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[Abhishek Nag](https://orcid.org/0000-0002-1394-5105), [Luciano Zinni](https://orcid.org/0000-0002-9165-9251), Jaewon Choi, J. Li, [Sijia Tu](https://orcid.org/0009-0003-6260-6534), [A. C. Walters](https://orcid.org/0000-0001-9822-3859), S. Agrestini, [S. M. Hayden](https://orcid.org/0000-0002-3209-027X), [Matías Bejas](https://orcid.org/0000-0003-4254-0622), Zefeng Lin, [H. Yamase](https://orcid.org/0000-0003-0328-5657), [Kui Jin](https://orcid.org/0000-0003-2208-8501), [M. García-Fernández](https://orcid.org/0000-0002-6982-9066), [J. Fink](https://orcid.org/0000-0002-5286-1684), [Andrés Greco](https://orcid.org/0000-0001-5958-5080), Ke-Jin Zhou

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[Impact of electron correlations on two-particle charge response in electron- and hole-doped cuprates](https://mdr.nims.go.jp/datasets/299ef475-f289-4d72-a253-3650086297b9)

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Impact of electron correlations on two-particle charge response in electron- and hole-doped cupratesPHYSICAL REVIEW RESEARCH 6, 043184 (2024)Impact of electron correlations on two-particle charge response in electron- and hole-doped cupratesAbhishek Nag ,1,2,3,* Luciano Zinni ,4 Jaewon Choi,1,5 J. Li,1,6 Sijia Tu ,7,8 A. C. Walters ,1 S. Agrestini,1S. M. Hayden ,9 Matías Bejas ,10 Zefeng Lin,7,8 H. Yamase ,11 Kui Jin ,7,8 M. García-Fernández ,1J. Fink ,12,13,† Andrés Greco ,10,‡ and Ke-Jin Zhou1,§1Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom2SwissFEL, Paul Scherrer Institute, 5232, Villigen-PSI, Switzerland3Department of Physics, Indian Institute of Technology Roorkee, Uttarakhand 247667, India4Facultad de Ciencias Exactas, Ingeniería y Agrimensura (UNR), Avenida Pellegrini 250, 2000, Rosario, Argentina5Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Daejeon 34141, Republic of Korea6National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA7Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China8University of Chinese Academy of Sciences, Beijing 100049, China9H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom10Facultad de Ciencias Exactas, Ingeniería y Agrimensura and Instituto de Física Rosario (UNR-CONICET),Avenida Pellegrini 250, 2000, Rosario, Argentina11Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan12Leibniz Institute for Solid State and Materials Research Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany13Institut für Festkörperphysik, Technische Universität Dresden, D-01062 Dresden, Germany(Received 20 July 2024; accepted 6 November 2024; published 21 November 2024)Estimating many-body effects that deviate from an independent particle approach has long been a keyresearch interest in condensed matter physics. Layered cuprates are prototypical systems, where electron-electroninteractions are found to strongly affect the dynamics of single-particle excitations. It is, however, still unclearhow the electron correlations influence charge excitations, such as plasmons, which have been variously treatedwith either weak or strong correlation models. In this work, we demonstrate the hybridized nature of collectivevalence charge fluctuations leading to dispersing acoustic-like plasmons in hole-doped La1.84Sr0.16CuO4 andelectron-doped La1.84Ce0.16CuO4 using the two-particle probe, resonant inelastic x-ray scattering. We thendescribe the plasmon dispersions in both systems, within both the weak-coupling mean-field random phaseapproximation (RPA) and strong-coupling t-J-V model in a large-N scheme. The t-J-V model, which includesthe correlation effects implicitly, accurately describes the plasmon dispersions as resonant excitations outsidethe single-particle intraband continuum. In comparison, a quantitative description of the plasmon dispersionin the RPA approach is obtained only upon explicit consideration of renormalized electronic band parameters.Our comparative analysis shows that electron correlations significantly impact the low-energy plasmon excita-tions across the cuprate doping phase diagram, even at long wavelengths. Thus, complementary information onthe evolution of electron correlations, influenced by the rich electronic phases in condensed matter systems, canbe extracted through the study of two-particle charge response.DOI: 10.1103/PhysRevResearch.6.043184I. INTRODUCTIONInteractions among constituent entities leading to emer-gent phenomena are observed across disciplines, including*Contact author: abhishek.nag@ph.iitr.ac.in†Contact author: j.fink@ifw-dresden.de‡Contact author: agreco@fceia.unr.edu.ar§Contact author: kejin.zhou@diamond.ac.ukPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.superconductivity [1,2], active colloids [3], and neural func-tions [4]. In condensed matter systems, dynamic behaviorof the constituent entities is probed using spectroscopictechniques. For instance, in many-electron systems whereelectron-electron interactions dominate the low-energy phys-ical properties, angle-resolved photoemission (ARPES) ortunneling spectroscopy can assess the strength of “electroncorrelation.” These correlation effects arise from short-rangeinteractions between particles, which are seen in the low-energy quasi-particle properties. The direct observation ofdynamical charge susceptibility, representing the two-particlecharge-charge correlation function χ ′′c (q, ω), in comparison,is possible via spectroscopic techniques such as resonantinelastic x-ray scattering (RIXS) or electron energy-loss spec-troscopy (EELS).2643-1564/2024/6(4)/043184(14) 043184-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-1394-5105https://orcid.org/0000-0002-9165-9251https://orcid.org/0009-0003-6260-6534https://orcid.org/0000-0001-9822-3859https://orcid.org/0000-0002-3209-027Xhttps://orcid.org/0000-0003-4254-0622https://orcid.org/0000-0003-0328-5657https://orcid.org/0000-0003-2208-8501https://orcid.org/0000-0002-6982-9066https://orcid.org/0000-0002-5286-1684https://orcid.org/0000-0001-5958-5080https://ror.org/05etxs293https://ror.org/03eh3y714https://ror.org/00582g326https://ror.org/02tphfq59https://ror.org/05apxxy63https://ror.org/01q47ea17https://ror.org/05cvf7v30https://ror.org/05qbk4x57https://ror.org/0524sp257https://ror.org/02tphfq59https://ror.org/026v1ze26https://ror.org/04zb59n70https://ror.org/042aqky30https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.043184&domain=pdf&date_stamp=2024-11-21https://doi.org/10.1103/PhysRevResearch.6.043184https://creativecommons.org/licenses/by/4.0/ABHISHEK NAG et al. PHYSICAL REVIEW RESEARCH 6, 043184 (2024)One of the fundamental bosonic excitations in metal-lic systems is plasmon, originating from collective charge-density oscillations in the presence of long-range Coulombinteractions [5]. Typically, in isotropic electron systems,the long-wavelength plasmon energy associated with thecharge oscillations is finite. In layered three-dimensional (3D)electron systems, the 3D Coulomb interactions are