Strange metal behaviour from charge density ﬂ uctuations in cuprates

Besides the mechanism responsible for high critical temperature superconductivity, the grand unresolved issue of the cuprates is the occurrence of a strange metallic state above the so-called pseudogap temperature T * . Even though such state has been successfully described within a phenomenological scheme, the so-called Marginal Fermi-Liquid theory, a microscopic explanation is still missing. However, recent resonant X-ray scattering experiments identi ﬁ ed a new class of charge density ﬂ uctuations characterized by low characteristic energies and short correlation lengths, which are related to the well-known charge density waves. These ﬂ uctuations are present over a wide region of the temperature-vs-doping phase diagram and extend well above T * . Here we investigate the consequences of charge density ﬂ uctuations on the electron and transport properties and ﬁ nd that they can explain the strange metal phenomenology. Therefore, charge density ﬂ uctuations are likely the long-sought microscopic mechanism underlying the peculiarities of the metallic state of cuprates.

A mong the different phases and orders populating the phase diagram of superconducting cuprates, the region where the strange metal occurs has a preeminent role for this class of compounds over a rather wide doping range pivoting around optimal doping (see Fig. 1). Experimentally, the most evident benchmark of this region is represented by the linear behaviour of the electrical resistivity ρ(T) as a function of the temperature T, from above a doping-dependent pseudogap crossover temperature T * up to the highest attained temperatures. Such occurrence is less evident in the underdoped regime, where T * is almost as high as room temperature (e.g. at doping p ≈ 0.11, see Fig. 1), while it dominates the transport properties of the metallic state in its entirety above optimal doping (p ≈ 0.17-0.20, see Fig. 1), where T * decreases and eventually merges with the superconducting critical temperature T c . Beyond such occurrence, the main deviations from the paradigmatic behaviour dictated by the Landau Fermi-liquid theory of standard metals are the optical conductivity, following a non-Drude-like frequency dependence σ (ω)~1/ω, and the Raman scattering intensity, starting linearly in frequency and then saturating into a flat electron continuum, as expressed by the dependence of the susceptibility of the scattering mediator, Im PðωÞ $ ω= max ðT; jωjÞ. It was shown long ago 1 that the phenomenological assumption of this form for Im PðωÞ accounts for the above anomalous properties. In particular, the related low-energy excitations, mediating a momentumindependent electron-electron effective interaction, give rise to a linear dependence of the imaginary part of the electron self-energy both in frequency and temperature Im Σðk; ωÞ $ max ðT; jωjÞ: ð1Þ Although there are theories that do not rely on a specific mediator 2 , a huge effort has been devoted along the years to identify the excitations mediating this scattering, mostly based on the idea of proximity to some form of order: circulating currents 3 , spin 4,5 , charge order [6][7][8][9][10] or the phenomenological coupling to incoherent fermions 11 .
A step forward in the identification of low energy excitations that might be responsible for the strange-metal behaviour was recently taken by means of resonant X-ray scattering (RXS), performed on Nd 1+x Ba 2−x Cu 3 O 7−δ (NBCO) and YBa 2 Cu 3 O 7−δ (YBCO) thin films 12 19 , thereby indicating that these excitations may well provide a generic scattering mechanism in all cuprates.
In the following we will focus on NBCO or YBCO investigated in the precursor experiment. These experiments not only confirmed the occurrence of incommensurate charge density waves (CDWs), correlated over several lattice spacings, in the underdoped region and below T *20-28 , but, quite remarkably, also identified a much larger amount of very short-ranged (≈3 lattice spacings) dynamical charge density fluctuations (CDFs, see  In the red region encompassed between the pseudogap temperature T * and the upturn temperature T up of the resistance, above the superconducting critical temperature T c , in particular close to the optimally doped regime (e.g. at hole doping p ≈ 0.17), these compounds display a strange-metal behaviour. This is revealed in the experimental resistance R data by the presence of a linear temperature dependence, displayed as a red thick solid line in the R(T) curves above the phase diagram. In the underdoped regime (e.g. at p ≈ 0.11), below T * (blue region) a downturn from the linear-in-T resistance is observed, since additional mechanisms lead to deviations from the strange-metal regime. In the overdoped regime (e.g. at p ≈ 0.21), below T up (yellow region) the upturn from the linear-in-T resistance is due to the setting in of the Fermiliquid regime. Recent Resonant X-Ray Scattering experiments 12 showed that also the charge order phenomenon is widespread in the phase diagram. In particular, short-ranged dynamical charge density fluctuations (sketched by red waves