Unusual dynamic charge-density-wave correlations in HgBa$_2$CuO$_{4+\delta}$

TL;DR

This study uses high-resolution RIXS to reveal complex, energy-dependent charge-density-wave correlations in underdoped HgBa$_2$CuO$_{4+\delta}$, linking them to pseudogap phenomena and magnetic excitations.

Contribution

First high-resolution RIXS measurement of dynamic CDW correlations in HgBa$_2$CuO$_{4+\delta}$, revealing multiple energy scales and their relation to pseudogap and magnetic excitations.

Findings

Dynamic CDW correlations at ~40 meV above T_C

Emergence of a high-energy (~160 meV) feature in charge response

No significant coupling between CDW and magnetic excitations

Abstract

The charge-density-wave (CDW) instability in the underdoped, pseudogap part of the cuprate phase diagram has been a major recent research focus, yet measurements of dynamic, energy-resolved CDW correlations are still in their infancy. We report a high-resolution resonant inelastic X-ray scattering (RIXS) study of the underdoped cuprate superconductor HgBa$_{2}$CuO$_{4+\delta}$ ($T_c = 70$ K). At $T=250$ K, above the CDW order temperature $T_\mathrm{CDW} \approx 200$ K, we observe significant dynamic CDW correlations at about 40 meV. This energy scale is comparable to both the superconducting gap and the previously reported low-energy pseudogap. At $T = T_c$, a strong elastic CDW peak appears, but the dynamic correlations around 40 meV remain virtually unchanged. In addition, we observe a new feature: dynamic correlations at significantly higher energy, with a characteristic scale of about 160 meV. A similar scale was previously identified in other experiments as a high-energy pseudogap. The existence of three distinct features in the charge response is highly unusual for a CDW system, and suggests that charge order in the cuprates is closely related to the pseudogap phenomenon and more complex than previously thought. We further observe the paramagnon dispersion along [1,0], across the two-dimensional CDW wavevector $\boldsymbol{q}_\mathrm{CDW}$, which is consistent with magnetic excitations measured by inelastic neutron scattering. Unlike for some other cuprates, our results point to the absence of a discernible coupling between CDW and magnetic excitations.

Full text

I Introduction

The high-$T_{c}$ cuprates are doped charge-transfer insulators with lamellar structures that feature the quintessential CuO2 plane Keimer et al. (2015). At moderate and intermediate hole doping, these complex oxides exhibit a partial depletion of the density of states at the Fermi-level (the pseudogap, PG). The PG state is characterized by myriad ordering tendencies, including CDW order with a modulation direction along the planar Cu-O bond and a periodicity of 3-4 lattice units Tranquada et al. (1995); Hoffman et al. (2002); Howald et al. (2003); Wu et al. (2011); Ghiringhelli et al. (2012); Chang et al. (2012); Comin et al. (2014); Tabis et al. (2014); Croft et al. (2014); Hashimoto et al. (2014); Wu et al. (2015); da Silva Neto et al. (2014, 2015); Gerber et al. (2015); Chang et al. (2016); Peng et al. (2016); da Silva Neto et al. (2016); Tabis et al. (2017); Arpaia et al. (2019). Whereas CDW order competes with superconductivity Chang et al. (2012, 2016); Gerber et al. (2015), dynamic CDW correlations have long been argued to play a pivotal role in shaping the phase diagram (Fig. 1a) Castellani et al. (1997); Kivelson et al. (2003); Caprara et al. (2017). However, key questions remain unresolved. The mechanism of CDW formation could be related to Fermi-surface nesting (i.e., a reciprocal-space mechanism) Hanaguri et al. (2004); Shen et al. (2005); Wise et al. (2008), or to strong electronic correlations that lead to charge separation (i.e., a real-space mechanism) Tranquada et al. (1995); Abbamonte et al. (2005). Furthermore, the relation between CDW correlations and the PG is far from understood, with suggestions that either one is the underlying phenomenon Castellani et al. (1997); Caprara et al. (2017); Pelc et al. (2019a); Arpaia et al. (2019); Hamidian et al. (2016). The short correlation lengths indicate that disorder might play an important role Caplan et al. (2015), but it is still debated how the CDW in the cuprates becomes static on cooling. Finally, whereas a prominent early theoretical proposal argued in favor of instantaneous pairing Anderson (2007), ultrafast optical measurements indicate that the interaction in the cuprates is most likely retarded Dal Conte et al. (2012); Dal Conte et al. (2015). Although the relevant bosonic excitations are often argued to be of magnetic origin Abanov et al. (2003); Scalapino (2012), there exists indirect evidence that charge correlations play an important role Fanfarillo et al. (2016); Pelc et al. (2019a).

CDW correlations have been detected in numerous cuprates, primarily via resonant X-ray scattering (RXS) Ghiringhelli et al. (2012); Chang et al. (2012); Comin et al. (2014); Tabis et al. (2014); Croft et al. (2014); Hashimoto et al. (2014); da Silva Neto et al. (2014, 2015); Blanco-Canosa et al. (2014); Gerber et al. (2015); Wu et al. (2015); Chang et al. (2016); Peng et al. (2016); da Silva Neto et al. (2016); Chaix et al. (2017); Tabis et al. (2017), including HgBa2CuO4+δ (Hg1201) Tabis et al. (2014, 2017). In RXS, the incident photon energy is tuned to the Cu $L_{3}$ edge to discern modulations of the valence electron density in the quintessential CuO2 layers. The scattered photons can be directly measured in energy-integrated mode (conventional RXS), or additionally analyzed by a spectrometer in energy-resolved inelastic mode (RIXS). Conventional RXS has the benefit of relatively short counting times and enabled the efficient exploration of the doping and temperature dependence of CDW correlations. In principle, the charge dynamics can be measured via RIXS, yet until recently, the best available energy resolution was well above 100 meV.

