemental Material [85]).Figure 2 compares the measured in-plane plasmon dis-persions for eight hole dopings between δ = 0.05 and0.40 (open symbols) with the corresponding t-J-V calcu-lations (solid lines). For each calculated dispersion, onlythe hole doping, the temperature, and the momentumtransfer (H,K,L) are adjusted to match the experimen-tal conditions. The momentum coordinates are expressedin reciprocal lattice units (r.l.u.).Overall, the calculated in-plane dispersions are in goodagreement with the RIXS data throughout the full dopingrange. This includes cases in which the dispersion at fixedδ was measured for different values of L [Figs. 2(d),(e)], aswell as measurements performed at the O K- and the CuK-edges, and on both thin-film and single-crystal sam-ples (see also Fig. 1). In the overdoped regime, however,the RIXS data for δ = 0.30 and 0.40 reported in Ref. [51]deviate somewhat from the calculated t-J-V dispersions,particularly at large H. By contrast, the high-resolutionRIXS data for δ = 0.35 reported in Ref. [75] agree verywell with the calculated dispersion at every finiteH. Thispoint will be further discussed below.Figure 3 provides the complementary comparison forthe out-of-plane plasmon dispersion. Similarly to the in-400.20.40.60.81.01.20 0.5 1.0(a) δ = 0.12H = 0.10 0.5 1.0H = 0.03H = 0.05H = 0.08(b) δ = 0.1600.20.40.60.81.01.212 13 14(c) δ = 0.20H = 0.090 0.5 1.0L [r.l.u.]Plasmon energy [eV]H=−0.021H=−0.041(d) δ = 0.35FIG. 3. Out-of-plane plasmon dispersion. (a-d) Open sym-bols are plasmon energies determined in RIXS measurementson LSCO with hole dopings δ = 0.12, 0.16, 0.20, and 0.35,respectively. Measurements were performed by varying themomentum transfer along the out-of-plane L direction, whilethe in-plane momentum H was fixed. Solid lines are t-J-Vmodel calculations, using the same set of microscopic param-eters determined for LSCO with δ = 0.16 in Ref. [50], whileadjusting only δ and the temperature. RIXS data in panel (a)are from Ref. [49], in (b) from Ref. [48], in (c) from Ref. [51],and in (d) from Ref. [75].plane dispersion, the overall agreement between exper-iment and theory is satisfactory for all measured holedopings, δ = 0.12, 0.16, 0.20, and 0.35, with small devi-ations occurring for δ = 0.12 and 0.20 [Figs. 3(a),(c)].Taken together, Figs. 2 and 3 illustrate that the samemicroscopic parameter set captures the main featuresof the three-dimensional plasmon dispersion reported invarious RIXS experiments, without tuning model pa-rameters individually for each doping or introducingphase-specific corrections. This result is especially sig-nificant in the overdoped regime, where the continuedrelevance of strong correlations has been debated [17–19, 24, 51, 54, 91], signaling that a strongly correlatedframework retains predictive power even in that limit.Discussion. Notably, our results differ qualitativelyfrom those obtained within a random-phase approxima-tion (RPA) treatment. While RPA can reproduce theplasmon dispersion at a given doping level after intro-ducing ad hoc renormalized band parameters [54], thesame parameter set does not remain valid when the holedoping is changed. Specifically, as the hole doping movesaway from the value for which the RPA calculation wasadjusted, the discrepancy with experiment becomes in-creasingly pronounced (see Supplemental Material [85]).By contrast, the layered t-J-V model captures the plas-mon dispersion across the available doping range using asingle microscopic parameter set.This difference reflects how doping enters the two de-scriptions. In the t-J-V framework, the quasiparticle dis-persion contains the correlation-induced renormalizationfactor δ/2 in Eqs. (2) and (3), which controls both theeffective bandwidth and interlayer hopping. The plas-mon dispersion, however, is not set