highlighted in the red circle, and observed in the striped area) populate the strange-metal region, while in the underdoped region, below the onset temperature T CDW , they coexist with the usual longer-ranged charge density waves (sketched by blue waves in the blue circle, and observed in the wavy area). T N is the Néel temperature. The data of the R(T) curves are taken from Refs. 12,40 . CDFs are peaked at a wave vector, along the (1,0) and (0,1) directions, which is very close to that of the intermediate-range CDWs 12 , arising below a given temperature T CDW (p) for each measured doping p. We also notice that, when the temperature is raised towards T CDW (p), the CDWs correlation length decreases down to values close to those of the CDFs. These facts suggest that the two charge fluctuations have a common origin. One possibility is that they develop differently in different regions, with CDFs remaining noncritical, whereas CDWs evolve towards order. This is also supported by the possibility that the narrow peak (NP) of the RXS response function, customarily associated to the CDWs, arises at the expense of the broad peak (BP) due to CDFs. However, differently from CDWs, CDFs are quite robust both in temperature (they survive essentially unaltered up to the highest explored temperatures, T ≈ 270 K) and doping. These excitations are at low energy (≈15 meV in an optimally doped sample with T c = 90 K) and so short ranged that in reciprocal space they produce the BP observed in the RXS scans. CDFs not only provide a strong scattering channel for the electrons, but also overcome the difficulty of the CDWs, which, being so peaked, give rise to anisotropic scattering dominated by the hot spots on the Fermi surface. CDFs, instead, being so broad, affect all states on the Fermi surface nearly equally, resulting in an essentially isotropic scattering rate. This isotropy is a distinguished feature of the strange-metal state and we show below that it can account for the peculiar behaviour of the electronic spectra and for the linearin-T resistivity.

Results
Strange-metal behaviour of the electron self-energy. Figure 2a shows a qualitative explanation of the inherent isotropy of the scattering by CDFs. RXS experiments directly access the frequency and momentum-dependent charge susceptibility (see Methods) and find the above mentioned BP at a well-defined incommensurate wave vector Q c , but the large width of this peak means that a wealth of low-energy CDFs are present over a broad range of momenta. Therefore, an electron quasiparticle on a branch of the Fermi surface can always find a CDF that scatters it onto another region of the Fermi surface [see Fig. 2a)]. Thus the whole Fermi surface is hot in the sense that no regions exist over the Fermi surface that can avoid this scattering. This is visualized in Fig. 2a, where the overlap of the Fermi surface with its translated and broadened replicas (due to the scattered quasiparticles) is almost uniform, and no particular nesting condition is needed. Quite remarkably, the CDF-mediated scattering stays isotropic even in an energy window of ≈20 meV around the Fermi surface (see Supplementary note 1 and Supplementary  Fig. 2). On the contrary, since CDWs are quite peaked, only a few of them around Q c are available to scatter quasiparticles at low energy: Only quasiparticles at the hot spot are then significantly scattered by CDWs [see Fig. 2a]. In a quantitative way, this is shown in Fig. 2b, where the actual scattering rate along the Fermi surface has been separately computed for CDFs (solid red line) and CDWs (dashed blue line) with parameters suitable to describe a slightly underdoped NBCO sample (p ≈ 0.15), where CDF and CDW coexist (see Supplementary note 2 and Supplementary Fig. 5). This feature makes these CDFs an appealing candidate to mediate the isotropic scattering required by the original marginal Fermi-liquid theory. We therefore test this expectation by explicitly calculating how the CDFs dress the electron quasiparticles modifying their spectrum. In many-body theory, this effect is customarily described by the electron selfenergy. In particular, the imaginary part of the electron selfenergy, Im Σ, provides the broadening of the electron dispersion as measured, e.g. in angle-resolved photoemission experiments. We adopt the following strategy: (a) we extract from the experimental inelastic RXS spectra the information on the dynamics of the CDFs (see supplementary note 2) evaluated within the linear response theory; (b) we borrow from photoemission experiments the electron dispersion in the form of a tight-binding band structure 29 ; (c) we calculate the electron self-energy resulting from the coupling between CDFs and the electron quasiparticles, as discussed in Supplementary note 1 and represented as a diagram in Supplementary Fig. 1.
With the extracted parameters, using the coupling between quasiparticles and CDFs obtained from the resistivity fit (see below) and taking the frequency derivative of the real part of the self-energy, we also calculated the dimensionless coupling λ at T ≈ T * finding λ ≈ 0.35-0.5 (see Supplementary note 1).