We report a RIXS study of the charge dynamics of underdoped Hg1201 ($T_{c}=70$ K; see Fig. 1a) across $\bm{q}_{CDW}$ with energy resolution of 60 meV. Although this energy resolution is still an order of magnitude larger than that of, e.g., THz experiments Torchinsky et al. (2013); Hinton et al. (2013), it is very high for RIXS, and we are able to obtain qualitatively new energy- and momentum-resolved information. Hg1201 is a single-layer compound, with a simple tetragonal crystal structure and an optimal $T_{c}$ of nearly 100 K Eisaki et al. (2004); Baris̆ić et al. (2008); Li et al. (2011); Barišić et al. (2013a, b); Mirzaei et al. (2013); Chan et al. (2014); Barišić et al. (2019); Chan et al. (2016a, b); Popčević et al. (2018); Pelc et al. (2019a, b). The model nature of Hg1201 is exemplified, e.g., by the observations that the normal-state magnetoresistance in the PG state exhibits Kohler scaling Chan et al. (2014) and that low-temperature transport measurements reveal Shubnikov-de-Haas oscillations due to Fermi-surface reconstruction associated with the CDW order Barišić et al. (2013b); Chan et al. (2016b). The zero-field CDW order measured via RXS exhibits short in-plane correlation lengths, and there is no evidence for a phase transition Tabis et al. (2014, 2017). Gaining further understanding of these emergent correlations, and of their connection with the PG phenomenon and the pairing glue, is the primary goal of the present work. The comparatively high onset temperatures, reduced dimensionality, and short length scales suggest that significant dynamic CDW correlations might be present, in analogy with well-known systems such as NbSe2 in the presence of point disorder Berthier (1978); Arguello et al. (2014); Wu et al. (2015); Chatterjee et al. (2015). On the other hand, the CDW phenomenology in the cuprates is quite different from conventional systems, as it occurs in the part of the phase diagram that is dominated by the highly unusual PG phenomenon.

RIXS is a unique probe to study such effects. Indeed, at 250 K, well above the temperature ($T_{\mathrm{CDW}}\approx 200$ K) below which evidence for quasistatic short-range CDW order was previously observed in Hg1201 via RXS Tabis et al. (2014, 2017) and time-resolved optical reflectivity Hinton et al. (2016), we only discern $dynamic$ charge correlations, with a characteristic energy of about 40 meV. At 70 K ($\approx T_{c}$), on the other hand, we observe quasistatic, short-range CDW correlations, and the dynamic response now includes an additional feature in the 100-250 meV range that is broadly peaked at about 165 meV. The response centered at about 40 meV does not appear to be affected on cooling from 250 K to $T_{c}$ and may be of mixed charge/phonon character. These two energy scales appear in other observables, and have been associated with both the PG Honma and Hor (2008) and the superconducting pairing glue Dordevic et al. (2005); Hwang et al. (2007); van Heumen et al. (2009); Muschler et al. (2010); Ahmadi et al. (2011); Dal Conte et al. (2012). Moreover, the larger of the two scales has been linked to the strong electronic correlations that cause the charge-transfer gap of the undoped parent insulators Pelc et al. (2019a, b). Our experiment therefore potentially detects not only CDW correlations, but charge correlations in a broader sense. We demonstrate that energy integration of the new RIXS data yields the prior RXS result for Hg1201, including the high-temperature feature seen at $\bm{q}_{\mathrm{CDW}}$ in Hg1201 and other cuprates Tabis et al. (2014, 2017). This high-temperature feature, which is typically treated as background, is therefore shown to be associated with dynamic charge correlations. From a comparison with RXS data for other single-layer cuprates at the doping level for which the CDW order is most robust, we find that the strength of the high-temperature dynamic signal is consistently 35-50 % of the signal at $\sim T_{\mathrm{c}}$. Given the ubiquitous presence of the high-temperature feature in prior RXS data, even at doping levels where CDW correlations are weak, this points to the universal existence of significant dynamic charge (or mixed charge/phonon) correlations throughout the temperature-doping phase diagram. We discuss the possibility that these charge correlations are rooted in the doped-charge-transfer-insulator nature of the cuprates.

The present data naturally yield the paramagnon dispersion along [1,0], which we find to be consistent with magnetic neutron scattering results Chan et al. (2016a); Anderson et al. . The magnetic excitations are nearly unaffected by the unusual charge correlations, consistent with their somewhat different energy scales. However, comparison with the characteristic energy scales of the glue function extracted from optical and Raman spectroscopy points to the distinct possibility that both charge and magnetic excitations contribute to the superconducting pairing glue in the cuprates.

II Experimental Methods

The measurements were performed with the ERIXS spectrometer at beam line ID32 of the European Synchrotron Radiation Facility (ESRF), Grenoble, France Brookes et al. (2018). The incident X-ray energy was tuned to the maximum of the Cu $L_{3}$ absorption peak around 932 eV, and the X-ray polarization was set either parallel ($\pi$) or perpendicular ($\sigma$) to the scattering plane Sup . The scattered photons were analyzed without considering the final-state polarization. The energy resolution was approximately 60 meV, as determined from the full width at half maximum (FWHM) of the non-resonant spectrum of a standard polycrystalline silver sample. In order to prepare a clean, high-quality surface, the Hg1201 single crystal was cleaved ex situ to reveal a face parallel to the CuO2 planes.

The cleaved surface was uniform and large compared to the beam size, thus minimizing the possibility of the beam drifting off the sample. In order to avoid effects due to beamline energy drifts, the resonant energy was periodically checked via X-ray absorption spectroscopy at the Cu $L$ edge. At each momentum transfer, the energy zero was determined by measuring the non-resonant spectrum of a polycrystalline silver standard located near the sample. The data were collected in short time intervals of 45 min, and each scan was corrected for the energy shift, if any was observed. In order to avoid errors in the determination of the energy zero, the thickness of the silver was approximately the same as the thickness of the sample, and both were placed on the same axis of rotation. As a consistency check, we monitored the energy of $dd$-excitations and found no significant change with temperature in the effective $dd$ excitation energy, and hence in the energy zero Sup . Importantly, the energy resolution (60 meV, FWHM) and observed energy scales ($\sim 40$ meV and $\sim 165$ meV) are large compared to typical beamline energy drifts, thus minimizing the chance of an adverse affect on our conclusions.

