Quantum critical behavior of cuprate superconductors observed by inelastic X-ray scattering

Abstract

Progress toward a complete understanding of cuprate superconductors has been hindered by their intricate phase diagram, potentially linked to a quantum critical point (QCP). However, conclusive evidence for the QCP is lacking, as the presumed QCP is buried under the superconducting dome, disguising its presence. Here, we use high-resolution resonant inelastic X-ray scattering to examine the dynamical charge-charge correlation in La$_{2-x}$Sr$_x$CuO$_4$ and uncover the quantum critical scaling, a key feature required for a QCP. Specifically, \djh{we observed that the inverse correlation lengths for various dopings and temperatures collapsed onto a universal scaling curve, yielding a critical exponent $ν$ of $0.74 \pm 0.08$. The non-negativity of this exponent confirms the presence of a QCP. Remarkably, the value of $ν$ suggests that while the QCP is manifested through the charge-density wave, other orders also participate, such that the QCP appears to belong to the universality class characterized by the O(4) symmetry, reminiscent of the microscopic SO(4) symmetry in the Hubbard model at half-filling. Further analysis indicates that the QCP is highly dissipative with a short quasi-particle lifetime, reflecting the intertwined quantum fluctuations due to its being buried inside the superconducting state.

Figures (5)

• Figure 1: Phase diagram of La$_{2-x}$Sr$_x$CuO$_4$ plotted in the plane of temperature versus hole doping level $x$ and its anomalous quantum criticality. The black solid curve, denoted by $T_{_{\rm C}}$, and the red dashed curve, denoted by $T^*$, mark the SC transition temperature and the crossover temperature of the pseudogap phase, respectively. The black dashed line marked by $T_{\rm N}$ indicates the Néel temperature. The red circle denotes the putative QCP. The gray circles and squares indicate the onset temperature of the CDW, with a dashed curve providing a visual guide Croft2014WenNatComm2019Choiniere2018. Reports have indicated varying onset curves Miao2021LinPRL2020. Additionally, studies have observed different CDW orders (the CDW stripe orders) exhibiting different temperature dependencies across $T_c$. The CDW stripe order is in the low temperature regime with doping $x$ between 0.10 and 0.13 (see Refs. WenNatComm2019 and Croft2014). In the present work, we shall focus on the CDW order that is most often cited and is beyond the CDW stripe order. Note that different kinds of CDW order also appear in the stripe order region WenNatComm2019. The strange-metal phase is a region in which the coefficient of the $T$-linear term in resistivity is non-vanishing and much larger than that of the quadratic term Cooper2009ayres2021. LSCO exhibits anomalous quantum criticality: its strange-metal phase extends to the overdoped region of $x=$ 0.22.
• Figure 2: Doping-dependent O $K$-edge RIXS of La$_{2-x}$Sr$_x$CuO$_4$ with $x = 0.12,~0.15,~0.17,~\& ~0.18$. RIXS spectra were recorded with $\sigma$-polarized incident X-ray tuned to the Zhang-Rice singlet hole peak at 24 K. The momentum transfer is q$=(q_{\|}, 0, L)$ with $L$ varying between 0.47 and 1.03 in reciprocal lattice units. (a), RIXS intensity distribution maps in the plane of energy loss vs. in-plane momentum transfer q$_\|$ along $(\pi, 0)$. (b), RIXS intensity distribution maps after the subtraction of elastic scattering. The raw RIXS data for each momentum scan are plotted in Figs. S2-S5 of the Supplementary Information.
