Signature of quantum criticality in cuprates by charge density fluctuations

The strange metal phase in cuprate superconductors occupies a wide region of the temperature–doping phase diagram and is particularly robust near the doping level p* ≈ 0.19, where Tc is maximal. At this doping, transport and spectroscopic anomalies appear: resistivity linear in temperature down to Tc (or to the lowest temperatures under high fields), loss of quasiparticle character, anomalous optical conductivity, linear magnetoresistance, and nearly temperature-independent spin relaxation. Conventional Fermi-liquid theory fails to capture these features. Competing theoretical scenarios invoke strongly correlated, quasi-2D electron systems with either nearly local excitations or long-wavelength modes (e.g., loop currents, phase fluctuations of incommensurate charge order). A prevalent hypothesis posits a quantum critical point at p* governing the strange metal. In a QCP framework, a characteristic fluctuation energy scale Δ should diminish as p approaches p*, controlling the onset of strange metal behavior. Identifying the relevant fluctuations and establishing their energy’s doping dependence has been elusive, and their nature (charge, spin, or mixed) remains debated. Recently observed finite-energy, short-range charge density fluctuations (CDF), precursors to charge density waves, pervade the strange metal regime and are candidates for the critical fluctuations underlying QCP behavior. The present work probes whether CDF exhibit the expected signatures—enhanced intensity and reduced characteristic energy—at p*.

Prior studies have mapped the strange metal’s broad signatures in transport and spectroscopy near optimal doping in cuprates, including linear-in-T resistivity, anomalous optical responses, and spin relaxation anomalies, motivating quantum critical interpretations centered around p*. Competing theoretical frameworks attribute strange metal behavior to: (i) strong scattering from nearly local excitations (e.g., Sachdev-Ye-Kitaev-like pseudospin degrees of freedom), (ii) charge density fluctuations and short-range antiferromagnetic fluctuations, or (iii) long-wavelength modes such as loop currents and phase fluctuations of incommensurate charge order. While the pseudogap energy scale is comparable to that of various fluctuations, a direct causal link is not universally accepted. Resonant x-ray scattering and related probes have established short-range charge density wave correlations in multiple cuprate families and revealed dynamical CDF across wide doping and temperature ranges, including above and below the pseudogap temperature. Theoretical work on charge instabilities in correlated Fermi liquids predicts overdamped quantum critical charge fluctuations with characteristic dispersions and damping that can dominate carrier scattering. Nonetheless, the definitive identification of the critical modes governing the putative QCP at p*—and their quantitative energy scaling with doping—has remained unresolved, with some neutron and NMR studies suggesting spin-related quantum criticality in other cuprates.