Momentum scans were performed by rotating the sample about the axis perpendicular to the scattering plane, and the detector angle was set to $2\theta=150^{\circ}$. The scattering wave vector is $\bm{Q}=H\bm{a^{*}}+K\bm{b^{*}}+L\bm{c^{*}}=(H,K,L)$ in reciprocal lattice units, where $a^{*}=b^{*}=1.62$ Å*-1* and $c^{*}=0.66$ Å*-1*. $K$ was chosen to be zero, and the scans were taken along $[H,0,L]$, with $L$ coupled to $H$. Due to the short-range nature of the observed two-dimensional (2D) correlations, the $L$ dependence of the cross section is expected to be negligible, and we quote the 2D reduced wave vector $\bm{q}=(H,K)$. The intensity of the RIXS spectra was normalized to the integrated intensity of $dd$ excitations, following prior work Braicovich et al. (2010).

III Results

Figure 1(b,c) shows RIXS intensity contour plots at $T=70$ K and 250 K, respectively, as a function of $H$ and energy transfer $\omega$ (negative $\omega$ corresponds to energy loss), obtained with $\sigma$-polarized incident X-rays directly from individual energy scans such as those shown in Fig. 1(d) Sup . The maps clearly display signal around $\bm{q}_{CDW}\approx(0.28,0)$, consistent with prior RXS work Tabis et al. (2014, 2017). Whereas the dominant signal at 70 K is quasielastic, at 250 K ($>T_{\mathrm{CDW}}$) the response is dynamic and centered in the optic phonon range. Due to the larger phonon contribution, the 250 K spectra exhibit higher intensity away from $\bm{q}_{CDW}$ than the 70 K data. Although weaker, as expected, charge correlations are also observed in $\pi$ polarization Sup .

A more detailed data analysis reveals additional information. Figure 2(a-d) shows the momentum dependence of RIXS signal (obtained from energy scans such as those in Fig. 1(d), with $\sigma$ polarization) integrated over the FWHM energy resolution (60 meV) with different energy ranges: quasielastic (-30, +30) meV; inelastic (-90, -30) meV, (-150, -90) meV, and (-210, -150) meV. In order to arrive at a systematic estimate of the $q$-integrated signal strength, we perform a heuristic fit of the data to a Gaussian peak, with fixed center ($\bm{q}_{CDW}$) and width ($\sim 0.075$ rlu), plus a concave, polynomial contribution. For Hg1201, the optic phonon range extends to about 75 meV d’Astuto et al. (2003); Uchiyama et al. (2004); Rez , and for energy transfers in this range the “background” contribution invariably includes phonon scattering; for all energy-integration ranges, this smoothly-varying contribution is indistinguishable at 70 K and 250 K.

The “elastic line” in inelastic scattering mainly originates from scattering off defects and disorder. Due to the high quality of our cleaved sample, the true elastic line is almost impossible to resolve in the raw spectra (Fig. 1d and Fig. S2 in Sup ), as the quasi-elastic signal is always contaminated by other low-energy excitations (mostly phonons). The intensity increase around zero energy and $\bm{q}_{CDW}$ due to quasistatic CDW correlations is clearly distinct from the elastic response. Our analysis indicates that the tail of the energy resolution vanishes faster than a Lorentzian function. The increase of the elastic line at small $H$ at 250 K (Fig. 2a) does not increase the inelastic signal (Fig. 2b), which further supports our assessment that the high-energy signal is genuine.

Figure 3(a,b) shows the energy dependence of the Gaussian amplitude obtained in this manner for $\sigma$- and $\pi$-polarization, respectively. As expected for charge scattering, the signal is weaker for $\pi$-polarization Sup . The 250 K results show broad peaks centered at $\sim 40$ meV and no evidence for elastic scattering, as the energy dependence of the amplitude is fully captured by the sum of Stokes and anti-Stokes scattering. The peaks are considerably broader than the 60 meV energy resolution. Assuming a heuristic intrinsic Gaussian profile, we estimate an intrinsic width of at least 50 meV (FWHM) after resolution deconvolution. The RIXS signal may be interpreted either as a single charge mode, with a width that is comparable to its characteristic energy, or as the result of a distribution of charge modes. It is equally well described by a log-normal distribution Sup . We note that our analysis removes phonon contributions that vary smoothly with $H$. Whereas the broad peak in Fig. 1(c) contains low-energy excitations, especially phonons, the extracted intensity at $\sim 40$ meV shown in Fig. 3(a,b) signifies an enhanced charge response at $\bm{q}_{CDW}$. As discussed in detail below, this enhanced response may in part be due to anomalous phonon scattering.

At 70 K, the dominant charge response is quasistatic. The width of this peak is larger than the energy resolution, consistent with a dynamic contribution at $\sim 40$ meV that is unchanged from the response at 250 K. This distinct possibility is highlighted for both polarizations in the (Bose-factor-corrected) intensity difference plots in Fig. 3(c,d), which reveal resolution-limited elastic peaks centered, within error, at $\omega=0$.

Interestingly, at 70 K we also observe dynamic charge fluctuations well above the optic phonon range. This is directly seen from Fig. 2, especially Fig. 2(e), where the large binning range (-280,-120) meV was chosen in order to optimize signal-to-background. These data, obtained with $\sigma$-polarization, are contrasted in Fig. 2(f) with the equivalent result with $\pi$-polarization, which is more sensitive to magnetic scattering Sup . The convex momentum dependence can be attributed to paramagnons, which become prominent above $\sim 200$ meV at $\bm{q}_{CDW}$ (see below). This is seen from the comparison in Fig. 2(f) with the result obtained with narrower (-200,-120) meV integration, which yields an approximately linear background consistent with the $\sigma$-polarization result in Fig. 2(e). From Fig. 3(a), the high-energy charge signal is seen to be peaked at about 165 meV; it is not discerned in $\pi$-polarization (Fig. 3(b,d)) due to the higher background level (proximity to paramagnon excitations; Fig. 2(f)) and lower expected charge scattering cross section (by a factor of two Sup ). We note that the 250 K data in Figs. 2 and 3(a) are consistent with non-zero high-energy charge signal, although with a smaller amplitude and potentially larger momentum width than at 70 K.