• Figure 3: Curve-fitting analysis of O $K$-edge RIXS of La$_{2-x}$Sr$_x$CuO$_4$ with $x=0.12, 0.15, 0.17,~\&~0.18$. (a)-(d), RIXS spectra and their spectral components after curve fitting for $q_{\|} = 0.12$. The RIXS spectra were fitted to four components with a linear background: elastic scattering, acoustic phonons, the mix of buckling phonon and apical oxygen phonon, and half-breathing (bond stretching) phonons, represented in gray, pink, green, and blue shades, respectively. Each phonon component was fitted to a spectral function of damped harmonic oscillator. RIXS data are plotted as black circles, and the fitted curve is given as a gray line. Detailed information about the curve fitting is provided in the Supplementary Information. (e) $\&$ (f), Integrated spectral weights of the fitted elastic scattering and the lowest-energy RIXS excitation, i.e., acoustic phonon for $q_{\|}$ away from $|\textbf{Q}|$ and CDW fluctuations for $q_{\|}=|\textbf{Q}|$, as a function of $q_{\|}$ for various doping levels.
• Figure 4: Analysis of the RIXS spectral profile of CDW fluctuations in La$_{2-x}$Sr$_x$CuO$_4$ for ${q}_{\|}$ integrated from 0.23 to 0.24 at various temperatures and dopings. Selected RIXS spectra and spectral components from curve fitting are plotted in subplots: (a)-(b) for $x = 0.12$ at $T = 24~\&~50$ K; (c)-(e) for $x = 0.15$ at $T = 24, 50, \&~80$ K; (f)-(g) for $x = 0.17$ at $T = 24~\&~50$ K; (h) for $x = 0.18$ at $T = 24$ K. The plotted RIXS spectra are after the subtraction of elastic scattering. The curve-fitting scheme is the same as that of Fig. 3 except for the acoustic component, which was replaced by a component of CDW fluctuation. The dynamical structure factor due to CDW fluctuations is related to the charge susceptibility $\chi({\bf q}, {\omega})$ by $S({\bf q},\omega)=S_{0}(1-{\rm e}^{-\beta\omega})^{-1} {\rm Im} \chi({\bf q}, {\omega})$, in which $S_{0}$ is a proportion constant and $\beta = 1/k_{_B}T$, with $k_{_B}$ denoting the Boltzmann constant. Note that the reduced Planck constant $\hbar$ is set to 1 in the expression of $S({\bf q},\omega)$. The raw RIXS data of all doping levels and temperatures are plotted in Figs. S8 of the Supplementary Information.
• Figure 5: Scaling of the inverse correlation length of the CDW fluctuations and the evolution of relaxation rate $\Gamma$ in La$_{2-x}$Sr$_x$CuO$_4$. (a), Doping and temperature dependence of the characteristic energy $m$. Black circles depict the deduced $m$ for various temperatures at $x = 0.12$, $0.15$, $0.17$, and $0.18$. They were fitted to a power-law form: $m = A(x_{\rm c}-x)^{\nu}+BT^{\nu}$ with a fitting scheme described in the Supplementary Information. The fitted coefficients were: $A = 81.68~{\pm}~16.2$, $x_{\rm c} = 0.195~{\pm}~0.007$, $\nu = 0.74~{\pm}~0.08$, and $B = 0.25~{\pm}~0.11$, defining the energy of $m$ shown by the color surface. Solid lines plot power-law curves of $(x_{\rm c}-x)^\nu$ for $T = 24$ K and $T^\nu$ for $x = 0.12$, 0.15, and 0.17. (b), Plot of $m/T^{\nu}$ versus $(x_{\rm c}-x)/T$ for various doping levels $x$ and temperatures $T$. All data points, with minimal variations, collapse onto a curve that takes the power-law form of $Ay^{\nu}+B$, where $y = (x_{\rm c}-x)/T.$ The gray vertical bars represent the error bars of $m/T^{\nu}$, where the individual uncertainties of $m$ and $\nu$ were derived from the fitted errors and then combined through error propagation. (c), Doping-dependent relaxation rate compared with the doping dependency of superfluid density $\rho_s$, reproduced from Refs. Panagopoulos1999 (Panagopoulos et al.) and Tallon2003 (Tallon et al.). The uncertainties of $\Gamma$ were derived from the fitted errors. $\rho_s$ is expressed in terms of $1/\lambda_{ab}^{2}$, where $\lambda_{ab}$ is the in-plane magnetic penetration depth. The solid line serves as a guide for the eye.