poorlyscreened due to the confinement of charges to planes separatedby dielectric blocks. Thus, although 3D Coulomb interactionstend to forbid gapless plasmons [6], for particular momentaperpendicular to the layers (qz), charge oscillations that are outof phase may lead to formation of acoustic plasmons (ω → 0as q → 0) along with the gapped (optical) plasmons [7–10].In layered systems like cuprates, optical plasmons weredetected soon after the discovery of the high-TC superconduc-tivity using transmission-EELS (T-EELS) [11]. Acoustic-likeplasmons in the cuprates, however, have only recently beenobserved with the development of the RIXS technique[12–17]. Due to the low energy of acoustic plasmons, theirrole has been discussed sparsely since the discovery ofsuperconductivity [18–22]. More importantly, the cupratesuperconductors, exhibiting anomalous electronic propertiessuch as those observed in the pseudogap and the strange-metal phases [1,2], are widely studied for correlated electronphysics. The observation of long-wavelength low-energyquantum fluctuations of charges along with spins [12,15,23–25] has, therefore, renewed efforts to develop a unified un-derstanding of electron correlations, Coulomb and exchangeinteractions aiming toward a microscopic theory [26–29].Correlation effects in cuprates lead to an enhancementof the quasiparticle electron mass. The mass enhancementfactor can be denoted as m∗/m, where m∗ and m are theband mass in the presence of interactions and that predictedby tight-binding calculations, respectively [30]. In contrast, avariety of descriptions can be found for the plasmon excita-tions in the cuprates. The dispersion of the optical plasmonsobserved using T-EELS has been described within mean-field random phase approximation (RPA) theories withoutexplicitly considering electron correlations [31]. The dis-persion of the acoustic-like plasmons observed using RIXShas been described using both free-electron layered modelsand models incorporating strong electron correlations suchas the t-J-V model [12–15,28,29,32]. The charge carrierdoping dependence of plasmon energies could not be ex-plained within an RPA model, leading to the introduction ofa scaling factor to the plasmon energies [12,13]. Recently,the t-J-V model in a large-N approximation was employedto explain the low-doping range dependence of plasmons[15,29]. A similar doping dependence was also discussed inRef. [33], but only for the optical plasma frequency. In thestrange metal phase, momentum-independent broad continuaobserved using reflection-EELS [34,35] have been describedusing holographic theories [36], while a RIXS study has founddispersive excitations in this phase [15]. This multitude of de-scriptions raises a pertinent question: Do electron correlationsthat affect the single-particle excitations strongly have anyrole to play in the collective charge excitations in cuprates, andwhat should be the appropriate framework used to describe it?In this study, we provide a unified perspective on theimportance of electron correlations on the dispersion of theacoustic-like plasmons in electron- and hole-doped cupratesprobed by RIXS. We compare equal doping levels (δ =0.16) of archetypal hole-doped La1.84Sr0.16CuO4 (LSCO),and electron-doped La1.84Ce0.16CuO4 (LCCO). The similarlattice parameters of these systems enable investigation ofthe plasmons in the same momentum phase space. We ob-serve dispersive coherent excitations for both O K- and CuL3-edge RIXS in both systems. For the equal doping level,we find that plasmon velocity in LSCO is smaller than thatof LCCO, consistent with the former’s smaller Fermi velocityderived from bare band electronic dispersion. However, withina free electron model, the plasmon velocities are overesti-mated when considering the bare Fermi velocities for bothsystems. We demonstrate that an appropriate fit to experimen-tally observed plasmon dispersion is possible within an RPAmodel with the inclusion of a system-dependent band renor-malization parameter, and without which unrealistic values ofdielectric constants and incoherent excitations are obtained.The acoustic-like plasmons can be accurately described bythe t-J-V model, where bare band parameters provided asinput get implicitly renormalized by electron correlations. Ourfindings reveal that plasmon dispersion in cuprates is affectedby electron correlations like the single-particle excitations,and is accounted for by the band renormalization parameterin the RPA model. Thus, by comparing plasmon dispersionsand bare band electron dispersion parameters, it is possibleto assess the role and magnitude of electron correlations indifferent phases in the cuprates.II. RESULTSA. Electronic structure of LSCO and LCCOHole-doped LSCO and electron-doped LCCO belong tothe family of single-layered cuprates, obtained upon dop-ing parent systems La2CuO4. They crystallize in distinctstructures, the K2NiF4-type T (LSCO) and Nd2CuO4-type T ′(LCCO) [38]. In the T structure, O atoms form octahedralcages around Cu, while apical O atoms are absent in the Cu-Oplanes in the T ′ structure [see Fig. 1(a)], leading to differentelectronic ground states for the doped systems. A strong Cu-Ohybridization and on-site Coulomb interactions give rise tothe upper Hubbard band (UHB) and the Zhang-Rice singlet(ZRS) band in cuprates, as shown in Fig. 1(b) [37,39–46]. Theelectrostatic potential at the Cu sites is raised due to the lackof apical oxygen in LCCO compared to LSCO, resulting in areduced charge-transfer energy (�CT). Hole doping shifts thechemical potential (μ) to the ZRS, whereas electron dopingshifts it to the bottom of the UHB. The charge carrier dy-namics in these systems can therefore be investigated usingx-ray spectroscopy by tuning the photon energy to resonanttransitions to these bands. The x-ray absorption spectra (XAS)of LSCO and LCCO (δ = 0.16), obtained at the Cu L3- and OK-edge, respectively, are shown in Figs. 1(c) and 1(d). The CuL3-edge XAS peak corresponds to the transition to the UHBin both systems. In LSCO, the first peak in O K-edge XAScorresponds to the hole states, with the transition to the UHBoccurring 1.5 eV higher [44,46,47]. In LCCO, the first peakin the O K-edge XAS is the transition to the UHB, loweredin energy due to the reduced �CT and chemical shift of the043184-2IMPACT OF ELECTRON CORRELATIONS ON … PHYSICAL REVIEW RESEARCH 6, 043184 (2024)EFUHBHole-dopedElectron-dopedZRSNBCTCTT(b)(a)CuOdcLCCOt t' t''tzLSCO (c)(d)(f)(e)FIG. 1. Electronic structure of doped cuprates. (a) Schematic lattice structures of single-layered hole-doped LSCO and electron-dopedLCCO showing the Cu-O planes and hopping pathways. O atoms are shown only around the central Cu atom. (b) Schematic representationof electron transitions within the multiband structure of hole- and electron-doped cuprates [37]. UHB, ZRS, NB, and T represent the upperHubbard, Zhang-Rice singlet, nonbonding O, and the Zhang-Rice triplet bands, respectively. (c) Cu