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arXiv:2503.15646

Decoupling momentum and energy relaxation rates in cuprate strange metals via giant THz nonlinearities

Dipanjan Chaudhuri, David Barbalas, Fahad Mahmood, Jiahao Liang, Ralph Romero, Anaëlle Legros, Xi He, H. Raffy, I. Božović, N. P. Armitage

Published Mar 19, 2025• 1 citation
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Abstract

Understanding the $T$-linear normal-state resistivity of cuprates remains a central physics challenge. The associated momentum relaxation rate, $Γ_M$, saturates near the conjectured ``Planckian" bound $Γ_M\sim kT/\hbar$, but the mechanism underlying the anomalous scattering remains unresolved. Here we employ nonlinear terahertz spectroscopy to systematically study La$_{2-x}$Sr$_x$CuO$_4$ across a broad temperature and doping range. We measure the normal-state third-order susceptibility, $|χ^{(3)}|\approx 6\times10^{-9}$ m$^2$/V$^2$, among the largest in the THz regime, enabling direct access to the rarely measured electronic energy relaxation rate, $Γ_E$. Strikingly, $Γ_E$ is 10-40 times smaller than $Γ_M$, revealing that the scatterings responsible for momentum loss and $T$-linear resistivity do not remove appreciable energy from the electrons. While $Γ_M (T)$ is consistent with quasi-elastic scattering from bosonic modes above their characteristic energy scale, this is incompatible with the increasing temperature dependence of $Γ_E(T)$. Our results exclude phonons as the source of $T$-linear resistivity and impose strong constraints on possible mechanisms.

