{'fraction_non_alphanumeric': 0.037539376968848445, 'fraction_numerical': 0.024588729436471824, 'mean_word_length': 4.649530400395452, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Dispersive charge density wave excitations and temperature dependent commensuration in Bi2Sr2CaCu2O8+{\delta}

Experimental evidence on high-Tc cuprates reveals ubiquitous charge density wave (CDW) modulations, which coexist with superconductivity. Although the CDW had been predicted by theory, important questions remain about the extent to which the CDW influences lattice and charge degrees of freedom and its characteristics as functions of doping and temperature. These questions are intimately connected to the origin of the CDW and its relation to the mysterious cuprate pseudogap. Here, we use ultrahigh resolution resonant inelastic x-ray scattering (RIXS) to reveal new CDW character in underdoped Bi2Sr2CaCu2O8+{\delta} (Bi2212). At low temperature, we observe dispersive excitations from an incommensurate CDW that induces anomalously enhanced phonon intensity, unseen using other techniques. Near the pseudogap temperature T*, the CDW persists, but the associated excitations significantly weaken and the CDW wavevector shifts, becoming nearly commensurate with a periodicity of four lattice constants. The dispersive CDW excitations, phonon anomaly, and temperature dependent commensuration provide a comprehensive momentum space picture of complex CDW behavior and point to a closer relationship with the pseudogap state.

by the pivotal role that RIXS has recently played in revealing orbital and magnetic excitations in cuprates [16][17][18][19][20] . In addition, RIXS provided the first x-ray scattering evidence for an incommensurate CDW in the (Y,Nd)Ba 2 Cu 3 O 6+δ [4], owing to energy resolution that separated the quasi-elastic CDW signal (bright spot in Fig. 1b, limited by the instrumental resolution ~ 130 meV) from other intense higher-energy excitations.
Notably this quasi-elastic signal is asymmetric with respect to zero energy loss (Fig. 1c), which indicates the possible existence of additional low energy excitations near the CDW wavevector (Q CDW ).
In this work, we exploit the newly commissioned ultrahigh resolution RIXS instrument at the European Synchrotron Radiation Facility to reveal these low energy excitations near the CDW. We choose the double-layer cuprate Bi 2 Sr 2 CaCu 2 O 8+δ (Bi2212), whose electronic structure has been extensively studied by surface-sensitive spectroscopy, such as scanning tunneling microscope 21 and angle-resolved photoemission 22 and in which a short-range CDW order was recently reported 7, 8 . With improved energy resolution up to 40 meV, we see additional features in the previous quasi-elastic region (Fig. 1c).
Figure 2a presents an energy-momentum RIXS intensity map of our high resolution data. Two excitation branches are clearly observed with a momentum dependent intensity distribution, also evident in the energy-loss spectra at representative momentum (Fig. 2b). The first branch centered at zero energy should contain the scattering signal due to an underlying CDW. Indeed, as plotted in Fig. 2c, the momentum-distribution of the RIXS intensity averaged over a small energy window exhibits a symmetric peak at a finite momentum with a peak-to-background ratio of ~ 2, unambiguously confirming the existence of the CDW in Bi2212. The peak position is located at Q CDW ~ 0.3 reciprocal lattice units (r.l.u.), consistent with previous STM studies 8  Interestingly, the phonon intensity varies non-monotonically with momentum, having a maximum near Q A ~ 0.37 r.l.u. (Fig. 3a). Examining the momentum distribution curves (Fig. 3b), the peak at zero energy disperses and broadens with increasing energy loss, smoothly connecting to the phonon at Q A . To better visualize this connection, we remove the elastic contribution from the energy-loss spectra (Fig. 3c) to reveal a funnelshaped RIXS intensity emanating from Q CDW , suggesting that these excitations are 5 associated with the CDW. It appears that these CDW excitations disperse to high energy and intersect with the phonon at Q A , causing the phonon intensity anomaly.
It is important to note that the RIXS phonon cross-section directly reflects the momentum-dependence of the electron-phonon coupling strength 25-27 (Supplementary Text), in stark contrast with the phonon self-energy measured by other scattering techniques. As a consequence, RIXS is also directly sensitive to the interference effect between phonons and underlying charge excitations, i.e. the Fano effect, which can manifest as an intensity anomaly. Figure 3f illustrates this point, showing a calculated RIXS intensity map for a 1-D metallic system with coupling between electrons and a bond stretching phonon mode. Perfect Fermi surface nesting at 2k F (i.e. twice the Fermi momentum) creates a particle-hole continuum, whose presence softens the phonon dispersion and creates an intensity anomaly at the intersection between the continuum and the phonon (Fig. 3d-f). This result qualitatively agrees with our observations in Bi2212, lending further support to the notion that some form of charge excitations associated with the CDW exists in underdoped Bi2212 and causes the observed phonon anomaly. By connecting Q CDW and Q A , we deduce a characteristic velocity of these CDW excitations: V CDW ~ 0.6 ± 0.2 eVÅ. We find that V CDW is neither equivalent to the Fermi velocity seen in ARPES nor the quasi-particle-interference dispersion seen in STM 21 (Supplementary Text), indicating that the weak coupling fermiology may be irrelevant for describing the observed dispersion.
Temperature dependent measurements were conducted on another Bi2212 crystal with a similar T c (40 K) with results shown in Fig. 4. We first confirm that the low temperature measurement at 20 K reproduces all the observations shown in Fig. 2 6 ( Supplementary Fig. 3). Upon increasing the temperature to 240 K (near the pseudogap temperature 22 ), we find that the CDW signal in the quasi-elastic region persists (Fig. 4a).
Surprisingly, Q CDW changes from incommensurate 0.3 r.l.u. at 20 K to nearly commensurate ~ 0.26 r.l.u. at 240 K ( Fig. 4b and c), which corresponds to a real-space periodicity of approximately four lattice constants (~4a o ), similar to the stripe ordered cuprates 1,3 . We also observed that the intensity anomaly at Q A diminishes and broadens, as shown by the momentum-distribution of the phonon intensity in Fig. 4d Finally, the CDW survives up to at least 240 K ~ T*, which taken together with the similarities to striped nickelates indicates a close relationship between the CDW, spins, and the pseudogap.