L3-edge XAS of LSCO and LCCO. (d) OK-edge XAS of LSCO and LCCO (δ = 0.16). Arrows mark the photon energies used to probe the plasmons. (e) One-band tight-bindingelectron dispersion of LSCO and LCCO [see Eq. (4) in Sec. V B]. (f) Fermi velocity distribution in LSCO and LCCO.O 1s level, consistent with observations in electron-dopedNd2−xCexCuO4 (NCCO) [46,47].B. RIXSFigures 2(a)–2(e) show the RIXS energy-momentum mapscollected on LSCO and LCCO (δ = 0.16) along the Cu-Oin-plane direction h, with k = 0.0 and l = 1.0. We denotemomentum transfers along h, k, and l directions in recipro-cal lattice units, where q = (ha∗, kb∗, lc∗) (a∗ = 2π/a, b∗ =2π/b, c∗ = 2π/c, and a = b and c are the in-plane and out-of-plane lattice parameters, respectively, see Table I). Theincident photon energy for RIXS maps shown in Figs. 2(b)and 2(c) correspond to resonant transitions to the ZRS andthe UHB at O K-edge in LSCO, respectively [see arrowsin Fig. 1(d)]. The highly dispersive plasmon excitations areprominent at both incident energies. A hybridized nature ofthe doped charges in hole-doped cuprates was concludedbased on a similar observation in La1.88Sr0.12CuO4 recently[14]. Despite the Cu-O hybridized content of the ZRS andthe UHB states, study at only the O K-edge is insufficient.In Fig. 2(a) we show the RIXS map at Cu L3-edge for LSCO.We can identify faint spectral weight present which appearsto follow the plasmon dispersion extracted from O K-edgeRIXS on LSCO (shown by the blue dashed lines). Presenceof these features is also evident in the momentum distributioncurves shown in Fig. 2(f) for energy transfer between 0.225and 0.515 eV. In contrast, our earlier investigation on LSCOand Bi2Sr1.6La0.4CuO6+� [32], along with studies on otherhole-doped cuprates [23,48,49], did not reveal the presence ofplasmons at the Cu L3-edge. This is most likely because theplasmon spectral weight is expected to be the strongest at l =1.0 [29], where we have investigated in this work. The pres-ence of plasmons at the Cu L3-edge, although extremely weak,validates their hybridized nature in the hole-doped cuprate.In LCCO, the dispersive plasmons are observed clearly forexcitation at either the Cu L3- or O K-edge, as shown in theRIXS maps in Figs. 2(d) and 2(e), respectively.Representative fits to the plasmon excitations in the RIXSline profiles as described in Sec. V A are shown in Figs. 2(g)–2(j). The plasmon energies and widths extracted from thefits are presented in Fig. 3. It is clear from the similarityof plasmon energies and the widths that we probe the sameTABLE I. Parameters for LSCO and LCCO. In-plane lattice constant: a. Distance between the Cu-O layers: d = c/2. Doping concen-tration: δ. Superconducting transition temperature: Tc. Average bare Fermi velocity: 〈vF〉bare. Optical plasmon energy: �p. Plasmon velocityobtained using 〈vF〉bare in Eq. (1): vbarep . Experimental plasmon velocity: vRIXSp . Mass enhancement factor obtained using vRIXSp and renormalized〈vF〉 in Eq. (1): m∗/m.a (Å) d = c/2 (Å) δ Tc (K) 〈vF〉bare (eVÅ) �p (eV) vbarep (eVÅ) vRIXSp (eVÅ) m∗/mLSCO 3.77 6.55 0.16 38 2.86 0.8 [51] 3.31 2.79 2.0LCCO 4.01 6.23 0.16 7.87 4.58 1.2 [13] 4.94 4.20 1.7043184-3ABHISHEK NAG et al. PHYSICAL REVIEW RESEARCH 6, 043184 (2024)FIG. 2. Energy-momentum distribution of plasmons in LSCO and LCCO (δ = 0.16). RIXS intensity maps with incident photon energyat (a) Cu L3-edge, (b) O K-edge ZRS, and (c) O K-edge UHB, respectively, for LSCO. RIXS intensity maps with incident photon energyat (d) Cu L3-edge and (e) O K-edge UHB, respectively, for LCCO. For all the data, k = 0.0 and l = 1.0. The color scales indicate scatteredintensities in arbitrary units. The markers denote the extracted plasmon energies. In (a), the blue dashed line is the plasmon dispersion extractedfrom (b), and the black dashed line is the extended paramagnon dispersion from Ref. [25] for LSCO. (f) Momentum distribution curves forenergy transfer between 0.225 and 0.515 eV for Cu L3-edge RIXS on LSCO showing the plasmons. (g)–(j) RIXS line spectra from (b)–(e) ath = 0.06. The dashed lines are elastic, lattice, magnetic, and background components as described in Sec. V A. The shaded distributions arethe fitted plasmon peaks which can be compared with the calculated charge susceptibility line profiles in Figs. 4(f)–4(h).charge oscillations at Cu L3- and O K-edges for LCCO, andthe ZRS and UHB peak at O K-edge for LSCO [Fig. 3(a)].The plasmons exhibit a nearly linear dispersion for small hvalues; however, since we cannot resolve the plasmon peaksbelow h = 0.02, and a gap may exist at h, k = 0.0 due tointerlayer hopping (tz) [28,50], we describe these excitationsto be acoustic-like. Note that an upper limit of tz was es-timated to be 7 meV for LSCO and LCCO [32,50], whichis negligibly small to influence the analysis presented inthis work. We observe that the plasmon energies for LSCOare smaller than LCCO for the same doping level and atthe same h, k = 0.0, l = 1.0 values. The plasmons in LSCOare more damped than LCCO, as can be seen from Fig. 3(b),where the extracted damping factor (γ /ω0) is plotted. γ andω0 represent the plasmon width (damping ∼ inverse lifetime)and plasmon pole energy, respectively. γ /ω0 is found to beless than 1 in the probed momentum phase space, signifyingthe coherence of the plasmons.C. Effective masses and Fermi velocities in LSCO and LCCOFor the same amount of electron- and hole doping, the plas-mon energies extracted from RIXS for LSCO are smaller thanin LCCO [Fig. 3(a)], with plasmon velocities vLSCOp = 2.79 ±0.04 eVÅ and vLCCOp = 4.20 ± 0.01 eVÅ. We first attemptto qualitatively describe the observed plasmon dispersionusing the homogenous free-electron layered model, or Fetter-Apostol model [see Eq. (12)] [7,8]. Note that this free-electronmodel is in the hydrodynamic limit [7] or in RPA [8], andalso does not consider interlayer hopping, and as such isnot strictly applicable to the cuprates. Although less rigorouscompared to many-body models, its simple analytic formallows a rudimentary association of the electronic band pa-rameters to the acoustic plasmon dispersion and the opticalplasmon frequency �p. The acoustic plasmon velocity vp atl = 1.0 and small in-plane momentum can be related to theaverage Fermi velocity 〈vF〉 using Eq. (12) byvp =√〈vF〉22+ d2�2p4, (1)where d is the distance between planes. Assuming that theplasmons are unaffected by electron correlations, we can thenuse experimentally reported �p (see Table I) and bare 〈vF〉extracted from electron band dispersion to approximately es-timate the vp. We take tight-binding derived bare parametersfor LSCO and NCCO (for LCCO) from Ref. [52] [see Eq. (4)in Sec. V B], and compute the chemical potential μ for dopingδ = 0.16. In Fig. 1(e) we show the bare band dispersion forLSCO and LCCO. The 3d band is close to half-filling forhole-doped cuprates, while for electron-doped cuprates theband filling is about 70%. Due to the proximity to the van043184-4IMPACT OF ELECTRON