by the band structurealone. It emerges from the poles of the dressed chargepropagator D11(q, ω), whose doping dependence is gov-erned by the Dyson equation and the bosonic self-energyΠab(q, ω), which carries a nontrivial doping dependencebeyond the δ/2 factor in the quasiparticle dispersion (formore details, see Supplemental Material [85]). The factthat the same t-J-V parameter set describes the plasmondispersion across the full doping range thus demonstratesthat the collective charge response encoded in D11(q, ω)evolves consistently. This indicates that the t-J-V frame-work provides a more unified description of cuprate plas-mons than an RPA treatment, even if RPA can be ad-justed to reproduce individual cases.The ability of a single t-J-V parameter set to accountfor the plasmon dispersion across the cuprate phase di-agram has important physical implications. It revealsthat the acousticlike plasmon is not primarily controlledby the degrees of freedom that distinguish the pseudogap,strange-metal, and Fermi-liquid regimes, even though theacousticlike branches disperse down to energies compara-ble to those of the pseudogap, charge-density-wave fluc-tuations, spin excitations, and the superconducting gap.In particular, the measured plasmon dispersion remainsquantitatively accounted for as the system evolves from aregime with strong antiferromagnetic correlations, pseu-dogap, and superconducting ground state to one in whichthese features are absent or substantially weakened. Thisindicates that, for the description of the currently avail-able RIXS data, the leading-order t-J-V formalism re-mains adequate without explicit inclusion of pseudogapformation, superconducting pairing, or charge ordering.Nonetheless, as mentioned above, the RIXS data forδ = 0.30 and 0.40 reported in Ref. [51] deviate somewhatfrom the calculated t-J-V dispersion, especially at largeH [Figs. 2(f),(h)], whereas the more recent δ = 0.35 dataof Ref. [75] are captured very well [Fig. 2(g)]. In Ref. [51],this discrepancy was discussed in terms of three possi-ble scenarios: (i) enhanced structural disorder in over-doped samples, (ii) an overestimation of correlation ef-fects within the t-J-V model at high doping where thenon-double-occupancy constraint and nearest-neighborexchange become less restrictive, and (iii) the increas-ing relevance of additional orbital degrees of freedom,including Cu 3d3z2−r2 and 4s. The excellent agreementwith the more comprehensive and higher-resolution dataof Ref. [75], however, suggests that the latter two scenar-ios do not need to be invoked given the newly availableδ = 0.35 data, and that the t-J-V framework is capable5of describing the plasmon dispersion even in the heavilyoverdoped regime.While the overall agreement shows that the t-J-Vframework is transferable across the available dopingrange, exact quantitative agreement across all compiledRIXS data should not be expected, because real samplesdiffer in additional microscopic details beyond δ. In par-ticular, the in-plane and out-of-plane lattice constantsvary substantially between δ = 0.05 and 0.40 and differbetween bulk and thin film samples, which in principleshould affect the Coulomb interaction in the t-J-V frame-work through α = ε̃/(a/d)2. Residual deviations such asthose in Figs. 2(f),(h), and possibly also the mismatchin the out-of-plane scan of the single crystal in Fig. 3(c),are therefore naturally attributable to sample-dependentchanges and/or sample-quality issues not captured whenδ is taken as the only doping-dependent input. Accord-ingly, they should not be interpreted as an indication fora breakdown of the t-J-V description.Furthermore, the demonstrated robustness of the t-J-V description does not exclude weak modifications of theplasmon across particular electronic transitions. Specif-ically, optical