Figures 1(d) and 4(a) reveal an additional broad, dispersive peak in the 0.15 to 0.4 eV range. This feature, which has been observed in a number of cuprates, signifies paramagnon scattering that evolves from well-defined antiferromagnetic excitations in the undoped parent compounds Haverkort (2010); Ament et al. (2009). These excitations are more prominent in $\pi$- than in $\sigma-$polarization in the present scattering geometry, as expected for magnetic scattering Sup ; Le Tacon et al. (2011); Murakami and Ishihara (2017). We extract the paramagnon energy by fitting the raw RIXS spectra to a resolution-limited elastic peak, an effective phonon peak, and a damped paramagnon excitation Sup . Figure 4(b) summarizes our result for the paramagnon dispersion along [1,0] at 70 K. We compare the RIXS data with magnetic neutron scattering data near the antiferromagnetic wave vector for two Hg1201 samples (one with essentially the same doping level and $T_{c}=71$ K Chan et al. (2016a), and the other with $p\approx 0.064$ and $T_{c}=55$ K Anderson et al. ) and find that these results are highly consistent and complementary. From a heuristic fit of the combined neutron and X-ray data above $H=0.1$ rlu to simple linear spin-wave theory, we obtain an effective nearest-neighbor exchange of 123(3) meV. The relatively large uncertainty and limited data near the zone boundary prevent a more detailed analysis Coldea et al. (2001); Headings et al. (2010); Ivashko et al. (2017). Overall, the RIXS data for the paramagnon dispersion in Hg1201 are consistent with prior measurements for other hole-doped cuprates Le Tacon et al. (2011); Dean et al. (2013); Ivashko et al. (2017); Sup .

IV Discussion

We observe charge signal in three different energy ranges: (1) a quasistatic response at 70 K; (2) low-energy ($\sim 40$ meV) signal at 250 K and 70 K; (3) high-energy fluctuations in the 100 - 250 meV range at 70 K. Below, we first discuss how our observations differ from conventional CDW systems. We then consider the possibility that the 40 meV feature signifies a phonon-related effect. Finally, we discuss our findings within the broader context of the cuprate phase diagram and, in particular, the PG phenomenon and the superconducting pairing glue.

IV.1 Cuprates vs. conventional CDW systems

Analogous to conventional superconductors, conventional CDW systems are described by BCS theory Kohn and Sherrington (1970). The CDW state is a condensate of electrons and holes, characterized by a complex order parameter, whose net momentum corresponds to the wavelength of the charge order. In clean, weak-coupling systems, macroscopic phase coherence and amplitude formation occur simultaneously at the mean-field transition temperature. In the presence of disorder or at strong coupling, on the other hand, the transition temperature can be significantly depressed, and such systems may exhibit extended PG-like behavior, with a gap that persists to high temperature in the absence of long-range order. Such an extended regime has been demonstrated for the canonical CDW compound $2H-$NbSe2 intercalated with Mn and Co, i.e., in the presence of point disorder Berthier (1978); Arguello et al. (2014); Wu et al. (2015); Chatterjee et al. (2015).

The PG and charge-order behavior of the cuprates are more complex. These compounds transform from Fermi-liquid metal at high doping, with a large Fermi surface that encompasses $1+p$ carriers, to a Mott insulator at zero doping ($p=0$) via an intermediate state with carrier density $p$ Padilla et al. (2005); Ono et al. (2007); Barišić et al. (2013a). This evolution involves the localization of exactly one hole per planar CuO2 unit and the concomitant opening of a PG on the “antinodal” parts of the underlying Fermi surface, which leaves ungapped nodal Fermi arcs with carrier density $p$ below $T^{**}$ (Fig. 1), the lowest of three characteristic PG temperatures (the other two are $T^{*}$ and $T_{hump}$, with $T^{**}<T^{*}<T_{hump}$) Honma and Hor (2008); Pelc et al. (2019a). This Mott-pseudogap phenomenon is distinct from the behavior of a canonical CDW system in the presence of strong coupling and/or disorder Chatterjee et al. (2015). The onset of short-range CDW correlations is generally observed at or below $T^{**}$ (Fig. 1). Associated with the three characteristic PG temperatures are characteristic energy scales, which at the doping level of the present study are about 40-50, 60-80, and 180-220 meV Honma and Hor (2008); Alldredge et al. (2013). The lowest of these scales is the value of the PG near the tip of the arcs Alldredge et al. (2013), whereas the largest scale has been associated with the charge-transfer gap of the insulator, renormalized by the itinerant carriers Pelc et al. (2019a, b). Since the PG formation is gradual and precedes the CDW order Gomes et al. (2007); Tal , and because the wavevector $\bm{q}_{CDW}$ appears to connect the tips of the arcs Tabis et al. (2014); Chan et al. (2016b); Tabis et al. (2017), the 40-50 meV PG scale likely sets an upper bound for the CDW gap scale. From the BCS expression $2\Delta=3.5k_{B}T_{CDW}$, with $T_{CDW}\sim 200$ K (Fig. 1), one would expect a gap of $\Delta\sim$ 30 meV. Given the considerable point disorder exhibited by the cuprates Eisaki et al. (2004), it is perhaps not surprising to observe CDW correlations over an extended temperature range below $T_{CDW}$, along with a dynamic signature above this nominal ordering temperature, which in this scenario represents a pinning temperature rather than a true phase transition.

