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[Hiroyuki Yamase](https://orcid.org/0000-0003-0328-5657), [Matías Bejas](https://orcid.org/0000-0003-4254-0622), [Andrés Greco](https://orcid.org/0000-0001-5958-5080)

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[Retaining Landau quasiparticles in the presence of realistic charge fluctuations in cuprates](https://mdr.nims.go.jp/datasets/815c4938-7226-4f3f-a939-565e10a0f0bd)

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Retaining Landau quasiparticles in the presence of realisticcharge fluctuations in cupratesHiroyuki Yamase1, Mat́ıas Bejas2, and Andrés Greco21Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan2Facultad de Ciencias Exactas, Ingenieŕıa y Agrimensuraand Instituto de F́ısica Rosario (UNR-CONICET),Avenida Pellegrini 250, 2000 Rosario, Argentina(Dated: October 26, 2023)AbstractCharge excitation spectra are getting clear in cuprate superconductors in momentum-energyspace especially around a small momentum region, where plasmon excitations become dominant.Here, we study whether Landau quasiparticles survive in the presence of charge fluctuations ob-served in experiments. We employ the layered t-J model with the long-range Coulomb interaction,which can reproduce the realistic charge fluctuations. We find that Landau quasiparticles are re-tained in a realistic temperature and doping region, although the quasiparticle spectral weight isstrongly reduced to 0.08-0.24. Counterintuitively, the presence of this small quasiparticle weightdoes not work favorably to generate a pseudogap.1I. INTRODUCTIONAngle-resolved photoemission spectroscopy (ARPES) is a powerful tool to reveal the one-particle excitation spectrum [1]. In cuprate high-temperature superconductors, it shows agaplike feature—the spectral weight at Fermi momenta is suppressed at zero energy alreadywell above the superconducting transition temperature Tc, leading to a peak at a finite en-ergy. This phenomenon is known as the pseudogap [2, 3]. It develops already around theoptimally doped region and is pronounced in the underdoped region. Since the high tem-perature superconductivity is realized inside the pseudogap phase, the understanding of thepseudogap is one of the most important issues in the cuprate phenomenology and has beenstudied intensively. Despite many studies, however, the pseudogap is still a controversialissue and remains elusive.Recently, resonant inelastic x-ray scattering (RIXS) revealed charge excitation spectra inmomentum-energy space especially around the zone center in both electron-doped [4–6] andhole-doped [6–8] cuprates. They were identified as plasmon excitations specific to layeredmetallic systems—not only the usual optical plasmon but also acoustic-like plasmon modesare present [9]. Given that the plasmon energies are low [6–8], it is important to examinethe role of realistic three-dimensional charge fluctuations in the low-energy quasiparticleproperties, including the pseudogap phenomenology.Nonetheless, many theoretical studies were performed not only in two-dimensional modelsbut also for a short-range interaction—hence there are no plasmons. Reference [10] concludedthat charge fluctuations do not lead to a pseudogap in the dynamical cluster approxima-tion to the two-dimensional Hubbard model. This conclusion is shared with Refs. [11, 12],where the pseudogap is associated with antiferromagnetic fluctuations. However, chargefluctuations considered in Ref. [10] are qualitatively different from the actual experimen-tal data. Moreover, a recent work in the dynamical cluster approximation indicates thatantiferromagnetic fluctuations alone cannot capture the pseudogap [13].The situation is also similar in research of a strange metal physics, which currentlyattracts much interest especially in the context of Planckian dissipation in metals [14–16].Recent experiments [17–19] discussed that charge fluctuations can be responsible for thestrange metallic properties in cuprates. Theoretical studies in Refs. [20, 21] are in line withthis scenario. However, they missed realistic three-dimensional charge excitations including2plasmons.Realistic charge excitation spectra were reproduced in a large-N theory of the t-J modelwith the