When doped with holes or electrons, cuprates at low temperatures can be tuned from a Mott insulating phase to a superconducting phase and then to a Fermi-liquid phase. In addition, an enigmatic pseudogap phase exists in the underdoped regime of hole-doped cuprates above the superconductivity transition temperature T C with a crossover temperature T ! , which decreases monotonically when the doping is increased [1]. Several symmetry-breaking orders such as charge-density waves (CDWs) have also been discovered in the cuprates with comparable onset temperatures [2].
Charge fluctuations are one of the most fundamental collective excitations in matter. Recently, CDWs in cuprate superconductors have attracted renewed interest , but their mechanism and competition with superconductivity remain subjects of vigorous discussion. The wave vector of the CDW, for example, exhibits an inconsistent evolution with doping in various cuprate families [7,8]. This observation has lead to debate on whether real-space local interactions or Fermi surface nesting in momentum space underlies the CDW physics. Moreover, CDWs are coupled to phonons in cuprates [22][23][24][25][26][27][28][29][30][31][32][33], but the underlying mechanism of phonon softening remains a mystery [6], although lattice vibrations in metals can be damped or softened by electronic quasiparticles or excitations.
Upon tuning a nonthermal parameter through a critical value, quantum phase transitions occur at the absolute zero of temperature T. The putative quantum critical point (QCP) in a cuprate holds the key to understanding many profound phenomena related to its superconductivity [34][35][36][37][38][39][40]. Anomalous thermodynamic [38] and transport properties [39,40] of cuprates close to a QCP have been observed. For example, as temperature T approaches zero, the electrical resistivity varies linearly with T rather than the T 2 dependence of a Fermi liquid. In approaching a QCP, two quantum states of the system can exchange their energy ordering. As a result, a CDW state can be intertwined with the superconducting state. Also, because the energy difference is small, switching back and forth between CDW and superconducting ground states is energetically likely, resulting in quantum charge fluctuations. Interestingly, recent resonant inelastic x-ray scattering (RIXS) studies have found that the charge-density fluctuations permeate through a broad region of the cuprate phase diagram [10].
A crucial task in the scenario of the presence of QCP is to identify the ordered phases that terminates at the QCP. In the present work, we presume the ordered phases to be the CDW order and try to find evidence for the QCP associated with the CDW in the superconducting phase. Inelastic x-ray scattering (IXS) probes charge fluctuations by measuring the dynamical structure factor Sðq; ωÞ, which is the space and time Fourier transformation of the density-density correlation function. Here, ℏq and ℏω are the momentum and energy transferred to charge excitations, respectively, with ℏ being the Planck constant h divided by 2π. For a given electron system of charge density nðr; tÞ at position r and time t, the charge fluctuation is δnðr; tÞ ¼ nðr; tÞhnðr; tÞi, in which h% % %i denotes the quantum statistical average. The densitydensity correlation function hnðr; tÞnðr 0 ; t 0 Þi can then be written as hnðr;tÞnðr 0 ;t 0 Þi ¼ hnðr;tÞihnðr 0 ;t 0 Þi þ hδnðr;tÞδnðr 0 ;t 0 Þi: ð1Þ
In the presence of disorder, an additional average over disorders must be taken over Eq. (1). X-ray scattering data, thus, comprise contributions of the static charge distribution hnðr; tÞihnðr 0 ; t 0 Þi and its dynamical fluctuations hδnðr; tÞδnðr 0 ; t 0 Þi, corresponding to charge susceptibilities denoted by χ 0 ðq; ωÞ and χðq; ωÞ, respectively. In addition, through the fluctuation-dissipation theorem, Sðq; ωÞ is related to the charge susceptibility χðq; ωÞ by Sðq; ωÞ ¼ 2ℏð1e -βℏω Þ -1 Imχðq; ωÞ, in which β ¼ 1=k B T, with k B denoting the Boltzmann constant.
O K-edge RIXS probes certain characteristics of the dynamical charge fluctuations of CDWs, although additionally modulated by the effects of RIXS matrix elements, light polarization, and orbital characters [10,23,41,42]. In this work, we perform high-resolution O K-edge RIXS measurements and theoretical calculations through a diagrammatic framework to investigate the quantum fluctuations of charge order in superconducting cuprate La 2-x Sr x CuO 4 (LSCO) near the optimal doping. Our results elucidate the role of charge order in the LSCO compound, providing evidence for the quantum critical scaling associated with charge order.