However, in this case one would expect a temperature-dependent dynamic scale and a transfer of spectral weight from the dynamic to the static response on cooling, yet this is not what we observe. The difference plots in Fig. 3(c,d) indicate that the Bose-factor corrected weight of the $\sim 40$ meV inelastic contribution is temperature-independent, since it cancels out within error for the two temperatures. Equivalently, there seems to be little spectral weight transfer from the $\sim 40$ meV dynamic to the quasistatic contribution on cooling, which would not be expected if the CDW were simply pinned below $T_{CDW}$. Therefore, the quasistatic and $\sim 40$ meV contributions likely have different physical origins. In principle, one possibility is an elastic contribution due to the interference of Friedel oscillations around impurities, as suggested theoretically Dalla Torre et al. (2016) and tested in a quasi-one-dimensional CDW system Yue et al. . The true underlying CDW signal would then be inelastic even at $T_{c}$, and the CDW correlations would remain dynamic at this temperature. Yet even in this case one would expect a temperature-dependent dynamic CDW signal. A more likely possibility is that the CDW response is quasielastic at all temperatures, whereas the dynamic $\sim 40$ meV signal is of a different origin. This conclusion is supported by recent work for NdBa2Cu3O6+δ (NBCO) Arpaia et al. (2019), which found the amplitude and correlation length of the $\omega\approx 0$ signal to exhibit temperature dependences below $T_{CDW}$ that point to a gradual build-up of quasistatic correlations and to an underlying phase transition at a temperature below $T_{c}$. As further discussed in the following subsections, it is therefore a distinct possibility that the temperature-independent (at least up to 250 K) $\sim 40$ meV feature at $\bm{q}_{\mathrm{CDW}}$ is either a direct signature of some other (non-CDW) bosonic charge modes, or the result of anomalous phonon scattering, and hence an indirect signature of such modes.

In a conventional system, CDW order involves Fermi-surface nesting. The present data do not allow us to discern if the CDW in the cuprates forms predominantly due to nesting or a real-space mechanism. We can, however, make some inferences at this point. The doping dependence of the CDW wave vector is consistent with a nesting scenario, and quantum oscillation measurements show that the Fermi surface is reconstructed at low temperatures and high magnetic fields, with a resultant electron pocket whose size is consistent with a simple reconstruction scenario Barišić et al. (2013b); Chan et al. (2016b). However, the quasistatic short-range CDW correlations observed here at $T_{c}$ clearly do not induce a reconstruction, since transport properties are virtually insensitive to the CDW formation at this temperature Doiron-Leyraud et al. (2013). Furthermore, the presence of a significant dynamic component with high onset temperature and the emergent high-energy scale, potentially both signatures of the PG, suggest a strong-coupling scenario, making an underlying real-space mechanism more likely. This is to be expected if the CDW is indeed an emergent phenomenon, since the correlations that cause the cuprate PG appear to be local in real space, as further discussed below. It seems possible that, effectively, both $k$- and $r$-space effects contribute, which could be related to the deeper question of the existence of a well-defined reciprocal space in a material that is inherently inhomogeneous at the nanoscale Krumhansl (1992); Gomes et al. (2007); Alldredge et al. (2013); Pelc et al. (2019a)

IV.2 Nature of the 40 meV excitation

The 40 meV scale is consistent with superconducting gap and PG scales for cuprates with a comparable optimal $T_{c}$ Honma and Hor (2008); Alldredge et al. (2013), and with the lower bound of $\sim 20$ meV for the gap between the CDW-reconstructed pockets deduced from quantum oscillation experiments for Hg1201 Chan et al. (2016b). This scale lies in the optic phonon range and, in principle, the observed excitation might signify phonon scattering due to enhanced electron-phonon coupling, and thus be an $indirect$ signature of significant dynamic charge correlations. Although there is no evidence for anomalous phonon scattering at $\bm{q}_{\mathrm{CDW}}$ off resonance (see Fig. 5b for a prior RXS result for Hg1201 Tabis et al. (2014)), anomalous scattering involving valence electrons could be significantly enhanced at the Cu $L$ edge. In hole-doped Bi2Sr2CaCu2O8+δ (Bi2212) Chaix et al. (2017), charge correlations were identified up to about 60 meV. These correlations were suggested to be dispersive and associated with strong Fano interference at the intersection with the Cu-O bond-stretching phonon. At low temperature, a gradual phonon softening near $\bm{q}_{\mathrm{CDW}}$ was observed, with a minimum energy of 40-45 meV and an intrinsic width of 30-40 meV (FWHM). In Hg1201, at $\bm{q}_{\mathrm{CDW}}$, the Cu-O bond-stretching phonon lies in the 55-60 meV range, whereas the Cu-O bond-bending phonon lies in the 40-45 meV range d’Astuto et al. (2003); Uchiyama et al. (2004); Rez . It is therefore possible that a similar anomaly involving predominantly the bond-bending phonon is present in Hg1201.

Alternatively, the 40 meV feature could be a $direct$ signature of charge fluctuations, for a number of reasons. First, unlike the result for Bi2212 Chaix et al. (2017), our data indicate no change in the $\sim 40$ meV charge response (amplitude, characteristic energy and width) between high and low temperatures (Fig. 3). Second, we estimate the $intrinsic$ width of the excitation to be at least 50 meV (FWHM) (Fig. 3). This value is considerably larger than what might be associated with phonon linewidth broadening, and about 50-100% larger than the width of the peak ascribed to anomalous phonon scattering in Bi2212 Chaix et al. (2017). Third, we note that conventional RXS measurements generally reveal a local intensity maximum at/near $\bm{q}_{\mathrm{CDW}}$ already at high temperatures Ghiringhelli et al. (2012); Tabis et al. (2014); Comin et al. (2014); da Silva Neto et al. (2016, 2014). This feature is commonly regarded as a temperature-independent background and subtracted from the low-temperature data to extract the CDW signal, although it also has been associated with charge correlations da Silva Neto et al. (2016); Chaix et al. (2017), as clearly supported by our RIXS results in Figs. 2 and 3. By integrating over the whole energy range (-10, 2) eV of our experiment, we demonstrate in Fig. 5(a,b) that our RIXS data are highly consistent with the previous RXS study of Hg1201 at the same doping level Tabis et al. (2014). As seen from Figs. 2 and 3, and further discussed in Sup , most of the signal at/near $\bm{q}_{\mathrm{CDW}}$ originates from within the (-0.1,0.1) eV energy range. In Fig. 5c and 5d, we reproduce the corresponding RXS results for Bi2Sr2-xLaxCuO6+δ (Bi2201, $x=0.115$) Comin et al. (2014) and electron-doped Nd2-xCexCuO4 (NCCO, $x=0.145$) da Silva Neto et al. (2016), respectively. Like Hg1201, both of these cuprates are single-CuO2-layer compounds. The chosen doping levels are approximately those at which the CDW phenomenon is most robust. Remarkably, the ratios of the intensity amplitudes at/near $T_{c}$ and $T_{CDW}$ lie in the narrow 35-50% range for all three single-layer cuprates. Along with the insights gained here for Hg1201 (Figs. 2 and 3), this points to the existence of universal dynamic charge correlations, in addition to quasistatic correlations that emerge below $T_{CDW}$. Since phonon dispersions vary among the cuprates, and because any anomalous electron-phonon coupling can be expected to vary as well, this appears to rule out the possibility that the 40 meV feature we observe for Hg1201 is dominated by anomalous phonon scattering. Finally, we note that there exists indirect evidence for collective charge modes from neutron and X-ray scattering experiments of phonon anomalies throughout the superconducting doping range Reznik (2012); Park et al. (2014).