measurements on bismuth-based cuprateshave detected subtle renormalization effects on the op-tical plasmon across Tc [92]. It can therefore be ex-pected that the opening of the superconducting gap simi-larly affects the low-energy acousticlike plasmon branchesin LSCO, although this has not yet been resolved byRIXS and may require improved energy resolution andfiner temperature sampling compared to previous exper-iments. In addition, in layered nickelates a hardeningof the plasmon energy, which might be related to stripefluctuations, was observed [75]. This suggests that chargeand spin stripes in the underdoped regime of LSCO couldlikewise weakly affect the plasmon dispersion.Conclusion. In summary, we have shown that the lay-ered t-J-V model provides a consistent account of themeasured acousticlike plasmon dispersion in LSCO fromthe underdoped to the heavily overdoped regime usinga single microscopic parameter set while varying onlythe hole doping δ. This agreement includes the latestavailable RIXS data on the three-dimensional plasmondispersion in LSCO [75], and therefore supports a uni-fied description of the plasmon throughout the cupratephase diagram. Taken together, our results indicate thatthe low-energy acousticlike plasmon is a robust collectivecharge excitation of the correlated electron system, onlyweakly affected by the electronic phenomena that distin-guish the different regions of the cuprate phase diagram.Acknowledgments. We thank M. P. M. Dean and hiscollaborators for insightful discussions, and for makingthe LSCO RIXS data from Ref. [75] available prior topublication. We are also grateful to M. P. M. Dean fora careful reading of an early version of the manuscript.We further thank M. Minola and C. Falter for valuablediscussions. 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(1) inthe main text]. For a more comprehensive derivation ofthe formalism see Ref. [63]; here we summarize the keyexpressions needed to obtain the charge-charge correla-tion function χc(q, ω) presented in Eq. (5) in the maintext.Effective bosonic theoryThe starting point is the representation of the con-strained electronic operators in terms of Hubbard X̂-operators, which satisfy non-standard commutation rulesand are defined as X̂αβi = |iα⟩⟨iβ| with |α⟩ = |0⟩, | ↑⟩, | ↓⟩ [86]. In this basis, for instance, it follows thatc̃†iσ = Xσ0i and c̃iσ = X0σi . Besides, the non-double-occupancy constraint is encoded in the completeness re-lation X00i +∑σ Xσσi = 1, and the Hamiltonian [Eq. (1)in the main text] becomes quadratic in the X̂-operators.A path-integral treatment based on the Faddeev-Jackiw method [87] maps this problem onto an effec-tive field theory. After extending the spin index from2 to N and performing an expansion in powers of 1/N ,the resulting effective Lagrangian describes a theory offermions, bosons, and their interactions. The fermionicsector yields the quasiparticle dispersion εk given inEqs. (2) and (3) of the main text, where the hoppingintegrals acquire the correlation-induced renormalizationfactor δ/2 and the bond field ∆. The bosonic sector iscaptured by a six-component fieldδXi = (δRi, δλi, rxi , ryi , Axi , Ayi ) , (A1)whose components have the following physical origin.The field δRi parametrizes fluctuations of the on-sitehole doping δ around its mean-field value through X00i =N δ2 (1 + δRi). The field δλi represents fluctuations ofthe Lagrange multiplier that enforces the non-double-occupancy constraint beyond mean field. The remain-ing four components arise from a Hubbard-Stratonovichdecoupling of the exchange interaction J : the bond vari-able along the direction η = x, y is written as ∆ηi =∆(1+rηi +iAηi ), so that rηi and Aηi represent fluctuationsof its real and imaginary parts, respectively. Here ∆ isthe static mean-field