long-range Coulomb interaction [6, 7, 22, 23]—we may refer to it as the t-J-Vmodel [9]. Two theoretical studies were performed about the electron self-energy in thet-J-V model. First, plasmon excitations were found to generate a fermionic incoherentband—plasmaron dispersion—near the energy of the optical plasmon energy [24]. Second,quantum charge fluctuations, namely fluctuations at zero temperature, lead to a side bandwith an energy scale higher than the plasmarons, but on the opposite energy side across theFermi energy [25]. Considering that their energy scale is high, these features may not dependon temperature, which validates calculations at zero temperature in Refs. [24, 25]. However,both pseudogap and Planckian dissipation are related to not only finite temperature butalso a low-energy property of electrons close to the Fermi surface. These phenomena areassociated with the charge degree of freedom of electrons, suggesting a possibly pivotal roleof charge fluctuations.In this paper, we achieve accurate numerics even in a low-energy region at finite temper-atures in the large-N theory of the t-J-V model. This technical success allows us to studyclosely the electron self-energy by taking the realistic three-dimensional charge fluctuationsinto account. While the system loses approximately 75-90 % of the quasiparticle weight onthe entire Fermi surface, we find Landau quasiparticles in a realistic temperature and dopingregion. This implies that the charge fluctuations are not responsible for the pseudogap anda strange metal physics. We also find that the small quasiparticle weight is not effective togenerate a pseudogap even if additional self-energy corrections are considered, implying anintriguing role of charge fluctuations in the pseudogap state.II. FORMALISMThe t-J model is a microscopic model of cuprate superconductors [26–28]. To capturethe plasmon physics in cuprates and achieve realistic calculations, we employ the followinglayered t-J-V model:H = −∑i,j,σtij c̃†iσ c̃jσ +∑⟨i,j⟩Jij(S⃗i · S⃗j −14ninj)+12∑i ̸=jVijninj . (1)3Here c̃†iσ (c̃iσ) are the creation (annihilation) operators of electrons with spin σ(=↑, ↓) in theFock space without double occupancy at any site—strong correlation effects, ni =∑σ c̃†iσ c̃iσis the electron density operator, and S⃗i is the spin operator. The sites i and j run overa layered square lattice. The hopping tij takes the value t (t′) between the first (second)nearest-neighbor sites on the square lattice and is scaled by tz between the layers. Theexchange interaction Jij = J is considered only for the nearest-neighbor sites in the layer asdenoted by ⟨i, j⟩—the exchange term between the layers is much smaller than J (Ref. [29]).Vij is the long-range Coulomb interaction and describes plasmon excitations. The layeredstructure is a requisite to describe not only the usual optical plasmon but also acoustic-likeplasmons [30–32].In momentum space Vij is written as [33]V (q) =Vcα(2− cos qx − cos qy) + 1− cos qz, (2)where Vc = e2d(2ϵ⊥a2)−1 and α =ϵ∥/ϵ⊥(a/d)2; e is the electric charge of electrons, a the unitlength of the square lattice, d the distance between the layers, and ϵ∥ and ϵ⊥ are the dielectricconstants parallel and perpendicular to the planes, respectively. In Eq. (2), α describes theanisotropy between the in-plane and out-of-plane interaction.We analyze the model (1) by using a large-N technique in a path integral representationin terms of the Hubbard operators [34]. In this scheme, charge fluctuations associated withthe usual charge-density-wave and plasmons are described by a 2 × 2 matrix Dab(q, iνn)with a, b = 1, 2; q is the momentum of the charge fluctuations and νn a bosonic Matsubarafrequency. While D11 corresponds to the usual density-density correlation function, D22 isa special feature of strong correlation effects—it describes fluctuations associated with thelocal constraint. Naturally the off-diagonal component D12(= D21) is also present. Strictlyspeaking, there are also bond-charge fluctuations, which can be incorporated by enlargingDab to a 6×6 matrix. The bond-charge fluctuations are, however, less effective on the electronself-energy than the usual charge fluctuations [25] and are thus neglected for simplicity. Afterthe analytical