The most likely scenario is that there exist both types of contributions. This is supported by a time- and frequency-resolved optical spectroscopy measurement of optimally-doped Bi2212 that revealed bosonic modes of predominantly electronic character in the optic phonon range, along with a significant strongly-coupled phonon contribution and a small lattice contribution Dal Conte et al. (2012). The above observations point to universal dynamic charge correlations that extend to high energy and temperature, and that have little temperature dependence between room temperature and $T_{c}$. Moreover, given that prior RXS measurements have revealed a local intensity maximum at/near $\bm{q}_{\mathrm{CDW}}$ even at high and low doping levels where there is no evidence for quasistatic, low-temperature CDW correlations Blanco-Canosa et al. (2014); Sup , we conclude that there exist universal dynamic charge correlations in a very large portion of the temperature-doping phase diagram. As discussed in the next subsection, this conclusion is further supported by indications from Raman spectroscopy of underlying dynamic charge modes that may, in fact, extend throughout the entire Brillouin zone. These robust dynamic charge correlations therefore appear to be a precursor of the quasistatic CDW correlations that gradually develop below the doping-dependent temperature $T_{CDW}$ in a relatively narrow portion of the phase diagram, and that ultimately cause Fermi-surface reconstruction at very low temperatures, once superconductivity is suppressed with a $c$-axis magnetic field.

IV.3 Comparison with other results for the cuprates

RIXS measurements of high-energy charge correlations above the optic phonon range have been reported so far only for electron-doped NCCO, where they persist up to approximately 0.4 eV da Silva Neto et al. (2018). Figures 2 and 3 demonstrate for Hg1201 at the investigated (hole) doping level that the full dynamic response peaked at $\bm{q}_{\mathrm{CDW}}$ is captured by integrating up to about 0.3 eV. We estimate that at least 20% of the dynamic charge response at 70 K originates from above the optic phonon range Sup .

A recent RIXS study of hole-doped NBCO Arpaia et al. (2019) found dynamic charge correlations to persist to high temperatures (above $T^{*}$). The experiment was performed with 40 meV (FWHM) energy resolution, and the data were analyzed under the assumption that phonon contributions at q = ($H$,0) can be removed by subtracting data obtained at the same $q$ along q = ($H$,$H$), and by considering intensity differences for data obtained at different temperatures. Dynamic energy scales of 15 meV (at 150 K and 250 K) and 7 meV (90 K) were deduced for an optimally-doped sample with $T_{c}=90$ K ($p\approx 0.17$), whereas scales of 6 meV (150 K and 250 K) and 3 meV (90 K) were obtained for an underdoped sample with $T_{c}=60$ K ($p\approx 0.17$). The high-temperature signal was found to exhibit short spatial correlations and to be centered at a somewhat smaller wave vector than the quasistatic CDW response. Our data for Hg1201 are overall consistent with these findings, but do not allow us to discern an additional $\sim 10$ meV dynamic signal due to the relatively large energy resolution of 60 meV (FWHM) of our experiment. In turn, the NBCO data are consistent with our observation of significant dynamic charge correlations at higher energies Sup .

The large $\sim 165$ meV energy scale identified here for hole-doped Hg1201 is consistent with the high-energy PG (“hump”) scale seen in other observables Honma and Hor (2008); Pelc et al. (2019a), including the mid-infrared peak seen in optical spectroscopy experiments Uchida et al. (1991); Lee et al. (2005). A recent phenomenological model rooted in the dual empirical observations of a universal transport scattering rate Li et al. (2016); Barišić et al. (2019) and inherent inhomogeneity Pelc et al. (2018); Popčević et al. (2018); Pelc et al. (2019c), associates this scale with the charge-transfer gap at zero doping and with the delocalization of one hole per planar CuO2 unit above $T^{**}$ (Fig. 1) Pelc et al. (2019a, b). For Hg1201, at $p\approx 0.09$, the mean (de)localization gap and (Gaussian) gap distribution width that best capture the transport data are $\sim 180$ meV and $\sim 60$ meV (FWHM) Pelc et al. (2019a, b), respectively, consistent with the characteristic CDW scale of 163(12) meV and width of 77(16) meV established in the present work (Fig. 3(a,c)).