Samples and doping: The study investigates YBa2Cu3O7−δ (YBCO) thin films (p = 0, 0.06, 0.09, 0.185) and Ca-doped YBCO thin film (p = 0.22), grown by pulsed laser deposition on SrTiO3 and post-annealed to set doping; and a slightly overdoped Bi2Sr2CaCu2O8+δ (Bi2212) single crystal (Tc = 82 K) prepared by oxygen annealing. Doping p was determined from Tc and c-axis lattice parameters. Experimental techniques: Resonant inelastic x-ray scattering (RIXS) and energy-integrated resonant x-ray scattering (EI-RXS) were performed at the Cu L3 edge (~931 eV) with σ-polarized light. High-resolution RIXS with ΔE ≈ 38–41 meV was used at selected temperatures (e.g., 80 and 200 K for p ≈ 0.19; 20 K for p = 0.06), and medium-resolution RIXS with ΔE ≈ 62–65 meV enabled detailed temperature sweeps (20–290 K). Measurements mapped momentum transfers along (H,0) and (H,H) within the first Brillouin zone. To enhance the weak CDF signal and suppress elastic background from surface defects, difference maps (H,0) − (H,H) were analyzed. EI-RXS, integrating over elastic and inelastic scattering without energy discrimination, provided complementary momentum- and temperature-dependent intensity trends. Spectral analysis and fitting: High-resolution RIXS spectra near the CDF critical wavevector qCDF were decomposed into: (i) a resolution-limited elastic peak, (ii) a finite-energy CDF peak, (iii) bond-stretching phonon peak and its overtone, and (iv) a broad paramagnon/particle-hole continuum. Gaussian lineshapes were employed for elastic, CDF, and phonons; the paramagnon was modeled by an antisymmetrized Lorentzian. The CDF peak exhibited FWHM in energy of ~55–60 meV, exceeding instrumental resolution, and a broad peak in momentum with FWHM ~0.15 r.l.u. For medium-resolution data, the quasi-elastic spectral weight was integrated over an energy-loss window (−0.1 to 0.035 eV) to isolate the CDF contribution while minimizing bond-stretching phonon contributions; temperature evolution was analyzed at multiple q along both (H,0) and (H,H). Model for CDF dynamics: The CDF spectral intensity was modeled as I_CDF(q,ω) ∝ Im D(q,ω) × b(ω), where b(ω) is the Bose distribution, and D(q,ω) is the propagator of overdamped quantum critical charge fluctuations: D(q,ω) = 1 / [ω0(T) + v0 |q−qCDF|^2 − i γ ω − (ω^2/Ω)], with ω0(T) the minimum (gap-like) frequency linked to the CDF correlation length via ω0 = v0 ξ−2, v0 a dispersion coefficient, γ the Landau damping parameter, and Ω a high-energy cutoff making the dispersion quadratic away from qCDF. At q = qCDF, the maximum of Im D occurs at the characteristic energy Δ; away from qCDF, the peak occurs at a higher energy defining a bosonic scale Ω. Global fitting strategy: A global fit simultaneously considered all YBCO p ≈ 0.185 data along both (H,0) and (H,H) across 13 temperatures, including contributions from four equivalent qCDF vectors within the Brillouin zone. This yielded ω0(T), v0, γ, and Ω. An alternative procedure isolated the CDF peak via subtraction of (H,H) from (H,0), enabling determination of the CDF momentum FWHM and its temperature dependence. Far from qCDF, the temperature dependence of the quasi-elastic intensity was fit to a Bose function with a single characteristic energy Ω per doping, capturing the nearly isotropic, linear-in-T increase of intensity at high temperatures. Data corrections and calibrations: Spectra were corrected for self-absorption using measured XAS-derived absorption coefficients and experimental geometry, then normalized to dd-excitation integrals. Zero energy was established at Γ via the specular elastic peak. The fitting uncertainty on CDF and phonon energies was ~4 meV, smaller than instrumental bandwidth due to centroid-based determination.

• CDF exhibit quantum-critical-like behavior centered at p* ≈ 0.19: the CDF intensity is strongest near p ≈ 0.19 and diminishes with increasing temperature and when moving away in doping; the characteristic energy Δ(Tmin,p) is minimal at p ≈ 0.19 and increases both with temperature and with under/overdoping. • Direct measurements at p ≈ 0.19 (YBCO and Bi2212) show a finite-energy CDF peak at qCDF with energy position increasing from ~4–10 meV at 80 K to ~15 meV at 200 K; the CDF energy FWHM is ~55–60 meV; momentum profiles are broad with FWHM ~0.15 r.l.u., indicating short correlation lengths. • In strongly underdoped YBCO (p = 0.06), a CDF peak at qCDF ≈ (0.35,0) is observed with Δ(20 K) ≈ 25 meV, larger than at p ≈ 0.19, and with similar temperature trends in intensity and width. In overdoped Ca-YBCO (p = 0.22), no CDW is detected, but CDF persist as a broad, weakly temperature-dependent quasi-elastic peak, with a larger characteristic energy than at p ≈ 0.19. • The CDF wavevector HCDF follows the known doping dependence of the CDW wavevector in YBCO, extending its linear trend to p = 0.06 (HCDF ≈ 0.35) and p = 0.22 (HCDF ≈ 0.27). • The quasi-elastic resonant scattering intensity increases nearly isotropically and linearly with temperature at high T, consistent with a Bose distribution in the semiclassical regime; this behavior is steeper at p* than in underdoped samples. Far from qCDF, the extracted bosonic energy scale Ω reproduces the same doping trend as Δ and satisfies Ω > Δ at all dopings; Ω is ~15–20 meV larger at p = 0.06 than at p ≈ 0.19. • Global fits yield ω0(T) rising from ~5 meV near Tc to ~20 meV at room temperature; v0 ≈ 1.26 eV (r.l.u.)^2; and a Landau damping parameter γ ≈ 1 above Tc. Subtraction-based analysis indicates a mild slope change in CDF peak height and FWHM at Tc, suggesting interaction with superconductivity. • Bond-stretching phonon softening correlates with CDF: at p ≈ 0.19, the phonon energy drop near q just above qCDF is ~15 meV at 80 K and reduced at 200 K; at p = 0.06, softening persists but is weaker (~4 meV at 20 K). This indicates that high-energy phonon softening is associated not only with long-range CDW but also with CDF.