continuation iνn → ν+ iΓch, where Γch(> 0) is infinitesimally small, we obtainthe full charge excitation spectrum described by ImDab(q, ν), which contains both plasmonexcitations as well as gapless particle-hole excitations—see Ref. [35] for a comprehensiveanalysis of ImDab(q, ν).The charge fluctuations can renormalize the one-particle property of electrons, which can4be analyzed by computing the electron self-energy. This requires involved calculations inthe large-N theory because one needs to go beyond leading order theory. At order of 1/N ,the imaginary part of the self-energy is calculated as [25]ImΣch(k, ω) =−1NsNz∑a,b∑qImDab(q, ν)ha(k,q, ν)hb(k,q, ν) [nF (−εk−q) + nB(ν)] . (3)Here ν = ω − εk−q, εk is the electron dispersion obtained at leading order, ha(k,q, ν) avertex describing the coupling between electrons and charge excitations, nF and nB theFermi and Bose distribution functions, respectively, Ns the total number of lattice sites ineach layer, and Nz the number of layers; see Ref. [25] for the explicit forms of Dab(q, ν),εk, and ha(k,q, ν). The above self-energy has the same structure as a self-energy obtainedfrom the Fock diagram in a perturbation theory. However, in the large-N scheme, we haveboth Hartree and Fock diagrams in a nontrivial way at order of 1/N . Moreover, it includescharge fluctuations associated with the local constraint described by ImD12 and ImD22.The real part of Σch(k, ω) is calculated by the Kramers-Kronig relations. Since theelectron Green’s function G(k, ω) is written as G−1(k, ω) = ω + iΓsf − εk − Σch(k, ω), weobtain the one-particle spectral function A(k, ω) = − 1πImG(k, ω):A(k, ω) = − 1πImΣch(k, ω)− Γsf[ω − εk − ReΣch(k, ω)]2 + [ImΣch(k, ω)− Γsf ]2, (4)where Γsf(> 0) originates from the analytical continuation in the electron Green’s function.III. RESULTSA. Role of realistic charge fluctuationsWe choose parameters t′/t = −0.20, J/t = 0.3, tz/t = 0.01, Vc/t = 31, α = 3.5,Γch = Γsf = 0.03t, and Nz = 10, and put t = 1 as the energy unit. These parameterswere obtained to describe the plasmon dispersion observed in La2−xSrxCuO4 (LSCO) [6].In LSCO, the plasmon energy with a finite qz becomes less than 55 meV at the in-planezone center [6]. This low-energy plasmon seems to offer a favorable situation where plasmonexcitations could affect effectively the electron property around the Fermi surface. We thusfocus on a small energy window around ω = 0 in this paper. Because of the layered model,the Fermi surface depends on kz. Our conclusions, however, do not depend on kz and wehave presented results for kz = 0.500.050.100.150.200.250.300.35-0.2 -0.1 0 0.1 0.2kFNkFANδ = 0.145, kz = 0(a)-Im Σch(kF, ω) [arb. units]ωT = 00.030.05-0.6-0.4-0.2 0 0.2 0.4 0.60-0.10 -0.05 0.05 0.10Re Σ ch(kF, ω)ω 0 1 0  1ky/πkx/π 0 2 4 6 8 10 120-0.05 0.05 0.10 0.15 0.20 0.25 0.30 0.35(b)A(kF, ω) [arb. units]ω0.150.160.170.180 0.01 0.02 0.03 0.04 0.05ZTkFNkFANFIG. 1. Electron property for different temperatures T at the doping rate δ = 0.145. (a) Imaginarypart of the electron self-energy at the nodal point kNF and the antinodal point kANF on the Fermisurface for T = 0.03 and 0.05; results at T = 0 are shown only for kNF here. The Fermi momentaare defined in panel (b). The inset is the corresponding real part of the self-energy at T = 0 and0.05. (b) Corresponding spectral function for the temperatures in panel (a). The inset describesthe quasiparticle weight Z as a function of temperature at several choices of Fermi momenta.Figure 1(a) shows that ImΣch(kF , ω)—the imaginary part of the electron self-energy fromcharge fluctuations at the Fermi momentum kF—vanishes at energy ω = 0 and temperatureT = 0 and is characterized by ∼ ω2 dependence including the case at finite temperatures;see Appendix A for a further analysis. In addition, we can check that its temperaturedependence at zero energy is characterized by ImΣch(kF , 0) ∼ T 2. In the inset in Fig. 1(a),we plot the corresponding real part ReΣch(kF , ω). It shows a linear dependence with anegative slope at ω = 0, a typical feature of a Fermi liquid. As expected, the spectralfunction A(kF , ω) exhibits a single peak at ω = 0 as shown in Fig. 1(b). All these resultsdemonstrate that despite the presence of acoustic-like plasmon excitations as well as gaplessparticle-hole excitations, charge fluctuations do not yield a non-Fermi liquid feature, but thesystem retains the Fermi-liquid property.However, the quasiparticle weight is reduced substantially. To see this, we compute60 0.1 0.2 0.3 0.4 0.5-0.2 -0.1 0 0.1 0.2kFNkFANT = 0.05, kz = 0(a)-Im Σch(kF, ω) [arb. units]ωδ = 0.05= 0.10= 0.15= 0.20= 0.250-1.00-0.500.501.000-0.10 -0.05 0.05Re Σch(kF, ω)ω 0 1 2 3 4 50-0.05 0.05 0.10 0.15 0.20 0.25 0.30 0.35(b)A(kF, ω) [arb. units]ω0.050.100.150.200.250.05 0.1 0.15 0.2 0.25ZδkFNkFANFIG. 