CORRELATIONS ON … PHYSICAL REVIEW RESEARCH 6, 043184 (2024)FIG. 3. Plasmon energies and lifetimes in LSCO and LCCO (δ =0.16). (a) Plasmon energies extracted from fits to RIXS spectra at OK-edge ZRS and O K-edge UHB for LSCO and Cu L3-edge and O K-edge UHB for LCCO. (b) Plasmon damping factor (γ /ω0) extractedfrom the same fits.Hove filling, this results in a smaller average bare Fermi veloc-ity 〈vF〉LSCO,bare = 2.86 eVÅ than 〈vF〉LCCO,bare = 4.58 eVÅ[shown in Fig. 1(f)]. Using these values in Eq. (1), we ob-tain vLSCO,barep = 3.31 eVÅ and vLCCO,barep = 4.94 eVÅ. It isexpected that the plasmon velocities in hole-doped cupratesare smaller than the electron-doped cuprates with similar d ,due to smaller 〈vF〉; however, as shown in Fig. 4(a), the Fet-ter model with the bare band parameters overestimates theplasmon velocities by 17% compared to the values extractedfrom RIXS (see Table I). Conversely, if we use Eq. (1) andvp’s extracted from RIXS, we obtain 〈vF〉LSCO = 1.36 eVÅand 〈vF〉LCCO = 2.71 eVÅ. These values are nearly 50% of thebare band estimates for both systems. Therefore, to explainthe experimental results in this approximate model, one needsto use renormalized band dispersions which amount to massenhancement of m∗/m = 2.0 and m∗/m = 1.7 for LSCO andLCCO, respectively.D. Random phase approximationNext, we consider the explicit description of the plasmonswithin an RPA framework with long-range Coulomb inter-action for a layered lattice system (see Sec. V B for detailsFIG. 4. Comparison of plasmons to weak- and strong-electron coupling models. (a) Plasmon energies from RIXS for LSCO and LCCO(markers). Lines are plasmon energies calculated using the free-electron Fetter-Apostol model [Eq. (1)] and different 〈vF〉s for LSCO andLCCO. (b) Plasmon energies from RIXS for LSCO (markers). Lines are plasmon energies (continuous) and upper boundaries of electron-holecontinua (dashed) calculated using the weak-coupling RPA model with different m∗/m values for LSCO. (c) Same as in (b) for LCCO.(d) Plasmon energies from RIXS for LSCO and LCCO (markers). Lines are plasmon energies (continuous) and upper boundaries of electron-hole continua (dashed) calculated using the strong-coupling t-J-V model and bare band parameters for LSCO and LCCO. (e) Momentum-dependent deviation of the plasmon energies calculated using the RPA and the t-J-V models from experiments on LSCO. The shaded areasrepresent the propagated fitting errors from RIXS spectra. (f) Charge susceptibilities (continuous lines) and real part of the dielectric functions(dashed lines) at h = 0.06 obtained from the RPA model with different m∗/m values for LSCO. (g) Same as in (f) for LCCO. (h) Chargesusceptibilities obtained from the t-J-V model at h = 0.06 for LSCO and LCCO. The charge susceptibility line profiles in (f)–(h) can becompared to corresponding plasmon peaks in RIXS [Figs. 2(g)–2(j)]043184-5ABHISHEK NAG et al. PHYSICAL REVIEW RESEARCH 6, 043184 (2024)TABLE II. Parameters extracted by fitting experimental plasmondispersions to the different models.Model m∗/m Vc (eV) α ε‖/ε0 ε⊥/ε0LSCO RPA 1.0 0.49 0.5 14.1 85.1RPA 2.0 7.6 3.7 6.72 5.49t-J-V - 18.8 4.1 3.01 2.22LCCO RPA 1.0 0.9 0.6 10.1 40.8RPA 1.7 9.2 3.2 5.05 3.81t-J-V - 30.0 3.5 1.71 1.17of the implementation). Note that we have ensured that thecalculations are consistent with the experimentally reportedvalues of �p for both systems. In Figs. 4(b) and 4(c), weshow the plasmon dispersion extracted from plasmon peaksin χ ′′RPA(q, ω) calculated with bare and renormalized band pa-rameters so that m∗/m = 1.0 and m∗/m = 2.0 for LSCO, andm∗/m = 1.0 and m∗/m = 1.7 for LCCO. Also plotted are theupper boundaries of the electron-hole continua for the respec-tive m∗/m values. For both systems, in the long-wavelengthlimit, the agreement with the experimental results appearsto be slightly better for m∗/m = 1.0, while above h = 0.06,the calculated results for m∗/m > 1.0 have smaller deviationsfrom experiments. The momentum-dependent deviation fromthe experimental plasmon energies for LSCO is highlighted inFig. 4(e), showing the better agreement with m∗/m > 1.0 forlarger momenta and energies. It should also be noted fromFigs. 4(b) and 4(c) that for m∗/m = 1.0, the plasmons arewithin the continuum boundary, while for m∗/m > 1.0, theyare clearly above the continuum. Figures 4(f) and 4(g) showthe χ ′′RPA(q, ω) and the real part of the dielectric functionRε(ω) for h = 0.06, which undergoes a sign change onlyfor m∗/m > 1.0. This signifies that true plasmon resonanceswhich are long-lived are obtained only for m∗/m > 1.0. Wecan compare this observation to the experimentally extractedratio γ /ω0 [see Fig. 3(b)]. The γ /ω0 values are less than 1,which means that experimentally we observe the plasmons ascoherently propagating excitations. Additionally, the ratio ofin-plane to out-of-plane dielectric constants obtained from theRPA analysis (see Table II in Sec. V B) for m∗/m > 1.0 is1.22 for LSCO and 1.32 for LCCO, while for m∗/m = 1.0the respective ratios are unrealistic ( 1): 1/6 and 1/4. Thus,the layered lattice RPA model also suggests the use of renor-malized band parameters for both systems for describing theplasmons.E. t-J-V modelIn this section, we model the observed plasmon dispersionwith the t-J-V model with long-range Coulomb interaction ina large-N approximation for a layered lattice system, whereour inputs are the bare band parameters (see Sec. V B fordetails of the implementation). Once again, we have verifiedthat the calculations are consistent with the experimentallyreported values of �p for both systems. In Fig. 4(d), weshow that there is a good agreement between the plas-mon dispersions obtained experimentally and those extractedfrom plasmon peaks in the calculated χ ′′tJV(q, ω). The plas-mons appear as well-defined peaks [Fig. 4(h)] and above theelectron-hole continuum. This is because the bare band pa-rameters are implicitly renormalized by electron correlationswithin the theory. To have an estimation of the band renormal-ization one can see Eq. (25) in Sec. V B 4, which gives m∗/mof around 4.5 for both systems. Also, the ratio of the in-planeto out-of-plane dielectric constants obtained from the t-J-Vanalysis is found to be 1.35 for LSCO and 1.46 for LCCO(see Table II in Sec. V B). Thus, the strongly correlated elec-tron model also describes the plasmons appropriately, withoutexplicitly invoking renormalized band parameters.III. DISCUSSIONA. Plasmon dispersion and correlationsDespite the large diversity in material-dependent prop-erties, the correlated electron nature of cuprates is widelyacknowledged. While the single-particle electron excitationsin cuprates clearly show the effects of correlations like massenhancement and incoherence, charge excitations like plas-mons have been described using theories ranging from freeelectron to weak- and strong coupling. The optical plasmonenergy �p for zero momentum in the mean-field RPA of ho-mogeneous layered electron systems