bond-order parameter, determinedself-consistently together with the chemical potential µthrough∆ =J4Ns∑k(cos kx + cos ky)nF (εk) , (A2)(1− δ) =2Ns∑knF (εk) , (A3)where Ns is the total number of lattice sites and nF isthe Fermi-Dirac distribution function.The quadratic part of the Lagrangian in δXi definesa bare bosonic propagator D(0)ab (q, iωn), whose inversereads[D(0)ab (q, iωn)]−1 = Nδ22 [V (q)−J(q)] δ2 0 0 0 0δ2 0 0 0 0 00 0 4∆2J 0 0 00 0 0 4∆2J 0 00 0 0 0 4∆2J 00 0 0 0 0 4∆2J, (A4)2with matrix indices a, b = 1, . . . , 6 corresponding to thecomponents in Eq. (A1), and J(q) = J2 (cos qx + cos qy).The overall factor ofN implies thatD(0)ab is O(1/N). Notethat [D(0)ab ]−1 is frequency-independent and that the long-range Coulomb interaction V (q) [Eq. (4) of the maintext] enters exclusively in the (1, 1) element and is re-sponsible for the emergence of collective plasmon modes.Dressed propagator and bosonic self-energyThe dressed bosonic propagator is obtained throughthe Dyson equationD−1ab (q, iωn) = [D(0)ab (q, iωn)]−1 −Πab(q, iωn) , (A5)where Πab is the 6 × 6 bosonic self-energy. Its explicitform isΠab(q, iωn) = − NNs∑kha(k,q, εk−εk−q)× nF (εk−q)− nF (εk)iωn − εk + εk−qhb(k,q, εk−εk−q)− δa1δb1NNs∑kε̃k − ε̃k−q2nF (εk) , (A6)where ε̃k is equal to εk with ∆ = 0, and the six-component vertex isha(k,q, ν) ={2εk−q + ν + 2µ2+ 2∆[cos(kx− qx2)cos qx2 + cos(ky− qy2)cosqy2];1; −2∆ cos(kx− qx2); −2∆ cos(ky− qy2);2∆ sin(kx− qx2); 2∆ sin(ky− qy2)}. (A7)A distinctive feature of this vertex is that it does not orig-inate solely from the interactions in the Hamiltonian: thenon-trivial algebra of the Hubbard X̂-operators and theenforcement of the local constraint both contribute to itsstructure. In particular, the first component (a = 1) car-ries an explicit dependence on the fermionic frequencythrough ν, while the remaining five components dependonly on momenta. The out-of-plane momenta kz and qzenter exclusively through εk−q in the a = 1 component;the other components involve only the in-plane momen-tum q∥.Charge-charge correlation functionThe full propagator Dab(q, iωn) encodes the completespectrum of charge fluctuations in the t-J-V model [88].00.20.40.60.81.01.20-0.3 -0.2 -0.1 0.1 0.2 0.3Plasmon energyExpω [eV]H [r.l.u.]0>0.3Imχ(q,ω)K = 0L = 1 FIG. S1. Energy-momentum maps of Imχc(q, ω) for δ = 0.35computed within the t-J-V model by employing the same pa-rameter set previously determined for optimally doped LSCO(δ = 0.16) in Ref. [50]. White dots represent the experi-mental plasmon energies extracted from Ref. [75] on a non-superconducting LSCO sample at δ = 0.35; the bars are theplasmon width. Green lines indicate the theoretical plasmondispersion.Its structure reflects two physically distinct sectors: the2 × 2 charge sector (a, b = 1, 2), which describes usualcharge fluctuations, including plasmon modes; and the4× 4 bond sector (a, b = 3–6), which originates from theJ term and gives rise to low-energy charge excitations.However, both sectors are essentially decoupled [88], andto obtain plasmons it is sufficient to focus only on the2× 2 charge sector.The standard charge-charge correlation functionχc(ri − rj , τ) = ⟨Tτni(τ)nj(0)⟩ is written in the large-Nframework asχc(ri − rj , τ) =1N∑pq⟨TτXppi (τ)Xqqj (0)⟩ . (A8)Using the completeness condition∑p Xppi = N/2−X00iand X00i = N δ2 (1 + δRi), the charge-charge correlationfunction can be expressed in Fourier space asχc(q, iωn) = N(δ2)2D11(q, iωn) . (A9)While χc is formally of O(1), its evaluation requires allO(1/N) contributions entering D11 through the Dysonequation (A5). The retarded response is obtained viathe analytical continuation iωn → ω + iΓ and by settingN = 2. The broadening parameter Γ > 0 effectivelyincorporates both finite experimental resolution and in-trinsic incoherent scattering processes [89]. The