The strange metal phase in cuprate high-temperature superconductors is a perplexing phenomenon characterized by a linear temperature dependence of resistivity, a lack of quasiparticle behavior, and anomalous responses to magnetic fields. This unconventional behavior deviates significantly from the predictions of Fermi-liquid theory, which successfully describes conventional metals. Understanding the origin of this strange metal phase is crucial for a complete comprehension of high-temperature superconductivity in cuprates. The phase occupies a significant portion of the doping-temperature phase diagram and displays particular robustness near optimal doping (*p* ≈ 0.19), the doping level where the maximum superconducting critical temperature (*T_c*) is achieved. At this optimal doping, the linear-in-*T* resistivity persists down to *T_c*, or even to the lowest temperatures under strong magnetic fields. Other unusual characteristics include the anomalous power-law decay of the Drude peak in optical conductivity and the linear magnetoresistance. These observations strongly suggest that electron correlations play a dominant role in determining the electronic properties of the strange metal phase. Several competing theoretical models attempt to explain the strange metal's behavior, including those that invoke strong scattering from local excitations (such as charge density fluctuations or short-ranged antiferromagnetic fluctuations), from long-wavelength fluctuations (like loop currents or phase fluctuations of incommensurate charge order), or invoking entirely novel states of matter characterized by strong quantum entanglement and Planckian scattering times. A recurring theme in many of these theories is the presence of a quantum critical point (QCP) at or near optimal doping, where quantum fluctuations drive the unusual transport properties. However, direct experimental evidence for a QCP in cuprates remains elusive due to the interference of superconductivity. This study focuses on the investigation of charge density fluctuations (CDFs) as a potential signature of this quantum critical point, seeking to establish a direct link between these fluctuations and the strange metal behavior.

Previous research has extensively investigated the strange metal phase in cuprates, highlighting its anomalous properties. Several studies have reported a linear temperature dependence of resistivity in the strange metal regime, extending down to *T_c* at optimal doping. This linear behavior is inconsistent with Fermi liquid theory, suggesting the breakdown of conventional quasiparticle descriptions. The discovery of charge density waves (CDWs) in some cuprates fueled investigations into the role of charge order in the strange metal phase. However, the CDW phase is only observed in a limited region of the phase diagram, making it difficult to explain the ubiquity of the strange metal state. Some theories suggest that the strange metal behavior is linked to quantum criticality, proposing the existence of a quantum critical point (QCP) at or near optimal doping. A QCP is a zero-temperature phase transition driven by quantum fluctuations, which can lead to unusual transport properties. However, direct observation of a QCP in cuprates is challenging due to the onset of superconductivity. This study builds on prior work by investigating the role of charge density fluctuations (CDFs), precursors of CDWs, in the strange metal phase. The presence of CDFs across a broad range of doping and their link to the strange metal phenomenology are explored to determine whether they are indeed the quantum fluctuations associated with a QCP.

This research employed resonant inelastic X-ray scattering (RIXS) and energy-integrated resonant X-ray scattering (EI-RXS) techniques at the Cu L₃ edge to investigate the charge density fluctuations (CDFs) in two families of cuprate superconductors: YBa₂Cu₃O_7-δ (YBCO) and Bi₂Sr₂CaCu₂O_8+δ (Bi2212). The experiments covered a wide doping range (*p*), with a particular focus on three doping levels: *p* = 0.22 (overdoped), *p* = 0.19 (optimal doping, near the putative QCP), and *p* = 0.06 (underdoped). High-resolution RIXS measurements (ΔE ≈ 38 meV) were performed to directly measure the characteristic energy (Δ) of the CDFs as a function of doping and temperature. Medium-resolution RIXS (ΔE ≈ 62 meV) and EI-RXS measurements were also conducted to analyze the temperature dependence of the quasi-elastic scattering intensity, providing complementary information about the CDFs. The YBCO samples used were thin films grown by pulsed laser deposition, while a Bi2212 single crystal was prepared by the traveling solvent floating zone method. Careful sample characterization ensured accurate doping level determination. The RIXS data were corrected for self-absorption effects to ensure accurate comparison between different samples and temperatures. The data analysis involved fitting the RIXS spectra with multiple Gaussian peaks to isolate the CDF contribution from the elastic scattering, bond-stretching phonons, and other excitations. The doping and temperature dependence of the CDF intensity, energy, and correlation length were extracted from these fits. A theoretical model based on the theory of charge density instability in a highly-correlated Fermi liquid was used to interpret the experimental data, focusing on the temperature dependence of the characteristic energy and the quasi-elastic spectral weight. The temperature dependence of the CDF intensity, both near and far from the critical wave vector (*q*_CDF), was analyzed to understand the relationship between the CDFs and the putative QCP. The doping dependence of the CDFs was also investigated across the underdoped, optimally doped, and overdoped regions to determine their role in the strange metal phase. High magnetic fields were not used in this study, leaving the potential behavior under suppressed superconductivity as a future research direction. The energy-integral of the quasi-elastic intensity far from *q*_CDF was modeled with a Bose distribution function to obtain another characteristic energy, Ω, which is expected to be larger than Δ.