2. Electron property for different doping rates at T = 0.05. (a) Imaginary part of the self-energy at kNF and kANF for δ = 0.05, 0.10, 0.15, 0.20, 0.25. The inset shows the corresponding realpart of the self-energy at δ = 0.05 and 0.25. (b) Spectral function for the doping rates in panel(a). The inset is the doping dependence of Z at kNF and kANF .the quasiparticle weight Z = (1 − ∂ReΣch(kF ,ω)∂ω|ω=0)−1 as a function of temperature in theinset of Fig. 1(b). The value of Z depends weakly on temperature and is around 0.17,meaning that charge fluctuations leave tiny quasiparticle weight around the Fermi energy atall temperatures. It is interesting to note in Fig. 1(b) that the value of Z becomes smallerat lower temperature, but the spectral function becomes sharper at lower temperatures.In all panels in Fig. 1 (except for ImΣch(kF , ω) at T = 0), we plot results for twocharacteristic momenta, kNF and kANF , each of which corresponds to the nodal and antinodaldirection [see the inset in Fig. 1(b)]. Although a difference of ImΣch(kF , ω) between kNF andkANF is visible in Fig. 1(a), this is a small effect in the sense that ReΣch(kF , ω) is not affectedpractically as seen in the inset in Fig. 1(a). In fact, the results in Fig. 1(b) show a weakkF dependence. That is, the effect of the charge fluctuations is essentially isotropic, namelys-wave-like along the Fermi surface.Figure 2 highlights results of the self-energy for different doping rates at T = 0.05. In7Fig. 2(a), ImΣch(kF , ω) is characterized by ∼ ω2 dependence around ω = 0 for all dopingrates and the value of ImΣch(kF , 0) decreases with increasing doping. The correspondingresults of ReΣch(kF , ω) are shown in the inset of Fig. 2(a). The slope of ReΣch(kF , ω) atω = 0 becomes larger with decreasing doping, leading to smaller quasiparticle weight forlower doping—the value of Z varies from 0.08 to 0.24 in 0.05 ≤ δ ≤ 0.25 as shown in theinset of Fig. 2(b). Consequently, the spectral function exhibits a single peak around ω = 0and the peak area becomes smaller with decreasing doping [Fig. 2(b)]. Interestingly, thepeak has a smaller half width at half maximum in spite of a larger value of ImΣch(kF , ω)with decreasing doping. This counterintuitive behavior is due to a larger negative slope ofReΣch(kF , ω) around ω = 0. It is also intriguing that the self-energy effect from chargefluctuations is pronounced for lower doping in Fig. 2, although the charge degree of freedomtends to be quenched at half-filling. In all panels in Fig. 2, results do not depend practicallyon a choice of Fermi momenta.B. Interplay with the pseudogapWe have shown that the realistic charge fluctuations in cuprates do not destroy quasipar-ticles and leave the quasiparticle weight Z = 0.08–0.24 in 0.05 ≤ δ ≤ 0.25—Z increases withdoping. This feature does not depend on temperature nor a choice of Fermi momenta. Giventhat the present theory captures charge excitation spectra including plasmons [6, 7, 22, 23],we expect that the electron self-energy that we have obtained is rather reliable. However,the pseudogap is observed especially in the underdoped region in hole-doped cuprates andthe quasiparticle picture is destroyed. What is then a role of charge fluctuations in thepresence of the pseudogap?We may formally write the self-energy observed in experiments (Σex) asΣex = Σch + Σpg + Σothers , (5)where Σch is a contribution from the charge fluctuations computed above and Σpg is acomponent that yields the pseudogap in the spectral function. Σothers is the other contri-butions to the electron self-energy, which may be responsible for strange metallic behavior[17, 18, 20, 21], a marginal Fermi liquid [36], and other anomalous behavior