The gradual delocalization of one hole per CuO2 unit is a large effect that clearly manifests itself in, e.g., the evolution of the Fermi-surface with doping and the strong temperature and doping dependence of the Hall number Ono et al. (2007); Pelc et al. (2019a, b). In contrast, the CDW order involves a relatively small fraction of a hole per unit cell. While X-ray scattering experiments cannot provide the absolute CDW amplitude, a value of about 0.03 hole per CuO2 unit was estimated for La2-xBaxCuO4 Abbamonte et al. (2005). A separate, consistent estimate can be obtained from NMR: from the universal relation between oxygen hole content and NQR frequency Jurkutat et al. (2014) and the 17O NQR line broadening in the CDW phase Wu et al. (2015), we estimate 0.028 hole in the Cu $3d$ orbital for Hg1201 Reichardt et al. (2016). Although $\sim 0.03$ hole per CuO2 unit is a relatively large fraction of the nominal doping level, it is small compared to 1. It therefore seems likely that the CDW phenomenon in the cuprates is a secondary, emergent phenomenon related to the strong correlations that underlie the hole localization. This is supported by STM evidence for a qualitative change in the CDW form factor of Bi2212 at a characteristic scale comparable to the PG scale Hamidian et al. (2016). Furthermore, it is known that the cuprates are intrinsically inhomogeneous, with local gaps that vary at the nanoscale and persist well above the PG temperature Gomes et al. (2007); Tal . Puddles of localized charge that sustain significant dynamic correlations thus may already exist outside of the nominal PG region (above $T^{**}(p)$). This could account for the present observation for Hg1201 at 250 K and for the similar result for NBCO Arpaia et al. (2019). The relevant wavevector would span the antinodal regions and hence be close to, but smaller than $q_{\mathrm{CDW}}$.

We have argued that the high-temperature peak at/near ${\bf q}_{CDW}$ seen universally in the cuprates is dominated by dynamic charge fluctuations, which for Hg1201 at the studied doping level have a characteristic energy of $\sim 40$ meV. Although not apparent from the data summarized in Fig. 5, there exists evidence from RXS that dynamic charge correlations centered at/near ${\bf q}_{CDW}$ extend throughout a much larger portion of the Brillouin zone, with real-space correlations no larger than 1-2 lattice constants and little doping and temperature dependence (unlike the quasistatic CDW signal). This is perhaps most clearly seen from RXS measurements of YBa2Cu3O6+δ Blanco-Canosa et al. (2014), for which we estimate the approximately doping-independent 2D-integrated strength of the high-temperature signal to be at least an order of magnitude larger than the strongly temperature-dependent quasistatic CDW signal (at the doping level where CDW correlations are strongest) Sup , consistent with a separate estimate based on the recent RIXS data for NBCO Arpaia et al. (2019). In other words, whereas it appears from data such as those in Fig. 5 that the energy-integrated high-temperature signal is about 35-50% of the low-temperature response, it may in fact be dramatically larger. This signal might be a direct signature of the fluctuations associated with the localized hole Pelc et al. (2019a, b), and the characteristic wavevector may correspond to the distance between the antinodal regions of the Fermi surface on which the PG develops, and hence be somewhat smaller than $q_{CDW}$. The 2D-integrated strength of the signal is still considerably smaller than what would be expected if it were fully associated with the localization of one hole per CuO2 unit: naively, given that the CDW amplitude corresponds to about 0.03 hole per CuO2 unit, one might expect this signal to be $(1/0.03)^{2}\sim 10^{3}$ times larger than the quasistatic CDW response. It is possible that there exists a considerable incoherent (local in real space) contribution to the charge response up to 0.3 eV and even higher energies that is not captured by RXS and RIXS experiments. Similarly, the $\sim 165$ meV scale might not be seen at 250 K in our experiment because the high-energy charge fluctuations are incoherent at high temperatures. These observations are consistent with the fact that, whereas in optical spectroscopy experiments the spectral weight of the mid-infrared feature is relatively small Uchida et al. (1991); Lee et al. (2005) and the coherent contribution at temperatures below $T$** corresponds to $p$ Mirzaei et al. (2013), integration up to about 2.5 eV (on the order of the charge-transfer gap) approaches $1+p$, where 1 is attributed to the localized charge Mirzaei et al. (2013); Barišić et al. (2019). We note that recent momentum-resolved electron-scattering measurements of the charge fluctuations in Bi2212 revealed a featureless continuum up to $\sim 1$ eV Husain et al. (2019) for a wide range of hole concentrations, from underdoped to overdoped, consistent with the existence of significant local excitations.

The above considerations are supported by neutron scattering, Raman scattering, infrared spectroscopy, and tunneling results. First, we note that evidence for an underlying collective charge mode has been deduced from anomalies in the Cu-O bond-stretching vibration in the 65-85 meV range single-layer La2-xSrxCuO4 (LSCO) in the superconducting doping range Park et al. (2014). An anomalous line-width broadening was observed in the entire momentum range from the 2D zone center to the 2D zone boundary at (0.5,0) rlu. Given the large intrinsic width of the $\sim 40$ meV feature in Hg1201, these findings for LSCO are consistent with anomalous electron-phonon coupling and a potentially universal spectrum of bosonic charge modes. Second, the broad $\sim 165$ meV feature seen in our experiment appears to be related to features seen in Raman spectra with $B_{1g}$ Li et al. (2012) and $B_{2g}$ symmetry Loret et al. (2019). Raman scattering probes large portions of the Brillouin zone, with distinctly different form factors in $B_{1g}$ and $B_{2g}$ symmetry. In $B_{1g}$ symmetry, a broad feature with characteristic energy of $\sim 200$ meV ($\sim 1700$ cm*-1*) was identified for a Hg1201 sample with a slightly larger doping level ($p\approx 0.11$, $T_{c}=77$ K) than in the present work and associated with the two-magnon excitations of the undoped antiferromagnetic parent compounds. We note that both paramagnon and charge excitations might contribute to the $B_{1g}$ response. While a clear temperature dependence was observed below $\sim T^{*}$, this feature was seen to persist to higher temperatures Li et al. (2012). In $B_{2g}$ symmetry, a feature of width $\sim 60$ meV centered at $\sim 150$ meV ($\sim 1250$ cm*-1*) was observed for Hg1201 with $T_{c}=72$ K, i.e., at nearly the same doping level as the present study, and interpreted as a CDW energy scale Loret et al. (2019). These values are remarkably close to the intrinsic width of 77(16) meV (FWHM) and peak of 163(12) meV that we extract and associate with charge fluctuations (Fig. 3).