is proportional to√1/m∗[53,54]. The optical plasmon dispersion up to second orderin q in this model is �p + Aq2, where A is a dispersioncoefficient dependent on m∗. Even so, the optical plasmondispersion observed in Bi2Sr2CaCu2O8 using T-EELS couldbe described using the bare band parameters [53,54]. Notably,in Sr2RuO4, a system for which ARPES estimated m∗/m ≈ 4,optical plasmons observed using T-EELS have been modeledusing bare band parameters [55,56]. This was explained onthe basis of optical spectroscopy data [57] which found anenergy-dependent m∗/m: close to 4 below 0.2 eV and close to1 at higher energies. Resilient quasiparticles at high energieshave been predicted by density-functional theory extended bydynamical mean-field theory calculations [58], and it appearsthat in Sr2RuO4 the high-energy plasmons of 1.5 eV areunaffected by correlations.Our observations extend this discussion by focusing on thelow-energy acoustic-like plasmon dispersion in LSCO andLCCO, in which the situation seems to be different fromthe aforementioned. We observe that the plasmon velocity ofLSCO is approximately 1.5 times smaller than LCCO for adoping δ = 0.16. Since the plasmons are collective excitationsinvolving electrons near the Fermi surface, one can qualita-tively explain this observation by considering the 1.5 timessmaller 〈vF〉 in the hole-doped cuprate. However, when usingthe 〈vF〉s derived from bare bands, the plasmon energies areoverestimated for both systems in the free electron model.Thus, for cuprates it seems that the acoustic-like plasmonscannot be described using bare band parameters and it isnecessary to consider the effects of correlation for a quan-titative analysis. Within the RPA approach, the agreementof the dispersion with m∗/m = 1 worsens as q increases,while it improves for m∗/m > 1. Although it may seem thatthe effects of correlation may be fully relaxed in the long-wavelength limit, using the m∗/m = 1 band parameters resultsin plasmons appearing within the electron-hole continuumand unreasonable dielectric constant values for either system043184-6IMPACT OF ELECTRON CORRELATIONS ON … PHYSICAL REVIEW RESEARCH 6, 043184 (2024)FIG. 5. Momentum dependence of correlation effects on plas-mons. Plasmon energies from RIXS for LSCO and LCCO (markers).Continuous lines are plasmon energies calculated using the fullt-J-V model. Dense dashed lines are plasmon energies calculatedusing the RPA model with renormalized band parameters obtainedfrom the t-J-V model. The match between the two models at longwavelengths suggest that RPA with the renormalized band mass for-malism accounts for the effects of electron correlations in this regionof momentum space. Locally strong correlation effects stemmingfrom double occupancy prohibition that are absent in RPA lead todeviations only at large momenta. Sparse dashed lines are plasmonenergies calculated using a partial t-J-V model (see Sec. V B 4).The overlapping dispersions obtained from the RPA and the partialt-J-V model show that despite the apparent complication of the t-J-Vformalism with respect to RPA, it has a “hidden” RPA structureincluding the effects of the electronic correlations.in our model. The value of the mass enhancement factorm∗/m = 2.0 for LSCO is numerically equal to that measuredusing ARPES at the nodal point [59]. However, this matchshould not be overemphasised, given that the result from theplasmons represents an average effect over the entire Brillouinzone (BZ), which means including the antinodal region nearthe saddle point (π, 0) with a low vF, and the nodal region near(π, π )/2. Also, smaller m∗/m values are observed in the RPAmodels for LCCO than LSCO. Although weaker correlationsare expected in electron-doped than in hole-doped cuprates[38,39,60–65], it should be noted that the value of m∗/m forLCCO was obtained using the band parameters of NCCOin the calculations. This is due to unavailability of the bandparameters for LCCO.It can be seen from Fig. 4(e) that the deviation from theexperiments in the t-J-V model, in which the correlationeffects are implicit, is similar to that obtained from RPA form∗/m > 1 [smaller (larger) difference at high (low) q]. Inthe t-J-V model, the bare band parameters get renormalizedby the doping δ and J , and additionally the charge responsecontains fluctuations of the constraint that prohibit doubleoccupancy at a given site. To compare with the RPA model,we use the renormalized band parameters obtained from t-J-Vin the RPA and plot the calculated plasmon dispersions forLSCO and LCCO in Fig. 5. We observe that in the long-wavelength region, the t-J-V and RPA plasmon dispersionscoincide. However, at short wavelengths, the RPA plasmondispersions deviate from the t-J-V . In Fig. 5, we also plotplasmon dispersions obtained from the t-J-V excluding somebosonic self-energy components which carry information ofthe coupling between charge fluctuations and fluctuations ofFIG. 6. Broadening of plasmons due to a decay into single-particle excitation continua. The electron-hole continuum due tointraband transitions (I). The continuum due to Umklapp scatter-ing related to interband transitions (II). Energy-momentum pocketdevoid of continuum in electron-doped systems (III). White lines:Plasmon damping caused by decay into the continuum from single-particle excitations. (a) LSCO, close to half-filled conduction band.(b) LCCO with additional filling of the conduction band, causing apocket in the continuum of the interband continuum.the Lagrange multiplier that force the non double occupancyconstraint (see Sec. V B 4). Exclusion of these componentsfrom the t-J-V model results in a mathematically identicalform of charge susceptibility to RPA, and hence identicalplasmon dispersions are obtained for the two models. Thus,it is evident that the use of renormalized band parameters, i.e.,the inclusion of m∗, in RPA accounts for electron correlationsthrough the enhanced band mass at long wavelengths, whilethe locally strong correlation effects stemming from doubleoccupancy prohibition that are absent in RPA lead to devia-tions only at large momenta. It would be interesting to probethe acoustic-like plasmons till large momenta to (a) fit thelong-wavelength acoustic-like plasmon dispersion using RPAwith the inclusion of an effective mass m∗ and using the t-J-Vmodel, and (b) evaluate the nature of the predicted disagree-ment between RPA and the t-J-V model at large momenta(h > 0.25).B. Plasmon widthIn a free-electron model there is no momentum phasespace for the decay of an acoustic or an optical plasmoninto intraband electron-hole excitations (Landau damping) ifthe plasmon is above the continuum (region I in Fig. 6).Moreover, in the long-wavelength region, the plasmon shouldbe undamped. Nevertheless, non-resolution-limited plasmonsare observed in experiments. For finite q a q2 dependencewas predicted due to a decay into electron-hole pair excita-tions, but the theoretical estimates of the broadening werean order of magnitude smaller than the experimentally de-termined values [66]. Calculations for decay via phonon andimpurity-assisted intraband transitions were also found to beinsufficient [67,68]. Thus, the plasmon width had been apuzzle for long time. Finally, it was theoretically proposed[69] that plasmon width appears through a decay into thecontinuum formed due to interband excitations (region II inFig. 