except for thepseudogap. Σothers also contains usual Fermi-liquid corrections from bosonic fluctuations8observed in cuprates [37]. We shall neglect the last term Σothers to perform a transparentanalysis.By employing a realistic Σex from experimental data, we may estimate Σpg by modelingit asΣpg(k, ω) =c2kω + iΓk. (6)This form is a simplified version capturing consistently various models to describe the pseu-dogap [38], when focusing on a momentum close to the Fermi surface (see Appendix B formore details); Γk describes a broadening and ck has the physical meaning of a kind of gap.Note that as we shall discuss later (see Fig. 4), the interplay of ck and Γk is crucial toproduce a pseudogap, which has not been recognized much. Since we shall make an analysisby focusing on the antinodal Fermi momentum, we may write ckANF= c and ΓkANF= Γ forsimplicity below.Figure 3(a) is a recent experimental data of the electron spectral function for LSCO [39]with δ = 0.145 at T = 45 K. We choose the same δ and T (= 0.0055t) by assuming t/2 = 0.35eV [6, 40]. We then tune the parameters c and Γ as well as a broadening of the spectralfunction Γsf [see Eq. (4)] to reproduce experimental data [Fig. 3(a)]. Our obtained Σpg andΣch are shown in Fig. 3(b). ImΣpg has a sharp peak at ω = 0, which generates the pseudogapin Fig. 3(a). The corresponding ReΣpg exhibits a steep slope with a positive sign at ω = 0to overturn the negative slope of ReΣch. While we have used Γsf = 0.03 in Figs. 1 and 2, weobtain Γsf = 0.536 to get a better fit especially to the tails away from the Fermi energy inFig. 3(a). This large Γsf may also reflect broadening due to the other contributions Σothers.In Fig. 3(a) we also plot the spectral function in two different conditions, with only Σpgand with only Σch. While the latter case exhibits a broad, but coherent peak at ω = 0—a typical Fermi-liquid feature, the former case indicates that an intrinsic pseudogap hassizable weight away from ω = 0 and forms a very broad structure with a gap nearly doublethe pseudogap observed in experiments.A major surprise in Fig. 3 is that in spite of rather small quasiparticle weight from Σch(Z ≈ 0.17 at δ = 0.145), we need a very pronounced peak of ImΣpg at ω = 0 to reproducethe pseudogap observed experimentally. To explore this outcome more, we study a conditionof c and Γ to reproduce a pseudogap in the presence of Σch. We make a map of ωpg—a halfdistance of double peaks of A(k, ω)—in the plane of c and Γ in Fig. 4; ωpg = 0 means asingle peak at ω = 0. The pseudogap is realized below the white curve—this condition is9-0.2 -0.1 0 0.1 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7c = 63.2 meVΓ = 4.26 meVΓsf = 375 meVδ = 0.145, T = 45 K(a)Intensity [arb. units]A(kFAN, ω) [arb. units]ω [eV]exp.fitchpg0-1.2-0.8-0.40.40.81.2-0.2 -0.1 0 0.1 0.200.20.40.60.81.0(b)Re Σ(kFAN, ω) [eV]-Im Σ(kFAN, ω) [eV]ω [eV]chpgFIG. 3. Self-energy consistent with experimental data extracted from Ref. [39]. (a) Typical spectralfunction observed in underdoped cuprates at the antinodal region on the Fermi surface, showingthe pseudogap, namely the suppression of the spectral weight at ω = 0. The solid black curve isa fitting in terms of Eqs. (5) and (6); we use t/2 = 0.35 eV [6, 40]. The spectral function in twodifferent conditions, with only Σch and with only Σpg, is also plotted. (b) Σch and Σpg used in thefitting to the experimental data in (a).given approximately by (see Appendix C for an analytical understanding)Γ2 < 2ZFLc2 . (7)Here ZFL is the Fermi-liquid quasiparticle weight in the absence of Σpg and is given by 0.17in the present case. The same calculations are also performed for different doping and wesuperimpose in Fig. 4 the obtained boundary, below which a pseudogap is realized. It showsthat we need a severer condition of the choice of c and Γ to reproduce the pseudogap fora lower doping rate, where the quasiparticle weight becomes smaller [see Fig. 2(b)]—thecontribution Σothers in Eq. (5) that we have neglected would reduce further the value of ZFL,yielding a further severer condition of c and Γ to produce a pseudogap. From the oppositeviewpoint, Fig. 4 indicates that Σpg tends to create a single peak when the quasiparticleweight becomes smaller in the absence