Of course, this perturbative approach, although supported by the low-moderate value of λ, is based on the Fermi-liquid as a starting point in the overdoped region. Its applicability can be safely extended to lower doping at high temperatures, in the metallic state and above T * , where the phenomenology is only marginally different from that of a Fermi-Liquid. c Imaginary part of the electron self-energy as a function of the (negative) electron binding energy, at different temperatures above T CDW , below which the CDWs emerge to produce the narrow peak in resonant X-ray scattering. The coupling between fermion quasiparticles and CDFs is g = 0.166 eV. d Same as c, but with both frequency and self-energy axes rescaled by the temperature (k B is the Boltzmann constant), to highlight the approximate scaling behaviour at low frequency.
The result of our calculation for an optimally doped NBCO sample with T c = 90 K is reported in Fig. 2c, d. After an initial quadratic behaviour, the scale of which is set by the energy scale ω 0 of the CDFs 30 (see Supplementary note 1), Im Σ displays an extended linear frequency dependence up to 0.10-0.15 eV (comparable to the one reported in the photoemission experiments of refs. 31,32 ). The overall value of this self-energy is comparable to, but it always stays smaller than, the Fermi energy scale of order 0.3-0.4 eV. This is an intrinsic manifestation of a strange-metal state, where the width of the quasiparticle peak must be of the same order of its typical energy. At low frequencies Im Σ saturates at a constant value that increases linearly with increasing T. This is precisely the behaviour expected from the strange-metal expression of Eq. (1). This self-energy is reported along a specific (1,1) direction, but it is crucial to recognize that it is also highly isotropic in momentum space. Figure 2b indeed reports the scattering rate (i.e. the imaginary part of the selfenergy at zero frequency) Γ(ϕ) ≡ Γ 0 + Γ Σ (ϕ). An isotropic scattering rate Γ 0 representing the effect of quenched impurities has also been included. Our results in Fig. 2c, not only share with the data of ref. 31 a similar form, but also display a scaling behaviour, as reported in Fig. 2d. As mentioned in ref. 1 , the isotropic linear-in-frequency self-energy behaviour, stemming from CDFs, is sufficient to produce a strange-metal behaviour in physical quantities like optical conductivity and Raman scattering.
Below T CDW = 150 K, an additional scattering due to the CDWs is present. This additional scattering has a significant anisotropic component, which is confined in a small region of momentum space, as shown by the dashed blue curve of Fig. 2b. This anisotropic character eventually leads to the departure from the strange-metal behaviour 33 below temperatures comparable with T * .
CDFs produce linear resistivity. Once the dynamics of the CDFs is identified by exploiting RXS experiments, one can investigate their effects on transport properties. The calculation of the electron resistivity is carried out within a standard Boltzmannequation approach along the lines of ref. 34 Fig. 7). From the electron self-energy we obtain the zero frequency quasiparticle scattering rate along the Fermi surface Γ(ϕ) defined above, and we use Γ 0 as a fitting parameter, obtaining values (≈20-60 meV) that are reasonable for impurity scattering. We also use the anisotropic Fermi wave vector along the Fermi surface, as obtained from the same band structure in tight-binding approximation 29 used for the self-energy calculation. Figure 3a displays the comparison between the ρ(T) curve of the optimally doped NBCO film (T c = 90 K), studied in ref. 12 (yellow line) and the theoretical results (black line). At high temperatures, the famous linear-in-T behaviour of the resistivity is found and the data are quantitatively matched. This behaviour stems from the very isotropic scattering rate produced by the CDFs [red solid line in Fig. 2b], which, for this sample and in this temperature range, are the only observed charge excitations. At lower temperatures, below T * , a discrepancy emerges between the theoretical expectation and the experimental evidence, since the expected saturation, due to the onset of a Fermi-liquid regime and to (isotropic) impurity scattering Γ 0 , is experimentally replaced by a downturn of the resistivity. Such discrepancy occurs gradually in T when, entering the pseudogap state, the pseudogap itself and other intertwined incipient orders (CDWs, Cooper pairing,...) play their role. These effects, which are outside our present scope, obviously lead to deviations from our theory, which only considers the effect of CDFs. On the other hand, in the overdoped YBCO sample (T c = 83 K), the pseudogap and the intertwined orders are absent, while the CDFs are the only surviving charge excitations, even down to T c 12 . Here, our theoretical resistivity, related to the scattering rate produced by CDFs, matches very well the experimental data, in the whole range from room temperature almost down to T c [see Fig. 3b]. In particular, the agreement is rather good even at the lowest temperatures above T c . The data display an upward saturation due to the onset of a Fermi-liquid regime that is well described by our calculation: At temperatures lower than the characteristic energy of CDFs their scattering effect is suppressed and the strange-metal behaviour ceases. We find remarkable that our theory not only describes the linear-in-T regime, but also captures the temperature scale of upward deviation from it, without additional adjustments.