DOI
10.48550/arxiv.2503.15646

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Understanding the TTT-linear normal-state resistivity of cuprates remains a central physics challenge. The associated momentum relaxation rate, ΓMΓ_MΓM​, saturates near the conjectured ``Planckian" bound ΓM∼kT/ℏΓ_M\sim kT/\hbarΓM​∼kT/ℏ, but the mechanism underlying the anomalous scattering remains unresolved. Here we employ nonlinear terahertz spectroscopy to systematically study La2−x_{2-x}2−x​Srx_xx​CuO4_44​ across a broad temperature and doping range. We measure the normal-state third-order susceptibility, ∣χ(3)∣≈6×10−9|χ^{(3)}|\approx 6\times10^{-9}∣χ(3)∣≈6×10−9 m2^22/V2^22, among the largest in the THz regime, enabling direct access to the rarely measured electronic energy relaxation rate, ΓEΓ_EΓE​. Strikingly, ΓEΓ_EΓE​ is 10-40 times smaller than ΓMΓ_MΓM​, revealing that the scatterings responsible for momentum loss and TTT-linear resistivity do not remove appreciable energy from the electrons. While ΓM(T)Γ_M (T)ΓM​(T) is consistent with quasi-elastic scattering from bosonic modes above their characteristic energy scale, this is incompatible with the increasing temperature dependence of ΓE(T)Γ_E(T)ΓE​(T). Our results exclude phonons as the source of TTT-linear resistivity and impose strong constraints on possible mechanisms.
The anomalous strange metal phase found in high-T c cuprates does not follow the conventional condensed-matter principles enshrined in the Fermi liquid and presents a great challenge for theory.Highly precise experimental determination of the electronic self-energy can provide a test bed for theoretical models of strange metals, and angle-resolved photoemission can provide this as a function of frequency, momentum, temperature and doping.Here we show that constant energy cuts through the nodal spectral function in (Pb,Bi) 2 Sr 2-x La x CuO 6+δ have a non-Lorentzian lineshape, consistent with a selfenergy that is k dependent.This provides a new test for aspiring theories.Here we show that the experimental data are captured remarkably well by a power law with a k-dependent scaling exponent smoothly evolving with doping, a description that emerges naturally from anti-de Sitter/conformal-field-theory based semi-holography.This puts a spotlight on holographic methods for the quantitative modelling of strongly interacting quantum materials like the cuprate strange metals.The rich temperature versus doping diagram of the cuprate high-T c superconductors presents a cornucopia of non-conformity 1 .The superconductivity does not follow BCS theory anywhere in the phase diagram 2,3 , and in their normal state these unusual metals display qualitatively different experimental behaviour compared to the Fermi liquid, with the linear-in-T resistivity and quadratic-in-T Hall angle being among the most strange.One intriguing possible explanation of strange metal phenomenology is that it is controlled by a strongly interacting quantum critical phase.If so, the question follows whether their properties can be
Abstract The mysterious metallic phase showing T-linear resistivity and a universal scattering rate 1 / τ = α P k B T / ℏ with a universal prefactor α P ∼ 1 and logarithmic-in-temperature singular specific heat coefficient, the so-called ‘Planckian metal phase’ was observed in various overdoped high- T c cuprate superconductors over a finite range in doping. Revealing the mystery of the Planckian metal state is believed to be the key to understanding the mechanism for high- T c superconductivity. Here, we propose a generic microscopic mechanism for this state based on quantum-critical local bosonic charge Kondo fluctuations coupled to both spinon and a heavy conduction-electron Fermi surface within the heavy-fermion formulation of the slave-boson t–J model. By a controlled perturbative renormalization group analysis, we examine the competition between the pseudogap phase, characterized by Anderson’s Resonating-Valence-Bond spin-liquid, and the Fermi-liquid state, modeled by the electron hopping (effective charge Kondo effect). We find a quantum-critical metallic phase with a universal Planckian ℏ ω / k B T scaling in scattering rate near an extended localized-delocalized (pseudogap-to-Fermi liquid) charge-Kondo breakdown transition. The d-wave superconducting ground state emerges near the transition. Unprecedented qualitative and quantitative agreements are reached between our theoretical predictions and various experiments, including optical conductivity, universal doping-independent field-to-temperature scaling in magnetoresistance, specific heat coefficient, marginal Fermi-liquid spectral function observed in ARPES, and Fermi surface reconstruction observed in Hall coefficients in various overdoped cuprates. Our mechanism offers a microscopic understanding of the quantum-critical Planckian metal phase observed in cuprates and its link to the pseudogap, d-wave superconducting, and Fermi liquid phases. It offers a promising route for understanding how d-wave superconductivity emerges from such a strange metal phase in cuprates–one of the long-standing open problems in condensed matter physics since 1990s–as well as shows a broader implication for the Planckian strange metal states observed in other correlated unconventional superconductors.
Metals at the brink of electronic quantum phase transitions display high-temperature superconductivity, competing orders, and unconventional charge transport, revealing strong departures from conventional Fermi liquid behavior. Investigation of these fascinating intertwined phenomena has been at the center of research across a variety of correlated materials over the past many decades. A ubiquitous experimental observation is the emergence of a universal timescale that governs electrical transport and momentum relaxation. In this Letter, we analyze an equally important theoretical question of how the energy contained in the electronic degrees of freedom near a quantum phase transition relaxes to the environment via their coupling to acoustic phonons. Assuming that the bottleneck for energy dissipation is controlled by the coupling between electronic degrees of freedom and acoustic phonons, we present a universal theory of the temperature dependence of the energy relaxation rate in a marginal Fermi liquid. We find that the energy relaxation rate exhibits a complex set of temperature-dependent crossovers controlled by emergent energy scales in the problem. We place these results in the context of recent measurements of the energy relaxation rate via nonlinear optical spectroscopy in the normal state of hole-doped cuprates.
