Universality of linear in temperature and linear in field Planckian scattering rate in high temperature cuprate superconductors

TL;DR

This work identifies a universal origin for Planckian transport in high-temperature cuprates by linking linear-in-$T$ and linear-in-$B$ scattering within a quantum-critical framework. A spin-based, Kondo-like charge-fluctuation mechanism embedded in a heavy-fermion–styled slave-boson $t$-$J$ model, with Zeeman coupling, yields a universal $B/T$ scaling of the scattering rate and a common framework for both $T$-linear and $B$-linear resistivity. The theory fits LSCO data near optimal doping with a shared set of parameters, predicting Planckian coefficients $oldsymbol{1}=rac{8}{ ext{π}}$ and $oldsymbol{1}=rac{4}{ ext{π}}$ (hence $oldsymbol{1}/oldsymbol{1}=2$) and a doping-insensitive correlation between the two regimes. The results support a quantum-critical origin of Planckian transport, propose a hidden quantum critical point inside the superconducting dome, and point to a unified description of strange-metal behavior in cuprates with broad experimental relevance.

Abstract

One of the long standing puzzles in strongly correlated materials is the microscopic origin of the quantum critical Planckian strange metal phase with universal linear in temperature scattering rate from which unconventional superconductivity directly emerges by lowering temperatures. Recently, the linear in temperature and linear in field resistivity have been simultaneously observed in high temperature cuprate superconductors, manifested by the universal field to temperature scaling in magnetoresistivity. To date, there has been a lack of coherent and unified understanding of these coexisting linear behaviors and their possible link to quantum criticality. In this work, we establish the universality in linear in temperature and linear in field Planckian behaviors in underdoped LSCO near optimal doping. Experimentally, we observe the linear in field Planckian scattering rate and its relation to its linear in temperature counterpart. Theoretically, we propose a spin based common microscopic mechanism based on Kondo-like charge fluctuations near local quantum criticality of heavy fermion formulated tJ model subject to a Zeeman term. Similar to frequency to temperature scaling near quantum criticality, we find the magnetic field here effectively introduces a Zeeman energy, reminiscent of an external energy in the quantum critical regime, leading to field to temperature scaling. Our analytically predicted universal field to temperature scaling in isotropic scattering rate and the relation between the linear in temperature and linear in field Planckian coefficients, unifies these two phenomena over an extended doping range, pointing toward a unified quantum-critical origin of Planckian transport in cuprates.

Figures (20)