resultingspectral function Imχc(q, ω) can be compared directly toRIXS intensity maps. We note that in the large-N formal-ism the hopping parameters and interaction strengths arescaled by 1/N [35]. When setting N to its physical valuefor comparison with experiments, the natural unit of en-ergy becomes t/2. This is the origin of the energy scale300.20.40.60.81.00 0.1(a) δ = 0.05<L> = 0.740 0.1(b) δ = 0.10<L> = 0.740 0.1 0.2(c) δ = 0.12L = 0.70 0.1<L> = 0.74L = 0.6L = 0.8L = 1(d) δ = 0.1600.20.40.60.81.00 0.1<L> = 0.74L = 13 - K = H(e) δ = 0.200 0.1(f) δ = 0.30<L> = 0.740 0.1(g) δ = 0.35<L> = 1.080 0.1H [r.l.u.]Plasmon energy [eV](h) δ = 0.40<L> = 0.74FIG. S2. In-plane plasmon dispersion in LSCO with dispersions calculated within RPA using a single parameter set fixed tothe δ = 0.35 data. Open symbols are the same plasmon energies determined by RIXS as shown in Fig. 2 of the main text. Solidlines are RPA dispersions obtained with the parameter set reported in Ref. [75], which was introduced to describe the δ = 0.35data and is applied here unchanged to all dopings. While this parameter set reproduces the δ = 0.35 dispersion [panel (g)], itdoes not provide a comparably consistent description across the full doping range.t/2 = 0.35 eV employed in the main text. Finally, wemention that spin fluctuations do not appear at O(1/N)in the present formalism.PLASMON DISPERSION OVER THE FULLH-SCAN AT δ = 0.35In Fig. 2(g) of the main text, the plasmon dispersionat δ = 0.35 is shown for positive values of H so as to di-rectly compare with all available RIXS data from otherexperiments. Figure S1 extends this comparison to thefull H-scan reported in Ref. [75], demonstrating that thet-J-V model captures well the experimental dispersionover the entire range of in-plane momenta, including neg-ative H. As for all other dopings across the phase dia-gram, this agreement is achieved without any parameteradjustment: the same microscopic parameter set previ-ously determined for optimally doped LSCO (δ = 0.16)is employed throughout.RANDOM PHASE APPROXIMATION ANALYSISWe performed a complementary analysis within thestandard random phase approximation (RPA). The RPAcharge susceptibility is given byχRPA(q, ω) =χ0(q, ω)1− V (q)χ0(q, ω), (A10)where V (q) is the long-range Coulomb interaction[Eq. (4) of the main text] and χ0(q, ω) is the Lindhardfunction computed from a tight-binding dispersion. Incontrast to the t-J-V approach, the RPA employs a bareelectronic dispersion Ek = E∥k + E⊥k , withE∥k = −2t(cos kx +cos ky)− 4t′ cos kx cos ky −µ , (A11)andE⊥k = −2tz(cos kx − cos ky)2 cos kz , (A12)where hopping amplitudes are treated as doping-independent constants [54], and the charge-fluctuationvertex reduces to unity—in contrast to the more com-plex vertex of the t-J-V formalism, which encodes thenon-canonical algebra of Hubbard operators and the non-double-occupancy constraint.It is well established that RPA with bare band param-eters yields a doping dependence of the plasmon energyopposite to experiment [51], and that a quantitative RPAdescription at a given doping requires the use of ad hocrenormalized band parameters [54]. Figure S2 shows theRPA dispersion obtained with the parameters of Ref. [75]400.20.40.60.81.00 0.1(a) δ = 0.05<L> = 0.740 0.1(b) δ = 0.10<L> = 0.740 0.1 0.2(c) δ = 0.12L = 0.70 0.1<L> = 0.74L = 0.6L = 0.8L = 1(d) δ = 0.1600.20.40.60.81.00 0.1<L> = 0.74L = 13 - K = H(e) δ = 0.200 0.1(f) δ = 0.30<L> = 0.740 0.1(g) δ = 0.35<L> = 1.080 0.1H [r.l.u.]Plasmon energy [eV](h) δ = 0.40<L> = 0.74FIG. S3. In-plane plasmon dispersion in LSCO with dispersions calculated for RPA parameters adjusted for δ = 0.16. (a-h)Open symbols are the same plasmon energies determined by RIXS that are given in Fig. 2 of the main text. Solid lines areplasmon dispersions calculated by RPA, using the parameters adjusted such that the RIXS data at δ = 0.16 are matched.