The key findings of this study demonstrate a strong correlation between charge density fluctuations (CDFs) and the strange metal phase in cuprate superconductors. The experiments revealed that the CDF intensity peaks at *p* ≈ 0.19, the putative quantum critical point (QCP), coinciding with the optimal doping for superconductivity. The characteristic energy (Δ) of the CDFs is also found to be minimum at this doping level. This minimum in Δ at *p* ≈ 0.19, combined with the peak in CDF intensity, forms a wedge-shaped region in the phase diagram, consistent with the behavior expected near a quantum critical point. High-resolution RIXS measurements directly revealed the temperature dependence of Δ, showing that it increases with temperature at all doping levels, consistent with a quantum critical scenario. The quasi-elastic spectral weight, analyzed by both medium resolution RIXS and EI-RXS measurements, exhibited an almost isotropic increase with temperature. This was interpreted as the signature of a finite-energy CDF contribution to the quasi-elastic signal, further supporting the observed doping and temperature dependences of the CDFs. This isotropic increase was quantitatively interpreted in terms of the model of the charge density instability in a correlated Fermi liquid, demonstrating the pervasive influence of CDF across the reciprocal space. The study also analyzed the doping dependence of the CDF energy (Δ) from underdoped (*p* = 0.06) to overdoped (*p* = 0.22) samples. The results showed that Δ increases as the doping moves away from *p* ≈ 0.19, both toward the underdoped and overdoped regions. This behavior supports the notion that the CDFs are indeed linked to the QCP at *p* ≈ 0.19. The wave vectors at which CDF peaks are located follow the known doping dependence of the CDW wave vector, showing a remarkable connection between CDF and CDW even where the CDW is absent. Analysis of the CDFs’ temperature dependence suggested that they possess a finite correlation length even at low temperatures, contradicting the expectation of a diverging correlation length at a conventional QCP. This suggested the possibility of either a frustrated quantum criticality or an unconventional QCP where critical slowing down leads to a persistent glassy state of CDF droplets. Furthermore, the characteristic CDF energy shows good quantitative agreement with the temperature at which the linear-in-*T* resistivity signature of the strange metal phase is lost, suggesting a direct link between CDFs and the strange metal behavior. A clear connection is found between the CDF and bond-stretching phonon softening, supporting a coupling between the electronic and lattice degrees of freedom. The findings reveal a consistent picture across different cuprate families (YBCO and Bi2212), strengthening the robustness and generality of these observations.

The findings of this study strongly suggest a significant role for charge density fluctuations (CDFs) in the strange metal phenomenology of cuprate superconductors. The observed minimum in CDF energy and maximum in intensity at *p* ≈ 0.19, coinciding with optimal doping and the putative QCP, is a compelling indicator of a connection between CDFs and quantum criticality. The wedge-shaped region in the phase diagram further reinforces this interpretation. The quantitative agreement between the characteristic CDF energy and the temperature where the linear-in-*T* resistivity vanishes highlights the likely causal relationship between CDF scattering and the strange metal transport properties. While this work supports a prominent role for charge fluctuations, it does not rule out the possibility of spin fluctuations also contributing to the quantum critical behavior, as suggested by other recent studies. The observed finite correlation length of the CDFs at low temperatures suggests an unconventional form of quantum criticality, possibly a frustrated criticality or an anomalous QCP characterized by a glassy state of CDF droplets. This unconventional nature needs further investigation. Future studies incorporating high magnetic fields to suppress superconductivity would be crucial to fully resolve the low-temperature behavior of the CDFs and establish more conclusively their role in the strange metal phase. Moreover, exploring the interplay between CDFs, lattice dynamics, and other competing orders (e.g., pseudogap) would provide valuable insights into the complex interplay of electronic and structural properties governing high-temperature superconductivity in cuprates.

This comprehensive study using resonant X-ray scattering techniques provides strong evidence for the association of charge density fluctuations (CDFs) with the quantum critical point at optimal doping in cuprate superconductors. The minimum in characteristic CDF energy, maximum in intensity, and their correlation with the termination temperature of the strange metal phase at various dopings strongly suggest that CDFs are a key element of the quantum critical behavior driving the strange metal state. However, the finite correlation length at low temperatures suggests an unconventional QCP, requiring further investigations under high magnetic fields to resolve this issue. Future research directions include exploring the interplay between CDFs, spin fluctuations, lattice dynamics, and the pseudogap to obtain a complete understanding of the complex interplay of phenomena governing high-temperature superconductivity in cuprates.

The study did not utilize high magnetic fields to suppress superconductivity, which might alter the behavior of the CDFs at low temperatures. This leaves the low-temperature behavior of the CDFs under suppressed superconductivity as an open question that requires further investigation. While the study establishes a strong link between CDFs and the strange metal phase, it does not definitively prove causality. Although the model used to interpret the data is well-suited to the studied samples, further theoretical exploration and refined modeling might be needed to address the nuances of the observed phenomena. Additionally, the absence of detailed investigation of the interplay between CDFs and other competing orders (such as the pseudogap and spin fluctuations) limits the ability to establish a complete picture of the underlying physics.