• Figure 1: Magneto--resistivity data for LSCO at doping $p=0.19$, in which $T$--linear and $B$--linear coexist over the range of temperatures ($T<$ 50K) and magnetic fields ($B>$ 60T) indicated by the shaded regions. (a) Magneto--resistivity versus temperature at different magnetic fields. Colored symbols represent experimental data and black solid lines correspond to the theoretical prediction given in \ref{['eq:rho-tt']}, with $c_0=7.2 \mu\rm{\Omega cm}$, $c_1=0.654\mu\rm{\Omega cm K^{-1}}$, $\gamma=3.2 (\mu\rm{\Omega cm})^{-1}\rm{T}$ , $\zeta=3.06$. For visibility purposes, the data points and curves have been shifted by $30 \mu\rm{\Omega cm}$ in the y-axis successively for each value of magnetic field. (b) Magneto--resistivity data versus magnetic field at different temperatures. Colored symbols represent experimental data and black dotted lines correspond to the theoretical prediction given in \ref{['eq:rho-tt']} with same fitting parameters values as in (a). (c) Scaling of the field dependent part of the magneto--resistivity at different magnetic field values in function of $\zeta x/2 \coth(\zeta x/2 )$, with $x=\mu_B B/(k_B T)$. This is obtained by subtracting to the total magneto--resistivity the constant and temperature-only dependent ($\rho_0(T)$) and dividing it by temperature using the same parameters as in (a). Data is the same as in ref. Beobinger-2018-Science-LSCO for LSCO.
• Figure 2: (a) Slope of the magneto--resistivity with respect to magnetic field versus temperature at different magnetic fields for doping $p=0.19$. Colored symbols represent experimental data and colored solid lines correspond to the associated theoretical prediction [ \ref{['eq:slope']}] obtained by taking the derivative of \ref{['eq:rho-tt']} , with same parameter values as in \ref{['fig:rho']} : $\gamma=3.2 (\mu\rm{\Omega cm})^{-1}\rm{T}$ , $\zeta=3.06$. For visibility, the data points and curves have been shifted by $0.1 \mu\rm{\Omega cm \cdot T^{-1}}$ in the y-axis successively for each value of magnetic field. (b) $B/T$--scaling of the rescaled slope of the magneto--resistivity as a function of $x=\mu_B B/(k_B T)$ for doping $p=0.19$. The slope values are then rescaled by $\gamma_{p=0.19}=3.2 (\mu\rm{\Omega cm})^{-1}\rm{T}$ for $p=0.19$, such that at high magnetic field values $\mu_B B>>k_B T$, the product $\gamma ({\partial} \rho/{\partial} B)$ saturates to unity. Note that $\gamma$ depends on the carrier density $n$ and is expected to slightly vary for different doping values. The black dotted line corresponds to the theoretical prediction of the derivative of the magneto--resistivity given in \ref{['eq:slope']} with $\zeta=3.06$. Magneto--resistivity data are reproduced from the same data as in Beobinger-2018-Science-LSCO for $p=0.19$.
• Figure 3: The linear-in-temperature (down triangles) and linear-in-field (up triangles) Planckian coefficients versus doping for LSCO obtained from magneto--resistivity data reproduced from Ref. Beobinger-2018-Science-LSCO as detailed in Appendix \ref{['sec:planck_coeff']}. Horizontal purple and turquoise dashed lines correspond to theoretical prediction $\alpha_T=8/\pi\simeq 2.54$ and $\alpha_B=4/\pi\simeq 1.27$, respectively. Note that our theoretical framework finds quantitative agreement with the experimentally measured values of $\alpha_B$ and agrees within $\sim 25\%$ of the experimentally measured values of $\alpha_T$. The calculated ratio of $\alpha_T/\alpha_B =2$ is in close agreement with experimental data for $0.16<p<0.19$. Inset: A plot of two-dimensional Fermi surface of LSCO at $p=0.17$ (dark region guided by red and blue curves), as measured by ARPES and adapted from Ref. LSCO_FS. The green circle is a schematic plot of isotropic electronic states near the Fermi surface that contribute to the isotropic scattering rate within our theoretical framework in the experimental doping range $0.16<p<0.19$.
• Figure 4: The doping-field-temperature $(p,B,T)$ phase diagram of LSCO near critical doping $p_{cr} \sim 0.19$ based on magneto--transport data in Beobinger-2018-Science-LSCO. Green circles defining the superconducting $T_c$ dome are experimental data obtained from Ando-PRL-2004 at zero magnetic field, and from Caprara_SciPost_2020 for finite magnetic field $B=5 \rm{T}$. Blue diamonds and blue square are data from Ramshaw-MottPlanckian-arxiv-2024 and Ando-PRL-2004 respectively and correspond to the crossover temperature between the SM phase into the PSG phase and the FL phase. Note that the temperature axis is rescaled by $T^{Max}_c$, which is experimental group dependent, and is defined by the highest $T_c$ value of superconducting $T_c$ dome at zero field.
• Figure 5: The self-energies (a) $\Sigma_{\xi}$ and (b) $\Sigma_f$, where the solid lines represent the spinon fields $G_{f}^{\sigma}$, the dashed lines the slave--boson fields ${G}_{b}$, and the wavy lines the $\xi$--fermions fields $G_{\xi}^{\sigma}$.