Sample growth, preparation and characterization
The high quality UD-Bi2212 (Bi 2.2 Sr 1. 8

Low-resolution RIXS measurements on UD-NBCO
The low energy resolution RIXS data on the UD-NBCO (Nd 1.2 Ba 1.8 Cu 3 O 6+δ ) sample were taken at the ADRESS beamline of the Swiss Light Source at the Paul Scherrer Institut (PSI, Switzerland) using the SAXES spectrometer 31,32 . The RIXS spectra were measured at T = 60 K, just below Tc = 65 K to maximize the CDW peak intensity and using linear vertical (σ) polarized incident photons. The total energy resolution was ~130 meV, the scattering angle was 130°. The RIXS map is obtained by combining spectra measured every 2° in the rotation of the incidence angle on the sample surface.
The NBCO films, 100 nm think, were deposited by high oxygen pressure diode sputtering on SrTiO 3 (100) single crystals. The lattice parameters measured by x-ray diffraction are a = b = 3.84 Å and c =11.7 Å.

Ultrahigh resolution RIXS measurements on UD-Bi2212
The RIXS measurements on the UD-Bi2212 sample 1 and 2 were performed using the ERIXS spectrometer at the ID32 beamline of the European Synchrotron Radiation Facility (ESRF, France). The RIXS spectra were taken with the photon energy of the incident x-rays tuned to the maximum of the absorption curve near the Cu L 3 -edge ( Supplementary Fig. 1c). The scattering geometry is sketched in Supplementary Fig. 1d. surface. In our convention, positive Q // corresponds to grazing-emission geometry and negative Q // corresponds to grazing-incidence geometry.

Data analysis and fitting
The ultrahigh resolution data (UD-Bi2212) were normalized to I 0 and corrected for self-absorption effects using the formalism described in Ref. 33 Fig. 3a). At a high temperature (T = 240 K), the phonon softening is shifted to a lower wavevector. In addition, we note that the phonon intensity extracted from the fitting procedure agrees with the averaged intensity around the phonon energy ( Supplementary Fig. 4b).

Theory
The calculations depicted in

Data availability:
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Competing financial interests
The authors declare no competing financial interests.

Corresponding authors
Correspondence to W. S. Lee (leews@stanford.edu), T. P. Devereaux (tpd@stanford.edu), or Z. X. Shen (zxshen@stanford.edu).        is comparable with that in YBCO. It might be interesting to also compare the integrated CDW intensity, which presumably is more representative to the square of the amplitude of the CDW order parameter Δ 2 . To estimate the integrated peak intensity, we assume that the CDW is two-dimensional and isotropic (i.e. same correlation lengths along a and b). The CDW peak widths used for the estimation are 0.03 r.l.u. and 0.085 r.l.u. for YBCO and Bi2212, respectively. We found that Δ Bi2212 2 / Δ YBCO 2 ~ 5. We note that this estimate is crude, but nevertheless suggest that the CDW strength in Bi2212 is not weaker than that in YBa 2 Cu 3 O 6.6 , despite of shorter correlation length.

Comparison with the Cu-O bond stretching phonon of Bi2201 measured via IXS
The Supplementary Fig. 4a presents a second derivative analysis of the raw data plotted in Fig. 2a 4b). This emphasizes an important aspect of the RIXS phonon cross-section -it directly reflects the momentum-dependent electron-phonon coupling strength 27 rather than the phonon self-energy that is probed by inelastic neutron scattering and IXS.

Estimation of the quasi-particle-interference velocity
We find that the quasi-particle-interference (QPI) velocities, V QPI ~ 0.2 -0.4 eVÅ, which is deduced from linear fits to the dispersions of the QPI wavevectors q 1 from the Fourier transformed STM conductance maps reported in literatures 8,35,36 . In particular, we also estimate the QPI velocity from the energy-resolved STM conductance map of the Bi2212 sample having a doping concentration similar to our sample (T c = 45 K) plotted in Ref. 8. We found V QPI ~ 0.3 eVÅ at this doping concentration. In summary, V QPI is smaller than our observed V CDW ; thus they may not be directly related.

Estimation of the Fermi velocity
The Fermi velocities can be extracted from the analysis of the ARPES data. These measurements were performed at Stanford Synchrotron Radiation Lightsource (SSRL) beamline 5-4 with a SCIENTA R4000 analyzer and the measured UD50-Bi2212 (Bi2212 single-crystals with a Tc = 50 K) samples were from the same sample growth batch as those used for the RIXS measurements. The data were taken using 18.4 eV photons with linear polarization perpendicular to the analyzer slit (measured cuts). The total energy and momentum resolution were set to ~10 meV and ~0.25°, respectively. The samples were cleaved and measured at 10 K with a pressure better than 3x10 -11 Torr. The