6). The latter originate from Umklapp processes due tothe square of the Fourier transform of the pseudopotentialof the ions in neighboring BZs. Experimentally, this was043184-7ABHISHEK NAG et al. PHYSICAL REVIEW RESEARCH 6, 043184 (2024)supported by EELS on alkali metals, where the plasmon widthwas found to be proportional to the square of the pseudopo-tential [70,71]. From the γ /ω0 < 1 values extracted fromRIXS [Fig. 3(b)], we can see that the damping of acousticplasmons in LSCO is twice as large as LCCO. In the case ofa half-filled band such as in LSCO and a kF equal to half ofthe BZ, the interband continuum extends to (q, ω = 0). Thus,the acoustic plasmons are damped additionally regardlessof the intraband continuum. Upon changing the band filling, apocket appears in the interband continuum in the low-energylow-momentum region [region III in Fig. 6(b)]. In this case(e.g., in LCCO), the acoustic plasmon will be less damped.Here, we mention that such pockets causing nearly undampedplasmons at low energy were previously described in T-EELSstudies of K-doped graphite [72]. In Ref. [73] the authors con-sidered the Hubbard model in the presence of the long-rangeCoulomb interaction using dynamical mean-field theory, andplasmons were obtained if one-particle self-energy effectsand vertex corrections due to correlations are treated prop-erly. The inclusion of electronic self-energy effects leads to abroadening of the plasmons (along with mass enhancement),and an energy dependence of the mass enhancement cannotbe ruled out [57,74,75]. One-particle self-energy effects canbe expected from the interaction between carriers and therich variety of low-energy charge excitations in the energyscale of J [76–78], which may lead to further differences inthe plasmon lifetimes of LSCO and LCCO. Further contri-bution to the broadening of the plasmons may appear fromplasmon-phonon decays. Instead of material-specific tuningof the broadening, in our calculations we have considereda minimal momentum- and energy-independent broadening� = 0.04 eV (0.1t ) comparable to experimental energy reso-lution to study the plasmon behavior. It should be noted thatour analysis is performed in the context of the t-J-V model,i.e., the t-J model, which can be derived from the Hubbardmodel in the large-U limit, in the presence of the long-rangeCoulomb interaction. Presence of plasmons in our calcula-tions in spite of the strong correlations shows the consistencyof our results with those of Ref. [73].IV. CONCLUSIONSWe have investigated LSCO and LCCO at equal dop-ing using RIXS, and have observed acoustic-like plasmonshaving different velocities. We find that the plasmon param-eters (energy and lifetime) are identical for a given systemirrespective of the probed site (Cu or O). While the RIXScross section is typically dominated by local-site effects, thisobservation highlights that the probed charge excitations arenonlocal and site independent, similar to magnetic excitations,due to the hybridized nature of valence charge fluctuations.We show that to appropriately describe the acoustic-like plas-mon dispersions in cuprates in a mean-field RPA approach,one has to consider renormalized band dispersion parameters.A similar renormalization of the bare band parameters occursimplicitly in the strong-coupling t-J-V model. This holds truefor both sides of the cuprate doping phase diagram, where weobserve m∗/m > 1 for both LSCO and LCCO. The compari-son with the t-J-V model justifies the use of the renormalizedband parameters in the RPA approach to effectively representthe mass enhancement stemming from electron correlationsat long wavelengths. Therefore, the weak-coupling nature ofthe RPA should not be used to dismiss its practical usage incuprates without due consideration. The role of correlations inthe two-particle charge response extends beyond a simple ad-justment of band parameters. An enhanced band mass reducesthe average Fermi velocity and pushes the electron-hole con-tinuum below the plasmon energies, allowing the observationof plasmons as resonant collective excitations. Even thoughhere we have used a uniform mass enhancement contributionto the acoustic-like plasmon dispersion, optical and ARPESstudies on Sr2RuO4 have suggested the mass enhancementfactors to be dependent on the quasiparticle energy [57,74].Typically the spin exchange energies (∼0.2 eV) are muchsmaller than the optical plasmon energies of about 1 eV.This may rationalize the non-dependence of optical plasmondispersion on electron correlations observed using T-EELSin cuprates and ruthenates [31,55,56,79], in contrast to theacoustic plasmons. The difference in the influence of electroncorrelations on acoustic and optical plasmons will be thesubject of our research in the near future.V. METHODSA. Experimental detailsA single crystal of La1.84Sr0.16CuO4 (LSCO) grown bythe floating-zone method and used for a previous report onplasmons [32] was reused for this experiment. The crystalwas re-cleaved before measurement at each edge, in vacuum.Hole doping of δ = 0.16 was verified using magnetizationmeasurements of LSCO corresponding to a superconductingtransition temperature of 38 K.High-quality La2−xCexCuO4 films were grown on SrTiO3substrates via the pulsed laser deposition technique with100 nm thickness. The films have a linearly varying Ce con-centration (x = 0.1 to 0.19) along the surface of the substrate,fabricated by the continuous moving mask technique [80].The direction of varying concentration is aligned normal tothe RIXS scattering plane. The c-axis lattice constants andsuperconducting transition temperatures measured along theconcentration gradient direction are consistent with resultsfrom single-doping LCCO films [81]. For x = 0.16, a su-perconducting transition temperature of 7.87 K was observedusing resistivity measurements.The pressure inside the sample vessel was maintainedaround 5 × 10−10 mbar. The samples were cooled down to25 K. While this means that the LSCO was below and theLCCO was above TC, a recent article [15] did not find sig-nificant change in the plasmon dispersion in this temperaturerange. The XAS were collected as total electron yield innormal incidence geometry with σ polarization, so that theelectric field was in the Cu-O plane. High-energy-resolutionRIXS spectra were collected at Cu L3-(�E 
 0.045 eV) andO K- (�E 
