Critical spin fluctuations across the superconducting dome in La$_{2-x}$Sr$_{x}$CuO$_4$

TL;DR

The study addresses the origin of strange-metal behavior in overdoped cuprates by linking it to nearly critical, low-energy spin fluctuations across the superconducting dome. Using inelastic neutron scattering, the authors observe ω/T scaling of the dynamic susceptibility χ''(Qδ,ω) with α ≈ 0.32 and a spin-relaxation rate Γδ ∝ T, along with κ(ω) scaling yielding a dynamic exponent z ≈ 1.83; these features fit a disordered spin-density-wave quantum critical framework. Complementary Hertz-Millis theory with spatial disorder produces a quantum Griffiths phase that reproduces the observed scaling and yields linear-in-T resistivity with Planckian dissipation, consistent with transport measurements. Collectively, the results propose that disorder-tuned spin fluctuations drive strange metal behavior across the cuprate phase diagram, linking magnetism, Planckian transport, and the suppression of superconductivity with doping.

Abstract

Overdoped cuprate superconductors are strange metals above their superconducting transition temperature. In such materials, the electrical resistivity has a strong linear dependence on temperature ($T$) and electrical current is not carried by electron quasiparticles as in conventional metals. Here we demonstrate that the strange metal behaviour co-exists with strongly temperature-dependent critical spin fluctuations showing dynamical scaling across the cuprate phase diagram. Our neutron scattering observations and the strange metal behaviour are consistent with a spin density wave quantum phase transition in a metal with spatial disorder in the tuning parameter. Numerical computations using a theory of spin density waves in a disordered metal yield an extended `Griffiths phase' with scaling properties in agreement with experimental observations. Thus we establish that low-energy spin excitations and spatial disorder are central to the strange metal behaviour.

Figures (4)