The LSCO single crystals with the doping level x ¼ 0.15 are grown by the traveling-solvent floating zone method [43][44][45]. After growth, the crystals are annealed to remove oxygen defects. The value of x is determined from an inductively-coupled-plasma atomic-emission spectrometric analysis. The T C of the x ¼ 0.15 sample is 37.5 K.
We conduct O K-edge RIXS measurements using the AGM-AGS spectrometer of beam line 41A at Taiwan Photon Source of National Synchrotron Radiation Research Center, Taiwan [46]. This recently constructed AGM-AGS beam line is based on the energy compensation principle of grating dispersion. High-resolution RIXS data are measured with an energy resolution of 16 meV at an incident photon energy of 530 eV. See Supplemental Material [47] for the details of the AGM-AGS spectrometer and data analysis.
We first measure static charge distributions hnðr; tÞihnðr 0 ; t 0 Þi of LSCO with x ¼ 0.15 using energyresolved elastic x-ray scattering. Figure 1(a) plots the intensity of O K-edge elastic scattering as a function of in-plane wave vector change q k varied along the antinodal direction ðπ; 0Þ at various temperatures. We observe CDW correlations in LSCO with an in-plane modulation vector q CDW ¼ ð0.235; 0Þ, which is given in reciprocal lattice units (r.l.u.) throughout this paper. Figures 1(c) and 1(d) show, respectively, the temperature-dependent CDW intensity and correlation length (ξ) defined as the inverse of the half width at half maximum (HWHM) of the momentum scan. The charge order of optimally doped LSCO exhibits a short correlation length, ranging from 22 to 12 Å. Consistent with previous results of LSCO, the CDWs are slightly suppressed in the superconducting phase as the temperature is decreased across T C [5,9].
The observed charge correlations of LSCO persist over a wide temperature range, up to well above T ! and into the strange-metal phase. These correlations achieve no longrange charge order, however, in contrast with other La-214 cuprates. For example, La 1.48 Nd 0.4 Sr 0.12 CuO 4 (Nd-LSCO), which has a low-temperature tetragonal (LTT) structure, shows long-range and short-range charge orders, as plotted in Fig. 1(b). The short-range charge order of LSCO might be caused by spatial disorders or phase separation arising from a subtle balance between Coulomb interactions and kinetic energy. At this time, we cannot determine if the CDWs with q CDW ¼ ð0.235; 0Þ are truly incommensurate or instead appear incommensurate due to defects known as discommensurations in the commensurate CDW state [48][49][50][51]. In a discommensurated structure, locally commensurate CDW regions are separated by discommensurations within which the CDW phase varies rapidly.
To unveil the charge instability and the coupling between CDW fluctuations and phonons in LSCO, we perform high-resolution O K-edge RIXS measurements. Figure 2(a) presents a series of temperature-dependent RIXS intensity distribution maps in the plane of energy loss vs in-plane momentum transfer q k along ðπ; 0Þ; Fig. 2(b) shows those maps after the subtraction of elastic scattering. There exists a pronounced excitation feature of energy about 14 meV in the neighborhood of q CDW . Its intensity decreases with increasing temperature, unlike that of the static CDW intensity that has a maximum at T C as plotted in Fig. 1(c) with red squares. This low-energy RIXS excitation appears to shift away from q CDW when the thermal energy approaches the excitation energy. This reflects that, at low temperatures, the low-energy RIXS excitation is dominated by charge-density fluctuations and its contribution from the coupling of CDW to acoustic phonons increases with the increase of temperature, as discussed later.
For a system that exhibits a QCP associated with charge order, Sðq; ωÞ near q CDW measures the collective fluctuations sensitive to the proximity of the QCP. When the thermal energy k B T is less than the typical energy of order parameter fluctuations and less than energy ℏω, their quantum nature is important. For low temperatures, we expect that the finite frequency fluctuation is dominated by the amplitude fluctuation of the charge density, and the following phenomenological form for charge susceptibility describes the quantum fluctuations [10,35]:
Here, Δ is the characteristic energy of the fluctuations at T ¼ 0 and can be interpreted as the inverse correlation length for the amplitude fluctuations. One expects that the characteristic energy Δ vanishes when the doping level approaches the quantum critical point. c is a parameter that characterizes the speed of the excitations with dispersion ω ∼ cjq kq CDW j, and Γ CDW is a parameter that characterizes the lifetime of the CDW amplitude excitations. In contrast, the static charge susceptibility χ 0 ðq; ωÞ vanishes except for ω ∼ 0 and its nearby region of CDW phase fluctuations characterized by Eq. ( 2) with Δ ¼ 0 and parameters c and Γ CDW . This energy region is beyond our RIXS energy resolution.