 0.043 eV) edges with σ polarization at the I21-RIXS beamline, Diamond Light Source, United Kingdom[82]. The zero-energy transfer position and energy resolutionwere determined from subsequent measurements of elasticpeaks from an adjacent carbon tape. Negative and positive043184-8IMPACT OF ELECTRON CORRELATIONS ON … PHYSICAL REVIEW RESEARCH 6, 043184 (2024)values of h represent the grazing-incident and grazing-exitgeometries, respectively.RIXS data were normalized to the incident photon flux,and subsequently corrected for self-absorption effects prior tofitting. A Gaussian line shape with the experimental energyresolution was used to fit the elastic line. Gaussian line shapeswere also used to fit the low-energy phonon excitations at∼0.045 eV and their overtones. The scattering intensities S(q,ω) of the plasmons, bimagnons, and paramagnons, dependenton the imaginary part of their respective dynamic susceptibil-ities χ ′′(q, ω), were modeled asS(q, ω) ∝ χ ′′(q, ω)1 − e−h̄ω/kBT, (2)where kB, T , and h̄ are the Boltzmann constant, temperature,and the reduced Planck constant. A generic damped harmonicoscillator model was used for the response functionχ ′′(q, ω) ∝ γω[ω2 − ω20]2 + 4ω2γ 2, (3)where ω0 and γ are the undamped frequency and the damping,respectively.First, we extracted the zone-centre energy, amplitude, andwidth of the broad incoherent mode at h = 0.01 and, con-cluding this to be a bimagnon, fixed its amplitude and widthfor the whole momentum range [32]. The energy values ofthe bimagnons were allowed to vary within ±20 meV. Anadditional paramagnon component was added for the RIXSspectra at the Cu L3-edge for LCCO. Significant correlationswere found below h < 0.02, between the elastic, phonon,and plasmon amplitudes and energies, and hence the plasmonenergy values determined in these regions are less conclusiveand not reported. A high-energy quadratic background wasalso included in the fitting model to account for the tailingcontribution from dd excitations above 1.5 eV. Representativefits using this model are shown in Figs. 2(g)–2(j).B. Theory details1. Band dispersion and Coulomb repulsionBased on ab initio calculations, the electron band disper-sion for the cuprates was proposed asEk = E‖k + E⊥k , (4)where the in-plane dispersion E‖k (Ref. [52]) and the out-of-plane dispersion E⊥k are given, respectively, byE‖k = −2t (cos kx + cos ky) − 4t ′ cos kx cos ky− 2t ′′(cos 2kx + cos 2ky) − μ, (5)E⊥k = − tz4(cos kx − cos ky)2 cos kz , (6)with μ as the chemical potential. The different hopping path-ways in the materials are shown in Fig. 1(a). We use the bareparameters [52]: t = 0.4 eV, t ′/t = −0.09, and t ′′/t = 0.07for LSCO, and t = 0.4 eV, t ′/t = −0.24, and t ′′/t = 0.15 forLCCO. The parameters used for LCCO are those given inRef. [52] for NCCO, due to unavailability of data for LCCO.In the out-of-plane dispersion E⊥k we have replaced cos kz with1 in the calculation, i.e., the contribution of E⊥k is independentof kz, which leads to a vanishing plasmon gap at the zonecenter, even for a finite value of tz. This is justified by the factthat the plasmon gap in LSCO and LCCO, if it exists, is smalland at present inaccessible experimentally, a topic which wasdiscussed in depth in Ref. [50]. In other words, the presenceof cos kz, as it was in Refs. [15,32,50], and the E⊥k dispersionis nearly irrelevant to the present analysis. Without losinggenerality, we have assumed tz/t = 0.01 for both systems.We have also neglected t ′′′ and t ′z. Finally, we compute thechemical potential μ for each case for doping δ = 0.16, whichgave μ = −0.24 eV and μ = 0.038 eV for LSCO and LCCO,respectively.Earlier works [7,10,83] considered the long-rangeCoulomb interaction V (q) for homogeneous layered electrongas asV (q) = V (q||, qz ) = q||d2sinh(q||d )cosh(q||d ) − cos(qzd ). (7)Here, we use the long-range Coulomb interaction V (q) for alayered lattice system for the RPA and t-J-V models:V (q) = VcA(qx, qy) − cos qz, (8)where Vc = e2d (2ε⊥a2)−1 andA(qx, qy) = α(2 − cos qx − cos qy) + 1. (9)These expressions are easily obtained by solving Poisson’sequation on the lattice [84]. Here, α = ε̃/[(a/d )2], ε̃ = ε‖/ε⊥,and ε‖ and ε⊥ are the dielectric constants parallel and perpen-dicular to the planes, respectively. It is important to note thatin the present V (q) model we have two dielectric constantsinstead of one as in Refs. [7,10]. e is the electric charge ofelectrons; a is the in-plane lattice constant and the in-planemomentum q‖ = (qx, qy) is calculated in units of a−1; sim-ilarly, d is the distance between the Cu-O planes, and theout-of-plane momentum qz is calculated in units of d−1. Inthe present work, we consider Vc and α as independent param-eters, and from them we can estimate ε‖ and ε⊥ and discusstheir reliability.2. Random phase approximationIn RPA the charge correlation function is given by the well-known expression [85]χRPA(q, iωn) = χ (0)(q, iωn)1 − V (q)χ (0)(q, iωn), (10)where χ (0)(q, iωn) is the usual Lindhard function,χ (0)(q, iωn) = 2Ns∑knF (Ek−q) − nF (Ek )iωn − Ek + Ek−q, (11)which accounts for the particle-hole continuum. q is a three-dimensional wave vector, ωn is a boson Matsubara frequency,and the factor 2 comes from the spin summation. Ns is thenumber of sites in each plane and nF is the Fermi distribution.The denominator in Eq. (10) is the RPA dielectric functionε(q, iωn) = 1 − V (q)χ (0)(q, iωn).After performing the analytical continuation iωn → ω +i� in χRPA(q, iωn), we obtain the imaginary part of the charge-charge correlation functions χ ′′RPA(q, ω), which can be directly043184-9ABHISHEK NAG et al. PHYSICAL REVIEW RESEARCH 6, 043184 (2024)compared with RIXS. � influences the width of the plas-mon, and its effect on the plasmon peak position is strongestwhen it becomes comparable to undamped plasmon energy(overdamped condition). From Fig. 3(b), we see that thiscondition may be applicable only close to the zone center.As discussed in the main text, several factors can affect theplasmon width. Here, we consider a minimal momentum- andenergy-independent broadening � = 0.04 eV (0.1t ) compara-ble to experimental energy resolution to study the plasmonbehavior [29,33,50,86]. The RPA calculation is a weak cou-pling approach and, in principle, the electron dispersion Ek[Eq. (4)] is given by the bare band [52]. However, as discussedin the text, the electron hopping parameters t , t ′, and t ′′ arerenormalized to account for m∗/m > 1.The analytical relation between plasmon and Fermi veloci-ties [Eq. (1)] in the homogeneous free-electron layered modelby Fetter-Apostol [7,8] is derived from the plasmon energy:ωp =√〈vF〉22q2‖ + �2pq‖d2sinh(q‖d )cosh(q‖d ) − cos(q⊥d ), (12)obtained by using the Coulomb potential in Eq. (7) and thedenominator in Eq. (10) [7]. Note that in the negligible con-duction dissipation limit, the factor 〈vF〉2q2‖/2 is absent (forω2p � 〈vF〉2q2‖/2) [87]. This is not the case for the energy-momentum range of plasmons probed in this work.3. The layered t-J-V model and the large-N formalismThe large-N approach for the t-J model was origi-nally developed in Ref. [88], and extensively used in thecontext of charge excitations in cuprates, among others,Refs. [28,29,32,50,86,89–91]. The aim of this section is togive a brief description of the main formulas.The layered t-J-V model is written asH = −∑i, j,σti j c̃†iσ c̃ jσ +∑〈i, j〉Ji j(�Si · �S j − 14nin j)+∑〈i, j〉Vi jnin j, (13)where the sites i and j run over a three-dimensional lattice.The hopping ti j takes a value t , t ′, and t ′′ between the first,second, and third nearest-neighbor sites on a square lattice,respectively. The hopping integral between layers is scaled bytz (see later for the specific form of the electronic dispersion).