The data establish that charge density fluctuations have the expected hallmarks of quantum critical fluctuations centered at p* ≈ 0.19: maximal intensity and minimal characteristic energy scale. The extracted Δ(p) (converted to temperature) aligns with the boundary where the strange metal’s linear-in-T resistivity terminates, reinforcing a causal link between charge fluctuations and strange metal transport. The nearly isotropic, Bose-like increase of quasi-elastic spectral weight across reciprocal space indicates that CDF affect carrier scattering broadly, consistent with the dissipation underlying strange metal behavior. The continuation of CDF above p*—despite the absence of a pseudogap—further connects charge fluctuations, rather than pseudogap physics, to the strange metal regime. Nonetheless, anomalies arise relative to conventional criticality: the CDF correlation length does not diverge at p* as T → 0, and superconductivity influences CDF below Tc (evidenced by slope changes at Tc). Two scenarios could reconcile this: (i) frustrated criticality where Δ remains finite as T → 0, or (ii) divergent damping γ leading to critical slowing down with finite ξ, producing a glassy ensemble of CDF domains. Either case suggests an unconventional, possibly anomalous QCP. While RIXS singles out charge modes, complementary neutron and NMR studies in other cuprates indicate spin-related criticality near the pseudogap endpoint, so a mixed or subdominant spin contribution cannot be excluded. The observed coupling of CDF to bond-stretching and likely acoustic phonons highlights an interplay between charge order fluctuations, lattice dynamics, strange metal transport, and superconductivity.

By combining high- and medium-resolution RIXS with EI-RXS across multiple cuprate families and dopings, this work quantitatively maps the temperature and doping dependence of charge density fluctuations and identifies their quantum-critical-like enhancement at p* ≈ 0.19. The characteristic CDF energy Δ is minimized at p*, and its magnitude tracks the boundary of the strange metal phase, supporting a central role of charge fluctuations in the anomalous normal state and their entanglement with superconductivity and phonons. The persistence of CDF beyond the pseudogap region and the correlation with phonon softening further strengthen this link. Future directions include: suppressing superconductivity with high magnetic fields to probe zero-temperature critical scaling of Δ, ξ, and γ; extending RIXS mapping in other cuprate families to disentangle charge versus spin contributions; momentum-resolved studies of CDF–phonon coupling (including acoustic modes); and exploring the relationship between CDF dynamics, the pseudogap, and superconducting pairing.

• Superconductivity masks potential zero-temperature critical scaling; correlation length divergence at p* could not be established. High-field, low-temperature measurements were not performed. • Near overlapping CDW and CDF wavevectors, disentangling static CDW and dynamic CDF at very low energies is challenging; analyses away from qCDF were used to extract Ω. • The Landau damping parameter γ was constrained primarily above Tc; its precise low-T behavior remains uncertain. • Energy resolution, while high, limits the precision of very low-energy CDF features; some parameters rely on model-dependent global fits. • The study focuses on selected cuprate families; potential spin contributions suggested by other probes in different compounds cannot be excluded. The causal hierarchy between CDF and the pseudogap remains unresolved.