Figure 3(a) plots RIXS spectra after the subtraction of elastic scattering for momentum integrated from q k ¼ 0.22 to q k ¼ 0.25 at different temperatures. This low-energy RIXS intensity decreases when the temperature is increased as plotted in Fig. 3(b). On comparing the intensity, spectral profile, and peak energy of the low-energy RIXS excitation near 14 meV with those of calculated Sðq CDW ; ωÞ, in which the contribution of the dispersion parameter c vanishes, one can obtain the evolution of Δ and Γ CDW for various temperatures. Figures 3(c) and 3(d) show the estimated temperature-dependent Δ and Γ CDW , respectively. The evolution of Δ, which is inversely proportional to the CDW correlation length [36], is accordant with the temperature dependence of static charge order shown in Fig. 1. At low temperatures, Γ CDW grows with increasing temperature. In the quantum critical regime, the scattering rate of quasiparticles reaches the Planckian limit, and the change of Γ CDW is comparable with k B T=ℏ, i.e., the Planckian dissipation [40,52]. Although the measured data cover only one temperature point below T C , this observation supports the quantum critical nature of CDW fluctuations in LSCO.
Figure 3(d) also shows that, as T is increased, the temperature dependence of Γ CDW exhibits a bending feature about 80 K. This suggests two possible scenarios: a crossover near the QCP and the coupling of CDW fluctuations to acoustic phonons. First, a crossover of Γ CDW behavior near the QCP [53,54] indicates that Planckian linear-T dependence works only at the critical regime (low-temperature regions) controlled by the QCP. In other words, only inside the critical regime, Γ CDW shows the linear-T dependence, which reflects the gapless nature at the QCP, i.e., 1=Γ CDW going to infinity. At higher temperatures beyond the critical regime, Γ CDW no longer reaches the Planckian limit. In this scenario, the quantum critical region is up to approximately 80 K. In the second scenario, the bending feature of Γ CDW suggests that, for the thermal energy k B T comparable to the charge fluctuation energy, CDW fluctuations alone do not explain the RIXS data at high temperatures, as shown in supplemental Figs. S8(e)-S8(g) [47]. This indicates that the low-energy RIXS is also contributed by the coupling of CDW fluctuations to phonons. We, hence, calculate the full RIXS intensity using a diagrammatic framework, in which the coupling between χ CDW and phonons renormalizes the phonon propagator.
The intensity of the phonon excitations is calculated by using the diagrammatic framework [55] generalized to the O K edge. The RIXS intensity is given by (ℏ ¼ 1) Here, n f ðx Þ is the Fermi factor; ϵ k is the electron band dispersion; g ν ðk; qÞ and D ν ðq; ξÞ are the electron-phonon vertex and phonon propagators for phonon branch ν, respectively; ϕ 2p α ðkÞ is the oxygen 2p α orbital character of the band; rα is a unit vector pointing along the direction of the oxygen orbital; Γ is the core-hole lifetime parameter; γ e is a lifetime parameter for the electrons; and Ω ¼ ω iω f and q ¼ k ik f are the energy and momentum, respectively, transferred to the sample. The phonon softening is determined by the total electronic polarizability.
Our model essentially assumes that the phonons respond to some strong CDW fluctuation caused by an external source like correlations. That is, the coupling of CDW fluctuations to phonons renormalizes the phonon propagator through charge susceptibility χ CDW ðq k ; ωÞ. See Supplemental Material [47] for the calculation details.
Figure 4(a) presents the calculated RIXS intensity distribution map. That the coupling of CDW fluctuations to acoustic phonons enhances the RIXS intensity significantly at 14 meV for momentum about q CDW agrees satisfactorily well with the measurements shown in Fig. 4(b), corroborating the phenomenological form of charge susceptibility given in Eq. (2).