Discussion and conclusions. The above results clearly show that the main features for the CDFs to account for the strange-metal behaviour are a) a short coherence length of 1-2 wavelengths to scatter the low-energy electrons in a nearly isotropic way and b) a rather low energy (ω 0 ≈ 10-15 meV) to produce a linear scattering rate down to 100-120 K. We emphasize here that ω 0 is only a characteristic minimal scale of CDFs, but these are broad overdamped excitations from ω = 0 (due to damping) up to about 0.1 eV, because they have a dispersion $ νðq À Q c Þ 2 with a stiffness energy scale ν % 1:0 À The question may also arises whether CDFs can also account for the so-called Planckian behaviour 35 : at some specific doping, when a strong magnetic field (several tens of Teslas) destroys superconductivity, the linear-in-T resistivity extends down to low temperatures of a few K. In order for our theory to account also for this behaviour, we should find CDFs with a lower characteristic energy of order 0.5-1.0 meV, while maintaining the correlation length short, to keep the scattering isotropic. Unfortunately at the moment no RXS experiments in the presence of such large magnetic fields are viable and we therefore cannot test these expectations. Nevertheless, we feel that it is not accidental that our theory accounts so well of the experiments done so far in the absence of magnetic field which show linearity up to very high temperature, well above T * , so far from the quantum region. No wonder if by lowering the temperature at special values of doping, other effects may come in to modify our parameters values.
One interesting question is why CDFs, even in the absence of the specific Planckian conditions have rather low characteristic energies ≈10-15 meV. In this regard, we notice that CDFs and CDWs have nearly the same characteristic wave vectors, indicating a close relationship. Since CDWs have a nearly critical character (that was theoretically predicted long ago 6,36 ), it is likely that CDFs are aborted CDWs, that for several possible reasons (competition with superconductivity, low dimensionality, disorder, charge density inhomogeneity, ...) do not succeed in establishing longer-range correlations. Still, this tight affinity with CDWs, which are nearly critical and therefore at very low energy, implies that CDFs also may have a broad dynamical range extending down to a rather low-energy scale ω 0 . In this scenario, where CDWs and CDFs coexist in the system, one and the same theoretical scheme accounts for both excitations.
In conclusion, although some issues are still open, like the effects of magnetic field on CDFs to possibly account for Planckian transport, or the origin of the pseudogap features in transport, we were able to show that CDFs account for the anomalous metallic state of cuprates above T * . Indeed, once the dynamics of the CDFs is extracted from RXS experiments, we can well explain, with the same parameter set, both the strange-metal behaviour of the electron self-energy (therefore all the related anomalous spectral properties observed, e.g. in optical conductivity and Raman spectroscopy, are also explained) and the famous linear-in-T resistivity in the metallic state of hightemperature superconducting cuprates. We thus believe that our results provide a very sound step forward in the long-sought explanation of the violation of the normal Fermi-liquid behaviour in cuprates.

Methods
Fitting procedure to extract the CDW and CDF dynamics. The CDW and CDF contributions to the RXS spectra are captured by a density response-function diagram as reported in Supplementary Fig. 1. In this framework, we carry out a twofold task: on the one hand, we show that dynamical CDFs and nearly critical CDWs account both for the RXS high-resolution, frequency-dependent, spectra, and for the quasi-elastic momentum-dependent spectra. On the other hand, from the fitting of these experimental quantities, we extract the dynamical structure of these excitations needed to calculate the physical quantities discussed above.
According to this scheme, the CDW or CDF contribution to the low-energy RXS spectra is where bðωÞ ½e ω=k B T À 1 À1 is the Bose distribution ruling the thermal excitation of CDFs and CDWs, and A is a constant effectively representing the intricate photon-conduction electron scattering processes 27,37 . In Eq. (2), Im Dðq; ωÞ is the imaginary (i.e. absorptive) part of the (retarded) dynamical density fluctuation propagator, which can describe either CDWs or CDFs. For both we adopt the standard Ginzburg-Landau form of the dynamical density fluctuation propagator, typical of overdamped quantum critical Gaussian fluctuations 6,7,36 , where ω 0 ¼ ν ξ À2 is the characteristic energy of the fluctuations, νðqÞ % ν jq À Q c j 2 , ν determines the dispersion of the density fluctuations, Q c ≈ (0.3, 0), (0, 0.3) is the characteristic critical wave vector (we work with dimensionless wave vectors, measured in reciprocal lattice units, r.l.u.) and Ω is a frequency cut-off. This form of the charge collective mode propagator is typical of metallic systems where the collective modes have a marked overdamped character at low energy, where they can decay into particle-hole pairs (Landau damping). At larger energies, above Ω, they acquire a more propagating character. In both regimes, however, the maximum of their spectral weight is dispersive with a definite relation between ω and momentum, as it should be for well-defined collective modes. This is valid for both CDFs and CDWs, although the coherence length of the formers is weakly varying in doping and temperature and is generically very short (of the order of the wavelength itself). The sharper CDWs have a nearly critical character, with a marked temperature dependence of the square correlation length, ξ 2 NP ðTÞ. In particular, if these fluctuations had a standard quantum critical character around optimal doping 6,7,36,38 , one would expect ξ 2 NP ðTÞ $ 1=T. The CDFs have a similar Q c , the main difference being in the behaviour of the correlation length, that, according to RXS experiments, increases significantly with decreasing the temperature and reaches up to 8-10 lattice spacings for the nearly critical CDWs, while the CDFs have correlation length in the range 2-3 lattice spacings, independently of the temperature.