〈i, j〉 denotes a nearest-neighbor pair of sites. The exchangeinteraction Ji j = J is considered only inside the plane; theexchange term between the planes (J⊥) is much smaller thanJ [92]. Vi j is the long-range Coulomb interaction on the latticeand is given in momentum space by Eq. (8). c̃†iσ (c̃iσ ) is the cre-ation (annihilation) operator of electrons with spin σ = (↑,↓)in the Fock space without double occupancy. ni = ∑σ c̃†iσ c̃iσis the electron density operator and �Si is the spin operator.In the large-N theory [28] the electronic dispersion EkreadsEk = E‖k + E⊥k , (14)whereE‖k = −2(tδ2+ �)(cos kx + cos ky) − 4t ′ δ2cos kx cos ky− 2t ′′ δ2(cos 2kx + cos 2ky) − μ, (15)E⊥k = − tz4δ2(cos kx − cos ky)2 cos kz. (16)For a given doping δ, the chemical potential μ and � aredetermined self-consistently by solving� = J4Ns∑k(cos kx + cos ky)nF (Ek ), (17)and(1 − δ) = 2Ns∑knF (Ek ). (18)We have obtained for δ = 0.16 the values μ = −0.044 eVand � = 0.024 eV for LSCO, and μ = −0.010 eV and � =0.024 eV for LCCO.In the context of the t-J model using a path-integral rep-resentation [88] for Hubbard operators [93], a six-componentbosonic field is defined asδX a = (δR, δλ, rx, ry, Ax, Ay), (19)where δR describes fluctuations of the number of holes at eachsite, thus, it is related to on-site charge fluctuations, δλ is thefluctuation of the Lagrange multiplier introduced to enforcethe constraint that prohibits the double occupancy at any site,and rx and ry (Ax and Ay) describe fluctuations of the real(imaginary) part of the bond field from the J term.The inverse of the 6 × 6 bare bosonic propagator associ-ated with δX a is[D(0)ab (q, iωn)]−1= N⎛⎜⎜⎜⎜⎜⎜⎝δ22 [V (q) − J (q)] δ/2 0 0 0 0δ/2 0 0 0 0 00 0 4J �2 0 0 00 0 0 4J �2 0 00 0 0 0 4J �2 00 0 0 0 0 4J �2⎞⎟⎟⎟⎟⎟⎟⎠,(20)where J (q) = J2 (cos qx + cos qy). We use J/t = 0.3.At leading order, the bare propagator D(0)ab is renormal-ized in O(1/N ). From the Dyson equation the renormalizedbosonic propagator is[Dab(q, iωn)]−1 = [D(0)ab (q, iωn)]−1 − �ab(q, iωn). (21)Here, the 6 × 6 boson self-energy matrix �ab is�ab(q, iωn) = − NNs∑kha(k, q, Ek − Ek−q)× nF (Ek−q) − nF (Ek )iωn − Ek + Ek−qhb(k, q, Ek − Ek−q)− δa 1δb 1NNs∑kẼk−q − Ẽk2nF (Ek ), (22)043184-10IMPACT OF ELECTRON CORRELATIONS ON … PHYSICAL REVIEW RESEARCH 6, 043184 (2024)where Ẽk is equal to Ek with � = 0 and the six-componentinteraction vertex is given byha(k, q, ν) ={2Ek−q + ν + 2μ2+ 2�[cos(kx − qx2)cos(qx2)+ cos(ky − qy2)cos(qy2)]; 1;− 2� cos(kx − qx2); −2� cos(ky − qy2);2� sin(kx − qx2); 2� sin(ky − qy2)}. (23)In the writing of this manuscript we noted a misprint in thelast term of of Eq. (22) in previous works, which has beencorrected here.As discussed previously [88,89], the element (1,1) ofDab is related to the usual charge-charge correlation func-tion χtJV (ri − r j, τ ) = 〈Tτ ni(τ )n j (0)〉, which in the large-Nscheme is computed in the q-ω space asχtJV(q, iωn) = N(δ2)2D11(q, iωn). (24)It is important to remark that the charge-charge correla-tion function is nearly unaffected by the value of J [77]. Asfor χRPA(q, iωn), after performing the analytical continuationiωn → ω + i� in χtJV(q, iωn) we obtain the imaginary partof the charge-charge correlation functions χ ′′tJV(q, ω). Theplasmon excitations are obtained for the resonant peaks ofχ ′′tJV(q, ω).4. Correlations in the t-J-V and RPA modelsLooking at the large-N formalism in Sec. V B 3, it is notclear why the plasmon excitations obtained in the context ofthe t-J-V model are similar to those obtained in RPA. Al-though the large-N formalism for the t-J-V model seems to becomplicated and rather different from the usual RPA, here weshow that inside this framework an RPA structure is contained.The large-N formalism within the t-J-V model renormalizesthe band parameters due to electron correlations, which canbe seen in the band dispersion directly [Eqs. (14)–(16)]. Com-paring it to the usual tight-binding dispersion [Eqs. (4)–(6)],we obtainteff = tδ + �,t ′eff = t ′δ,t ′′eff = t ′′δ,tzeff = tzδ, (25)where the hopping parameters t , t ′, t ′′, and tz are thetight-binding bare ones. We introduced these effective param-eters into the RPA model and plotted the obtained plasmondispersion in Fig. 5. To understand the deviation at largemomenta between RPA and t-J-V , we consider only the 2 × 2sector (a, b = 1, 2) in the Dab(q, iωn) [Eq. (21)]. If in Eq. (21)we set manually the bosonic self-energy components �11 and�12 to zero, the only relevant component is �22, and fromEqs. (22) and (23) it can be written as�22(q, iωn) = −N∑knF (Ek−q) − nF (Ek )iωn − Ek + Ek−q= −Nχ0(q, iωn)2. (26)In spite of χ0 [Eq. (11)] representing the particle-hole con-tinuum within the RPA and �22 appearing in the large-Nformalism within the t-J-V model as only one componentof the bosonic self-energy carrying the information of thefluctuations of the Lagrange multiplier associated with theconstraint that prohibits the double occupancy, both havea similar mathematical form. In this context, we computeχtJV(q, iωn) in Eq. (24) using the physical value N = 2 [28],which givesχtJV(q, iωn) = χ0(q, iωn)1 − V ′(q)χ0(q, iωn), (27)where V ′(q) = 2[V (q) − J (q)]. Equation (27) shows that thecharge-charge correlation function in the large-N formalismconsidering only the contribution from �22 has an RPA-likemathematical form. This shows the presence of a “hidden”RPA structure with electronic correlations within the t-J-Vformalism with respect to RPA. The contribution of J (q) canbe neglected because V (q) is significantly larger, and thefactor 2 accounts for the transformation to electron volts usingthe renormalized value of t in the large-N formalism. In fact,this supports the picture that the effective mass m∗ introducedin RPA has an electronic correlated origin.ACKNOWLEDGMENTSWe thank A. A. Aligia, M. Hepting, C. Falter and W.S. Lee for discussions and comments. The RIXS experi-ments were primarily supported by user research programof Diamond Light Source, Ltd., U.K. through beam-timeProposal No. MM27872. A part of the results presentedin this work was obtained by using the facilities of theCCT-Rosario Computational Center, member of the HighPerformance Computing National System (SNCAD, MincyT-Argentina). J.C. acknowledges financial support from theNational Research Foundation of Korea (NRF) funded bythe Korean government (MSIT) through Sejong ScienceFellowship (Grant No. RS-2023-00252768). 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