We now discuss the possibility that the dynamic charge and magnetic scales observed here correspond to features in the bosonic pairing glue function deduced from other spectroscopic techniques. While RIXS does not measure the glue function, it is a probe of both charge and magnetic excitations. In an early optical conductivity study, it was suggested that deviations of the in-plane conductivity from the conventional Drude form may originate from coupling to a bosonic mode Thomas et al. (1988). Subsequent optical spectroscopy efforts to extract the pairing glue revealed a robust peak around 50-60 meV and a second feature in the 100-300 meV range in a wide range of cuprates, including optimally-doped Hg1201 Dordevic et al. (2005); Hwang et al. (2007); van Heumen et al. (2009); Dal Conte et al. (2012). Given that optical spectroscopy yields an average over the Brillouin zone, whereas our RIXS experiment focuses on the vicinity of $\bm{q}_{\mathrm{CDW}}$, this result is fully consistent with an interpretation of our data in terms of charge modes that contribute to the pairing glue. This is demonstrated in Fig. 6, which compares the glue function extracted at 100 K for Hg1201 ($T_{c}=97$ K) with our RIXS result for the dynamic response at $T=T_{c}=70$ K. As noted in Section III, the results in Fig. 3(a,b) are equally well captured by intrinsic heuristic Gaussian and log-normal functional forms, and the latter is used for the comparison. Also shown in Fig. 6 is our result for the energy dependence of the strength of the paramagnon response at 70 K (obtained from the data in Fig. 4), as well as the momentum-integrated, local dynamic magnetic susceptibility obtained from neutron measurements near the antiferromagnetic wavevector for a sample with nearly the same doping level Chan et al. (2016a) (see also Fig. 4b). Assuming azimuthal symmetry, the former is proportional to the 2D momentum-integrated paramagnon strength. The neutron data are available in absolute units, and the RIXS paramagnon result is scaled to match the neutron data just below 100 meV. The RIXS data for the charge response, in turn, are scaled up by a factor of 10 for ease of comparison and, as noted, because the fully-2D-integrated dynamic charge response may indeed be an order of magnitude larger than what we observe in the immediate vicinity of $\bm{q}_{\mathrm{CDW}}$ Sup ; Arpaia et al. (2019). We also note that up to about 20% of the nominal paramagnon signal could be dynamic charge correlations Sup . As seen from Fig. 6, there exists an intriguing connection between the charge and magnetic scales obtained from our RIXS experiment and those of the pairing glue function extracted via inversion of optical spectroscopy data for Hg1201 van Heumen et al. (2009). Qualitatively similar results have been obtained from Raman scattering, and Fig. 6 includes a result for Bi2212 Muschler et al. (2010). As noted, these spectroscopies yield complex averages over the Brillouin zone and do not distinguish charge from magnetic response. RIXS has a key advantage over these and other techniques, as it is a rather direct, momentum- and energy-resolved measure of the charge and magnetic correlations. We also note that an analysis of Raman spectra for LSCO revealed a relatively narrow peak in the glue function below $\sim 100$ meV and a broad hump up to $\sim 500$ meV, seen in both B2g and $B_{1g}$ symmetry channels Fanfarillo et al. (2016). It was concluded that, in B1g symmetry, spin modes contribute more strongly and at all frequencies, whereas charge modes are particularly prevalent at low/intermediate frequency in the $B_{2g}$ channel. Tunneling data for optimally-doped Bi2212 ($T_{c}=95$ K), the lowest doping level investigated, indicate a strong local maximum at $\sim 40$ meV and weaker, broad feature peaked just above 100 meV Ahmadi et al. (2011).

Finally, we note that a recent RIXS study of electron-doped NCCO suggests a coupling between dynamic magnetic and charge correlations da Silva Neto et al. (2018). In hole-doped cuprates, on the other hand, there is no clear evidence for such a coupling, except for the special case of the “214” family of materials that exhibit charge-spin stripe order Miao et al. (2017). While we can not rule out a small (20% or less) charge contribution to the nominal paramagnon response for underdoped Hg1201 Sup , we find no evidence of a significant coupling: the extracted paramagnon energy (Fig. 4) and damping coefficient (Fig. S4 in Sup ) are insensitive to the dynamic charge correlations, consistent with the fact that the charge and magnetic excitations have somewhat different characteristic energy scales at/near $\bm{q}_{CDW}$.

V Conclusion

In conclusion, we have measured the charge dynamics and paramagnon response along the Cu-O bond direction of underdoped Hg1201 using RIXS at the Cu $L_{3}$ edge with high energy resolution. This has allowed us to discern dynamic charge correlations from the quasistatic CDW response. Above $T_{\mathrm{CDW}}$, the temperature previously identified with the onset of CDW correlations, the response is purely dynamic, with an energy scale comparable to both the superconducting gap and the low-energy PG. As expected, quasistatic CDW correlations are observed at low temperature. However, there also exists an additional dynamic signature with a remarkably high energy scale that appears to be an imprint of the high-energy PG and associated with the strong electronic correlations that cause the charge-transfer gap of the undoped parent insulators. It is a distinct possibility that the dynamic charge correlations identified here have a different underlying physical mechanism than the CDW phenomenon and that they significantly contribute to the superconducting pairing glue. The present work sets the foundation for future RIXS measurements of the detailed temperature and doping dependence of the dynamic CDW correlations in the model cuprate Hg1201 and, more broadly, for efforts to identify the pairing glue and understand the CDW phenomenon of the cuprates in the context of large local charge fluctuations.

VI Acknowledgements

We thank the ESRF for the allocation of beam time at ID32, D. Pelc for extensive discussions, and N. Baris̆ić, A. V. Chubukov, G. Ghiringhelli, J. Haase, M.-H. Julien, Yuan Li, D. Pelc, D. Reznik, and E. H. da Silva Neto for valuable comments on the manuscript. The work at University of Minnesota was funded by the Department of Energy through the University of Minnesota Center for Quantum Materials, under DE-SC-0016371. The work at the TU Wien was supported by FWF project P27980-N36 and the European Research Council (ERC Consolidator Grant No 725521).