• Figure 1: Spin fluctuations in La$_{\mathbf{2-x}}$Sr$_{\mathbf{x}}$CuO$_{\mathbf{4}}$ (x=0.22). (A) Schematic phase diagram of LSCO showing the possible extent of the critical spin fluctuations observed here. The dashed line shows the extent of spin freezing for $B=80$ T Frachet2020_FVZ where superconductivity is fully suppressed and the magenta region for $B=0$ from Ref. Kofu2009_KLF+. (B) Schematic of spin excitations in LSCO($x=0.22$) based on (Refs. Lipscombe2007_LHVRobarts2019_RBK) as represented by contours of the imaginary part of the susceptibility $\chi"(\mathbf{Q},\omega)$. Reciprocal space is labelled as $\mathbf{Q}=H\mathbf{a}^{\star}+K\mathbf{b}^{\star} +L\mathbf{c}^{\star}\equiv (H,K)$. The gray region shows the broad high-energy spin excitations Lipscombe2007_LHVRobarts2019_RBK. Pink regions are the low-energy excitations studied here. (C-F) Slices of $\chi^{\prime\prime}(\mathbf{Q},\omega)$ for $\hbar\omega=4$ meV, $L \in [-1,1]$ and various temperatures (see Methods for details). The white rectangle in (D) shows the region of integration used to produce the points in 1-D cuts such as those in Fig. \ref{['fig:cuts']}A, with integration along (1,1). Dashed lines represent the path of the 1-D cuts.
• Figure 2: The temperature dependence of the low-energy spin fluctuations in La$_{\mathbf{2-x}}$Sr$_{\mathbf{x}}$CuO$_{\mathbf{4}}$ (x=0.22). (A) The scattering intensity versus wavevector $\mathbf{Q}$ for different temperatures (bottom to top: 26, 40, 80, 155, 300 K) for energy transfers $\hbar\omega=4,8$ meV. The cuts are through the incommensurate $\mathbf{Q}_{\delta}$ positions as shown by the dashed lines in Fig. \ref{['fig:slices_and_phase_diagram']}D. (B) $E-\mathbf{Q}$ map of the magnetic response function $\chi^{\prime\prime}(\mathbf{Q},\omega)$ for $T=26, 80$ K showing its evolution with temperature. The trajectory of $\mathbf{Q}$ is the same as that in (A).
• Figure 3: Scaling behaviour of the dynamic magnetic susceptibility and correlation length in La$_{\mathbf{2-x}}$Sr$_{\mathbf{x}}$CuO$_{\mathbf{4}}$ (A) Temperature dependence of $\chi"(\mathbf{Q},\omega)$ at ordering wavevector $\mathbf{Q}_{\delta}$. Lines are fits to Eqn. \ref{['eqn:chi_qE']}. Open and closed symbols are two separate experiments. Inset shows the $T$-dependence of the spin relaxation rate $\Gamma_{\delta}$ and real part of the susceptibility $\chi'(\mathbf{Q}_{\delta})$ obtained when data are modelled with Eqns. \ref{['eqn:chi_qE']}-\ref{['eqn:Gamma_T']}. (B) $\omega/T$-scaling plot of the $x=0.22$ data in (A) using Eqn. \ref{['eqn:omega_over_T_scaling']} and the procedure described in Methods. The inset shows how the quality of the collapse varies with exponent $\alpha$. (C) Scaling plot for underdoped LSCO $x=0.14$ data from Ref. Aeppli1997_AMH. This data shows a similar collapse to the $x=0.22$ data in (B). (D) The width in $\mathbf{Q}$ of $\chi"(\mathbf{Q,\omega})$ is denoted by the inverse dynamic correlation length $\kappa(\omega)$. Here we plot $\kappa(\omega)$ versus temperature for $\hbar\omega=1.25,2.25,4,8$ meV. (E) Data in (D) can be collapsed onto a single trend line with suitable choice of $z$ and $\omega/T$-scaling. (F) Data in (D) can be scaled onto a single line using (\ref{['eqn:omega_T_quad']}) allowing the dynamic critical exponent $z$ to be determined. Dashed line is Supplementary Equation 15.
• Figure 4: Numerical theoretical results for low energy spin fluctuations. (A-D) (A-B) Dynamic spin susceptibility and its scaling plot, with best-fit exponent $\alpha = 0.84$. The inset in panel A provides a schematic phase diagram of the model. (C-D) Inverse correlation length $\kappa$ and its scaling plot, with best-fit critical exponent $z = 2.128$. $\kappa$ is in units of the inverse lattice spacing. We estimate $\approx 33$ meV as the unit of energy/frequency/temperature in the numerics and $\beta = 1/T$. Theory plots should be compared with the experimental plots in Fig. \ref{['fig:chi_T']}A, B, C, D, E. Plots based on Hertz-Millis model with spatial disorder are obtained from the results of Ref. Patel2024_PLS (with a mean-field treatment of interactions, but exact treatment of disorder), at the critical value of $\lambda=\lambda_c = -0.4586$, after exact analytic continuation to real frequencies (See Supplementary Methods); results for $\lambda > \lambda_c$ are in the Supplementary Methods. (E) Computations of imaginary frequency ($\Omega_m$) susceptibility by (in principle, exact) quantum Monte Carlo (QMC) results for the YSYK model in Ref. Patel2025_PLA, at the end of the strange metal quantum Griffiths phase that is present for $\lambda_s = 5.0 \le \lambda \le \lambda_G = 5.5$. In the QMC results, the transition into an ordered phase at the critical point $\lambda_c$ is replaced with a transition into a phase with glassy short-range order at $\lambda_s$. The dashed line in the inset is a fit to $\sim T^{-\alpha}$. Both plots yield $\alpha=0.20$. The energy unit for the QMC computations is the fermion hopping $t \approx 0.3$ eV. (F) Resistivity from spin fluctuations. The resistivity is computed from the spin susceptibility, as described in Ref. Patel2024_PLS. Note the distinct slopes of linear-in-$T$ resistivity at low and high $T$. Using the energy unit of $\approx 33$ meV and the parameters of Ref. Li:2024kxr, with the fermion hopping $t \approx 0.3$ eV and the variance of the random Yukawa coupling ${g'}^2 = 4t$, the largest resistivity shown in the plot is $\approx 0.15~h/e^2$.