of Σpg. Whether this can be related with the strange10Γ [eV]c [eV]δ = 0.250= 0.200= 0.145= 0.100= 0.05000.020.040.060.080.100 0.05 0.10 0.15 0.20 0 20 40 60 80ωpg [meV]δ = 0.145, T = 45 KΓsf = 0.375 eVhigher Thigher δFIG. 4. Condition to realize the pseudogap in the presence of charge fluctuations in the plane ofc and Γ in Eq. (7); we use t/2 = 0.35 eV [6, 40]. The pseudogap (ωpg ̸= 0) is realized below thewhite curve. Similar curves are also superimposed for other doping rates. To be consistent withthe pseudogap observed experimentally, c and Γ may depend on doping and temperature as shownby arrows schematically. The solid circle corresponds to the values of c and Γ used in Fig. 3.metal state in cuprates [3] is an interesting open issue. See Appendix D for a further analysis.IV. CONCLUSION AND DISCUSSIONSRecently charge fluctuations were proposed to be responsible for a strange metal and themarginal Fermi-liquid phenomenology [17, 18, 20, 21]. However, we have found that theself-energy from the realistic charge fluctuations is essentially isotropic and yields a Fermi-liquid contribution (Figs. 1 and 2). We have also found the small quasiparticle weight Z,which varies from 0.08 to 0.24 with increasing doping from 0.05 to 0.25 (Fig. 2). One mightexpect that a small Z at low doping would work favorably to form a pseudogap because thequasiparticles could be easily destroyed. However, the obtained theoretical insight is theopposite—the smaller the quasiparticle weight is, the more intense additional contributionsleading to the pseudogap should be [Eq. (7) and Figs. 3 and 4]. Furthermore, to be consistentwith experiments, c and Γ should exhibit a special doping and temperature dependence assketched with arrows in Fig. 4: the gap tends to be closed with decreasing c and to befilled with increasing Γ—the former feature like a gap-closing may be caused mainly byincreasing doping [1] and the latter one like a gap-filling by increasing temperature [1, 41, 42](see Appendix E). The microscopic origin of c and Γ is a challenge for understanding thepseudogap in cuprates.11In Ref. [10], a pseudogap very similar to the experimental data was obtained in thedynamical cluster approximation with eight sites to the two-dimensional Hubbard model.However, charge fluctuations in Ref. [10] are very different from those reported in RIXS [4–8]and also very weak. It is interesting to check whether the reported pseudogap in Ref. [10]practically remains even when the realistic charge fluctuations are taken into account.In the overdoped region, we expect Σpg → 0, but charge fluctuations survive. The factthat Σch is essentially isotropic on the Fermi surface (Fig. 1) and ImΣch ∼ T 2 may indicatethat Σch is promising to describe the transport properties in overdoped cuprates where thescattering rate is isotropic [43–45] and shows a dominant T 2 dependence [43–48]. In addition,ImΣch decreases with increasing doping [Fig. 2(a)], which is also in line with the behaviorof the resistivity [2, 49].Our value of Z is around 0.25 in the overdoped region [see the inset of Fig. 2(b)], implyingthe mass enhancement is around 4. Quantum oscillation measurements of Tl2Ba2CuO6+δfound a value of 3.1 - 5.1 [50], which is consistent with the present work. On the otherhand, ARPES measurements for overdoped Bi2Sr2CaCu2O8+δ reported a value around 1.5[51]. This difference might be related to the difference of the energy scale between quantumoscillation and ARPES.The pseudogap in cuprates, namely Σex, has been frequently studied by focusing on Σpgin Eq. (5) alone. However, we have demonstrated that the other contributions Σch andΣothers can be crucially important as shown in Fig. 3 and Eq. (7). Hence it is important todisentangle the source of the pseudogap from Σex. A valuable insight may be obtained asfollows. We first approximate Σpg = 0 around the nodal Fermi momentum, leading to thecomponents of Σch+Σothers. Then assuming those components are isotropic, we may obtainΣpg by subtracting the component Σch +Σothers from Σex at Fermi momenta away from thenodal point. This procedure may also be performed in numerical calculations as those inRefs. [11] and [12].ACKNOWLEDGMENTSThe authors thank L. Manuel, W. Metzner, and T. Schäfer for valuable