Although high-resolution spectra provide a wealth of information, they are experimentally very demanding, so that RXS data are more often available in the form of quasi-elastic spectra corresponding to the frequency integration of the inelastic spectra, Eq. (2), Our first goal is to extract from the experiments all the parameters entering the CDW and CDF correlators, ω 0 ; ν; Q c and Ω. Since high-resolution and quasi-elastic spectra provide different complementary information, we adopted a bootstrap strategy in which we first estimated the dynamical scale ω 0 from high resolution at the largest temperatures, where the NP due to CDWs is absent and all collective charge excitations are CDFs. Then, we used this information to fit the quasi-elastic peaks to extract the relative weight (intensity) of the narrow and broad contributions. Once this information is obtained, we go back to high-resolution spectra, since we now know the relative weight of the CDFs and CDWs contribution at all temperatures.
More specifically, the quasi-elastic peak has a composite character and, once the (essentially linear) background measured along the (1, 1) direction is subtracted (see, e.g. Fig. 2 A-D in ref. 12 ), the peak may be decomposed into two approximately Lorentzian curves, corresponding to a narrow, strongly temperature dependent, peak due to the standard nearly critical CDWs arising below T ≈ 200 K and to a BP due to the CDFs. This is the main outcome of the RXS experiments reported in ref. 12 . We thus fitted each of the two peaks with Eq. (4). From the fits, one can extract the overall intensity parameter A and the ratio ω 0 = ν ¼ ξ À2 . Since only this ratio determines the width of the quasi-elastic spectra, we need a separate measure to disentangle ω 0 and ν, so we used the high-resolution information on ω 0 for the BP at T = 150 K and T = 250 K to extract ν BP % 1400 meV(r.l.u.) −2 at these temperatures. The same procedure cannot be adopted for the narrow CDWs peaks, which always appear on top of (and are hardly unambiguously separated from) the broad CDFs contribution. Nevertheless, to obtain a rough estimate, we investigated the high-resolution spectra at low temperature (see Supplementary note 2), where the maximum intensity should mostly involve the NP to extract the characteristic energy of the quasi-critical CDWs obtaining, as expected, much lower values ω NP 0 % 1 À 3 meV (although these low values are less reliable, due to the relatively low resolution of the frequency-dependent spectra). These estimates allow to extract values of ν NP % 800 meV(r.l.u.) −2 for the CDWs, comparable with those of the CDFs, suggesting a common electronic origin of the two types of charge fluctuations. To reduce the fitting parameters to a minimum, although subleading temperature dependencies of the high-energy parameters ν and Ω over a broad temperature range can be expected, we kept those parameters constant. We also assumed a constant ω 0 for the CDFs, to highlight the noncritical nature of these fluctuations.

Data availability
The experimental resistivity and RXS data (see Fig. 3 of the main text and supplementary figures 4-7) have already been published in ref. 12 and are therefore available in the related data repository 39 . They are also available from one of the corresponding authors [M.G.] on reasonable request. The datasets (resistivity curves, fitted RXS spectra and electron self-energy) generated during the current study are available from one of the corresponding authors [M.G.] on reasonable request.

Code availability
The theoretical analysis was carried out with FORTRAN codes to implement various required numerical integrations [Eq. (4) in Methods to fit the RXS data, supplementary Eq. (1) for the self-energy, in the supplementary note 1, and supplementary Eq. (6) for the resistivity, in supplementary note 3]. Although the same task could easily by performed with Mathematica or other standard softwares, the FORTRAN codes we used are available from one of the corresponding authors [M.G.] on reasonable request.