(t = 0.39 eV, t′/t = −0.3, tz/t = 0.017, Vc/t = 15,α = 3.5, and Γ/t = 0.013), which were determined tofit the data at δ = 0.35. While the RPA reproduces theplasmon dispersion at that doping, it systematically over-estimates the plasmon energy at lower dopings and failsto capture the momentum dependence across the phasediagram, as anticipated in Ref. [75]. Figure S3 presents arevised RPA calculation with parameters determined tofit the data at δ = 0.16 (t = 0.25 eV, tz/t = 0.005, andΓ/t = 0.02, with the remaining parameters unchanged).This parameter set recovers good agreement at δ = 0.16,confirming that the RPA should not be dismissed as apractical tool for describing plasmons at any given dop-ing, provided the band parameters are appropriately ad-justed [54]. However, the same parameters cannot de-scribe the dispersion across the other dopings, exposingthe fundamental limitation of the RPA approach: the in-ability to transfer a single parameter set across differentdoping regimes. By contrast, the t-J-V results presentedin the main text agree well with the experimental dataover the full doping and momentum range using a sin-gle set of microscopic parameters, consistent with thefact that the plasmon energy scale is governed by thecorrelation-renormalized electronic structure rather thanby the bare band parameters alone.TEMPERATURE DEPENDENCE OF THEPLASMON DISPERSIONIn the main text, all t-J-V model calculations were per-formed at the temperature corresponding to the respec-tive RIXS experiment for each doping. Here we examinethe temperature dependence of the computed plasmondispersion in more detail.RIXS studies on LSCO found that the plasmon exci-tation exhibits only a marginal temperature dependencefor doping levels δ = 0.16, 0.2, 0.3, and 0.35 in the tem-perature range between 15 and 300 K [51, 75]. Withinthe t-J-V model, previous calculations near optimal dop-ing and within a similar temperature range have alsoshown weak sensitivity of the plasmon energy [63]. How-ever, the situation is qualitatively different at low dop-ing. Figure S4 shows the computed plasmon dispersionalong the in-plane H direction for several dopings be-tween δ = 0.05 and 0.40 at representative temperatures.For low doping levels, the plasmon energy exhibits an ap-preciable temperature dependence compared to the op-timal and overdoped cases, with the plasmon energy de-creasing upon increasing temperature, and the effect be-ing most pronounced at large H. This enhanced sensitiv-ity may be understood from the fact that at low dopingthe correlation-renormalized bandwidth, which scales asδ/2, is narrow, so that thermal effects produce a strongershift of the plasmon pole in D11(q, ω). For δ ≥ 0.16,the effective bandwidth is sufficiently large so that tem-500.20.40.60.81.00 0.1 0.2(a) δ = 0.05K = 0L = 0.740 0.1 0.2T = 0 KT = 150 KT = 300 K(b) δ = 0.100 0.1 0.2(c) δ = 0.120 0.1 0.2 0.3(d) δ = 0.1600.20.40.60.81.00 0.1 0.2(e) δ = 0.200 0.1 0.2(f) δ = 0.300 0.1 0.2(g) δ = 0.350 0.1 0.2 0.3H [r.l.u.]Plasmon energy [eV](h) δ = 0.40FIG. S4. Temperature dependence of the computed plasmon dispersion along the in-plane H direction for several dopingsbetween δ = 0.05 and 0.40 at fixed K = 0 and L = 0.74. Solid lines correspond to the plasmon dispersion calculated with thet-J-V model. The same microscopic parameter set as in the main text is employed. Blue, green and red lines are for T = 0 K,T = 150 K, and T = 300 K, respectively.perature variations induce only negligible changes in theplasmon dispersion, consistent with the experimental ob-servations in Refs. [51, 75].∗ yamase.hiroyuki@nims.go.jp† m.hepting@fkf.mpg.de‡ agreco@fceia.unr.edu.ar[1] B. Keimer, S. A. Kivelson, M. 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