The strange metal is an enigma within the field of many-body quantum physics, as it defies the standard Fermi liquid theory of metals due to the absence of well-defined, electron-like quasiparticles. It has been presumed that strange metal behavior arises due to strong correlations between electrons that are not present in ordinary Fermi liquids, but the exact nature of these correlations is unknown. To this end, this thesis aims to address the fundamental question: What, precisely, is so strongly correlated about charge in the strange metal? This question is tackled by experimentally measuring the dynamic charge response function, which directly encodes charge correlations in momentum and energy, using the technique of Momentum-resolved Electron Energy-loss Spectroscopy (M-EELS). In particular, two prototypical members of the cuprate and ruthenate strange metal families are studied: Bi-2212 and Sr2_22​RuO4_44​. By studying Bi-2212 at optimal doping, it is found that the charge response of the strange metal is characterized by a broad continuum of fluctuations that are momentum and energy independent over a large range of parameter space, unlike Fermi liquid charge correlations which exhibit propagating density waves of charge with well defined energy and momentum (i.e. plasmons). We argue that, in essence, the charge correlations of the strange metal are highly localized in both space and time, unlike the Fermi Liquid where they are localized in momentum and energy instead. To understand how charge correlations change as one leaves the strange metal regime, the M-EELS response of Bi-2212 is mapped out as a function of doping and temperature. At high temperature, it is found that the continuum is largely doping independent. However, upon cooling, very large changes in spectral weight are observed over a range of energy scales nearly two orders of magnitude larger than the thermal scale kBTk_B TkB​T. Moreover, this change in spectral weight undergoes a sign reversal as a function of doping, with spectral weight enhancement on the underdoped side, suppression in the overdoped regime, and no change at optimal doping. These changes in spectral weight are completely unlike the case of the Fermi liquid, where doping only acts to rigidly shift the plasmon energy and no significant changes occur as a function of temperature. Finally, we venture further and ask whether strange metal and Fermi liquid charge correlations can simultaneously coexist in a material. To explore this possibility, we measured the M-EELS response of Sr2_22​RuO4_44​, a material which exhibits strange and bad metal behavior at high temperature, but transitions to a well-defined Fermi liquid at low temperature. Surprisingly, in the strange metal regime, Sr2_22​RuO4_44​ exhibits both a broad continuum similar to Bi-2212 and a propagating Fermi liquid collective mode at low-energies and momentum. Upon cooling, low-energy spectral weight is suppressed in the strange metal continuum and the Fermi liquid collective mode velocity is strongly renormalized. These findings confirm the strange metal and Fermi liquid can coexist and demonstrate that Sr2_22​RuO4_44​ exhibits charge correlations with strange metal character at high energies and short length scales and Fermi liquid character at low energies and long wavelengths.
The mysterious metallic phase showing perfect TTT-linear resistivity and a universal scattering rate 1/τ=αPkBT/ℏ1/τ= α_P k_B T /\hbar1/τ=αP​kB​T/ℏ with a universal prefactor αP∼1α_P \sim 1αP​∼1 and logarithmic-in-temperature singular specific heat coefficient, so-called Planckian metal phase was observed in various overdoped high-TcT_cTc​ cuprate superconductors over a finite range in doping. Here, we propose a microscopic mechanism for this exotic state based on quantum-critical bosonic charge Kondo fluctuations coupled to both spinon and a heavy conduction-electron Fermi surfaces within the heavy-fermion formulation of the slave-boson ttt-JJJ model. Using a controlled perturbative renormalization group (RG) analysis, we examine the competition between the pseudogap phase, characterized by Anderson's Resonating-Valence-Bond spin-liquid, and the Fermi-liquid state, characterized by the electron hoping (effective charge Kondo effect). We find a quantum-critical metallic phase with a universal Planckian ℏω/kBT\hbar ω/k_B Tℏω/kB​T scaling in scattering rate near a localized-delocalized (pseudogap-to-Fermi liquid) charge Kondo breakdown transition. Our results are in excellent agreement with the recent experimental observations on optical conductivity (without fine-tuning) in Nat. Commun. 14, 3033 (2023), universal doping-independent field-to-temperature scaling in magnetoresistance in Nature 595, 661 (2021), and the marginal Fermi-liquid spectral function observed in ARPES (Science 366, 1099 (2019)) as well as Hall coefficient in various overdoped cuprates in Nature 595, 661 (2021) and Annu. Rev. Condens. Matter Phys. 10, 409 (2019). Our mechanism offers a microscopic understanding of the quantum-critical Planckian metal phase observed in cuprates d-wave superconducting, and Fermi liquid phases.
A one band Hubbard model with intermediate coupling is shown to describe the two most important unusual features of a normal state: linear resistivity strange metal and the pseudogap. Both the spectroscopic and transport properties of the cuprates are considered on the same footing by employing a relatively simple post-Gaussian approximation valid for the intermediate couplings U/t=1.5−−4U/t=1.5--4U/t=1.5−−4 in relevant temperatures T>100K.T>100\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}.T>100K. In the doping range p=0.1−−0.3\phantom{\rule{4pt}{0ex}}p=0.1--0.3p=0.1−−0.3, the value of UUU is smaller than that in the parent material. For a smaller doping, especially in the Mott insulator phase, the coupling is large compared to the effective tight binding scale and a different method is required. This scenario provides an alternative to the paradigm that the coupling should be strong, say U/t>6U/t>6U/t>6, in order to describe the strange metal. We argue that, to obtain phenomenologically acceptable underdoped normal state characteristics like T∗{T}^{*}T∗, pseudogap values, and spectral weight distribution, a large value of UUU is detrimental. Surprisingly the resistivity in the above temperature range is linear, $\ensuremath{\rho}={\ensuremath{\rho}}_{0}+\ensuremath{\alpha}\frac{{m}^{*}}{{e}^{2}n\ensuremath{\hbar}}T$, with the ``Planckian'' coefficient $\ensuremath{\alpha}$ of order 1.