In addition to the pronounced low-energy feature, RIXS also measures other phonon excitations [55,56]. The electron-phonon coupling involved with states near the Fermi level is weak for the apical mode [57]. The contribution of apical phonons to RIXS can be neglected. We, therefore, fit RIXS spectra with three components for highenergy phonons, i.e., bond-stretching (BS), A 1g buckling, and B 1g buckling phonons [30,32,[58][59][60], as shown in Figs. 4(c) and 4(d), which plot fitting spectra of q k ¼ 0.23 and 0.16 at 24 K, respectively. Results of other q k and temperatures are presented in Figs. S9-S11 in Supplemental Material [47]. Figure 4(a) also depicts the phonon dispersions from curve fitting. The diagrammatic calculations explain well the observed photon softening. In addition to phonon softening, the phonon energies from RIXS agree with those are from density-functional-theory (DFT) calculations [30] and measurements of Raman scattering [59], inelastic neutron scattering [58], IXS [28], and photoemission [60]. We also fit the RIXS data of T ¼ 24 K with three phonon modes, i.e., BS, buckling, and acoustic phonons; the fitting results plotted in supplemental Fig. S12 [47] show a phonon softening consistent with that from the four-phonon fitting. That is, our curve fitting results show that these phonons all exhibit the strongest softening at q k ¼ ð0.25; 0Þ rather than q CDW , supporting the scenario of discommensurate CDWs in LSCO at doping level x ¼ 0.15. In domains of discommensurate CDWs, phonons are strongly coupled to charge fluctuations, and the phase shift between x rays from separate domains is negligible. The phonon softening, therefore, occurs at q k ¼ ð0.25; 0Þ for CDW domains of period 4a, in which a is the in-plane lattice constant.
To further understand the characteristics of CDWs in LSCO, we examine the coupling of CDWs to phonons through performing DFT calculations in the scheme of frozen phonons, which are adopted to study qualitatively the coupling of longitudinal BS mode and apical phonon mode of A 1g symmetry with a periodicity of 4a. See Supplemental Material [47] for the details. The calculations show that the apical mode has an energy lower than that of the BS mode. Whereas the vibration of either the apical or the BS mode raises the total energy, the vibrations of these two modes together produce an energy gain for the CDW state as compared with the vibration of each single mode. This energy gain indicates that the coupling of these two phonon modes with CDWs is favorable; the coupling strength, measured by the energy gain, is found to be stronger than that with other periodicities, such as 6a. The coupling strength also enhances with increasing hole concentrations. These findings are in agreement with the RIXS results that strong phonon softening occurs about q k ¼ 0.25. One plausible scenario is that the CDW has effectively a period 4a and is short ranged [51]; a shift in the wave vector of a large sample appears in the Fourier space, giving rise to the formation of discommensurate CDWs.
Although the data presented in the present work are only from a sample near the optimal doping, quantum critical scalings of charge fluctuations in the temperature domain are clearly demonstrated. These results are consistent with the speculation on the existence of a QCP. On the other hand, by suppressing superconductivity in high magnetic fields, linear scalings in resistivity are observed in a more extended regime around the putative QCP near a doping level of 0.2 [39]. While the extended linear-resistivity region shows quantum scaling in temperature, it does not exhibit the quantum critical scaling in the nonthermal parameter, doping. Thus, the correspondence between quantum criticality and T-linear resistivity may not be of necessity [61,62]. The apparent extended critical region that is overlapped with the putative QCP may just be a non-Fermi-liquid phase [63]. In this scenario, the CDW phase becomes a non-Fermi-liquid phase as the doping level is tuned across the putative QCP. Clearly, to clarify these issues, future work on samples (both LSCO and Nd-LSCO) with different dopings around the critical point and inside the region of long-range charge-ordered phase (Nd-LSCO sample) is required in order to unequivocally prove the existence of QCP related to charge order in LSCO.
In conclusion, we observe short-range CDWs in LSCO near the optimal doping over a wide temperature range; these orders compete locally with the superconducting order. Through combining high-resolution RIXS measurements, theoretical modeling, and diagrammatic calculations, our results provide evidence for phonon softening in superconducting cuprates induced by charge order. The observation of phonon softening indeed results from quantum fluctuations of the charge order.