discussions.A part of the results presented in this work was obtained by using the facilities of theCCT-Rosario Computational Center, member of the High Performance Computing National12System (SNCAD, MincyT-Argentina). A.G. and H.Y. are indebted to warm hospitality ofMax-Planck-Institute for Solid State Research. H.Y. was supported by JSPS KAKENHIGrant No. JP20H01856 and World Premier International Research Center Initiative (WPI),MEXT, Japan.Appendix A: ω dependence of ImΣHere we provide an in-depth analysis of the results in Fig. 1(a) at T = 0.In Fig. 5(a), our numerical results ImΣch(kF , ω) at T = 0 for kF = kNF are fitted byusing two different functional forms, ω2 and ω2 log |ω|—the former is expected for the three-dimensional (3D) Fermi liquids and the latter for two-dimensional (2D) Fermi liquids [52].We see both nicely fit to the numerical results in the vicinity of ω = 0. Since our system is alayered model, we would expect a crossover from the 2D to the 3D character with decreasingω toward zero. Numerically, however, we cannot clearly distinguish them in the vicinity ofω = 0. Rather we could firmly say that the 2D character is more pronounced in a higher ωregion.One would expect that ImΣch(kF , ω) should have a symmetry with respect to ω = 0.However, a close inspection in Fig. 5(a) reveals that this is not exactly the case in thepresent model. We checked numerically that contributions from the saddle-point regionsin εk−q in Eq. (3) are larger in the positive ω side than in the negative ω side, which weinterpret as one of the main sources to yield an asymmetry of ImΣch(kF , ω) with respect toω = 0 in the low-energy region.This effect becomes more pronounced when we choose kF = kANF , much closer to thesaddle points for a small q, as shown in Fig. 5(b). This was the reason why we refrainedfrom presenting ImΣch(kANF , ω) in Fig. 1(a)—special care may be necessary for kF = kANF atT = 0. We thus consider the positive and negative energy regions separately and performthe fitting in each region. We find that the numerical results are well fitted to both ω2 andω2 log |ω| in the vicinity of ω = 0 and the higher energy region is fitted better to the latter.These technical subtleties, however, are special at T = 0 especially for kF = kANF and fadeaway at finite temperatures as seen in Figs. 1(a) and 2(a).13kFN(a)-Im Σ(kFN, ω)ω5.61 ω21.37 ω2 |log |ω|| 0 2 4 6 80-0.04 -0.02  0.02  0.04x10-3δ = 0.145, kz = 0kFAN(b)-Im Σ(kFAN, ω)ω12.9 ω26.35 ω23.14 ω2 |log |ω||1.55 ω2 |log |ω|| 0 4 8 120-0.04 -0.02  0.02  0.04x10-3FIG. 5. ω dependence of ImΣ(kF , ω) at T = 0. (a) Fitting of ImΣ(kF , ω) for kF = kNF by usingtwo different functional forms, ω2 and ω2 log |ω|. (b) Fitting of ImΣ(kF , ω) for kF = kANF in thepositive and negative energy regions separately.Appendix B: Modeling of the pseudogapThe origin of the pseudogap is still controversial and it is beyond the scope of the presentwork to pursuit it. Instead, from a practical point of view, we consider a self-energy thatcan reproduce the pseudogap observed by ARPES.Our modeling in terms of Eq. (6) is based on Ref. [38] and can be regarded as a simplifiedversion to cover different scenarios to capture the pseudogap phenomenology. The self-energywe consider is given byΣpg(k, ω) =c2kω + ε̃k + iΓ. (B1)In the case of a commensurate density wave with momentum Q = (π, π) such as the usualcharge- and spin-density-wave, ck corresponds to its gap andε̃k = −εk+Q . (B2)In the so-called Yang-Rice-Zhang (YRZ) model [53], ck controls the magnitude of a pseudo-gap and ε̃k is the nearest-neighbor term of the tight-binding dispersionε̃k = −2t(cos kx + cos ky) . (B3)14If the d-wave pairing fluctuations are responsible for the pseudogap formation, ck is theusual d-wave pairing gap and we haveε̃k = εk . (B4)We consider a momentum fulfilling ε̃k = 0. This condition determines the Luttingersurface, where the self-energy diverges at ω = 0 and Γ = +0. Therefore, the spectralfunction is expected to be strongly suppressed at zero energy when the Fermi surface crossesthe Luttinger surface. In order to capture a pseudogap feature, therefore, we consider asituation where ε̃kF≈ 0. This is typically realized close to a momentum of the antinodalregion in hole-doped cuprates, where a holelike Fermi surface crosses the Brillouin zoneboundary. This consideration leads to Eq. (6) in the main text after allowing a momentumdependence of Γ.While we have successfully fitted experimental data with Eq. (6) (see Fig. 3), this maynot necessarily indicate that the pseudogap should be explained in either of the above threescenarios. This is because the functional form of our simplified self-energy Eq. (6) might alsobe obtained in other scenarios [11, 12] and in this sense can be general phenomenologically.Appendix C: Analytical understanding of Eq. (7)The condition to produce a pseudogap is given approximately by Eq. (7). Here we provideanalytical grounds behind it.Since Σch exhibits a Fermi-liquid feature in Figs. 1 and 2, we may approximate it asΣch ≈ −aω − iaπ2ω0ω2 , (C1)where a is positive and we have assumed that ImΣch = 0 in |ω| > ω0. This approximationis expected to be good as long as we consider a low-energy property. We then obtainZ−1FL = 1− ∂ReΣch∂ω= 1 + a . (C2)Together with Σpg in Eq. (6), we may write the total self-energy asΣex ≈ Σch + Σpg (C3)≈ −(a− c2ω2 + Γ2)ω − iΓc2ω2 + Γ2. (C4)15Γ [eV]c [eV]00.020.040.060.080.100 0.05 0.10 0.15 0.20 0 20 40 60 80ωpg [meV]δ = 0.145, T = 45 KΓsf = 0.375 eVPGIPCP(a)δ = 0.145T = 45 Kc = 63.2 meVΓ = 50 meVΓsf = 375 meV(b)A(kF, ω) [arb. units]ω [eV]00.100.200.300.400.500-0.2  0.2  0.4  0.6Re Σ(kFAN, ω) [eV]-Im Σ(kFAN, ω) [eV]ω [eV]0-1.0-0.50.51.0-0.2 -0.1 0 0.1 0.200.040.080.120.160.20FIG. 6. Coherent and incoherent single peaks. (a) Reproduction of Fig. 4, but focusing on theresults for δ = 0.145. The non-pseudogap region above the white curve is divided into two regions,where a coherent or an incoherent single peak is realized. (b) Spectral function at the antinodalFermi momentum in the IP region marked by the cross in (a). The inset shows that the real partof the self-energy has a negative slope but the imaginary part has a peak at ω = 0.Here we have focused on a low-energy region so that ImΣch is negligible compared withthe contribution from ImΣpg. We then replace Σch in Eq. (4) with the above Σex and putΓsf = +0. At the Fermi momentum kF , we find that the spectral function A(kF , ω) has adouble peak around ω = 0 in the condition ofΓ2 < 2ZFLc2 . (C5)By comparing with numerical results, we checked that this formula is very precise for Γsf =0.001 eV and starts to have visible errors when a larger Γsf is invoked—yet it works as areliable guide to estimate the boundary of the pseudogap.16c = 80 meV(a)A(kFAN, ω) [arb. units]ω [eV]Γ [meV] = 20406080 0.1 0.2 0.3 0.4 0.5 0.6-0.1 -0.05  0  0.05  0.1Γ = 20 meV(b)A(kFAN, ω) [arb. units]ω [eV]c [meV] = 10204080 0.1 0.2 0.3 0.4 0.5 0.6-0.1 -0.05  0  0.05  0.1FIG. 7. Evolution of the spectral function at the antinodal Fermi momentum in the phase diagramshown in Fig. 4. (a) Several choices of Γ at a fixed c and (b) several choices of c at a fixed Γ.Appendix D: Coherent and incoherent single peaksAs shown in Fig. 4, a pseudogap is formed in Γ2 ≲ 2ZFLc2 [Eq. (7)]. If this condition isnot fulfilled, a single peak is realized. As indicated in Fig. 6(a), there are two kinds of singlepeaks. One is a coherent peak (CP) typical of the Fermi liquid as we already discussed inFigs. 1 and 2. The other is an incoherent peak (IP) shown in Fig. 6(b), for which ImΣexhibits a peak at ω = 0, but ReΣ retains a negative slope there as shown in the inset.There can be a different IP in that ImΣ exhibits a peak at ω = 0 and ReΣ has a positiveslope there, as actually obtained in a theoretical study of electronic nematic fluctuations[54]. However, we do not find this kind of IP in Fig. 6(a).Figure 6(a) indicates that the IP state intervenes between the PG and CP states. Re-calling the strange metal state also intervenes between the PG and Fermi-liquid states incuprates [3], it is interesting to explore further a possible connection between the IP stateand the strange metal state.Appendix E: Evolution of the spectral function with c and ΓWe have focused on the spectral function fitted to the experimental data [39] in the maintext. Here from a general point of view, we present how the spectral function evolves bychanging c and Γ in Fig. 4 at δ = 0.145. 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