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arXiv:2306.03681

Consistency between reflection M-EELS and optical spectroscopy measurements of the long-wavelength density response of Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$

Jin Chen, Xuefei Guo, Christian Boyd, Simon Bettler, Caitlin S. Kengle, Dipanjan Chaudhuri, Hoveyda, Farzaneh, Ali Husain, John A. Schneeloch, Genda D. Gu, Philip Phillips, Bruno Uchoa, T.‐C. Chiang, Peter M. Abbamonte

Published Jun 6, 2023• 0 citations
arXivHTMLPDF

Abstract

The density fluctuation spectrum captures many fundamental properties of strange metals. Using momentum-resolved electron energy-loss spectroscopy (M-EELS), we recently showed that the density response of the strange metal Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (Bi-2212) at large momentum, $q$, exhibits a constant-in-frequency continuum [Mitrano, PNAS $\textbf{115}$, 5392 (2018); Husain, PRX $\textbf{9}$, 041062 (2019)] reminiscent of the marginal Fermi liquid (MFL) hypothesis of the late 1980s [Varma, PRL $\textbf{63}$, 1996 (1989)]. However, reconciling this observation with infrared (IR) optics experiments, which show a well-defined plasmon excitation at $q \sim 0$, has been challenging. Here we report M-EELS measurements of Bi-2212 using 4$\times$ improved momentum resolution, allowing us to reach the optical limit. For momenta $q<0.04$ r.l.u., the M-EELS data show a plasmon feature that is quantitatively consistent with IR optics. For $q>0.04$ r.l.u., the spectra become incoherent with an MFL-like, constant-in-frequency form. We speculate that, at finite frequency, $ω$, and nonzero $q$, some attribute of this Planckian metal randomizes the probe electron, causing it to lose information about its own momentum.

DOI
10.48550/arxiv.2306.03681

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Consistency between reflection M-EELS and optical spectroscopy measurements of the long-wavelength density response of Bi2_22​Sr2_22​CaCu2_22​O8+x_{8+x}8+x​
The density fluctuation spectrum captures many fundamental properties of strange metals. Using momentum-resolved electron energy-loss spectroscopy (M-EELS), we recently showed that the density response of the strange metal Bi2_22​Sr2_22​CaCu2_22​O8+x_{8+x}8+x​ (Bi-2212) at large momentum, qqq, exhibits a constant-in-frequency continuum [Mitrano, PNAS 115\textbf{115}115, 5392 (2018); Husain, PRX 9\textbf{9}9, 041062 (2019)] reminiscent of the marginal Fermi liquid (MFL) hypothesis of the late 1980s [Varma, PRL 63\textbf{63}63, 1996 (1989)]. However, reconciling this observation with infrared (IR) optics experiments, which show a well-defined plasmon excitation at q∼0q \sim 0q∼0, has been challenging. Here we report M-EELS measurements of Bi-2212 using 4×\times× improved momentum resolution, allowing us to reach the optical limit. For momenta q<0.04q<0.04q<0.04 r.l.u., the M-EELS data show a plasmon feature that is quantitatively consistent with IR optics. For q>0.04q>0.04q>0.04 r.l.u., the spectra become incoherent with an MFL-like, constant-in-frequency form. We speculate that, at finite frequency, ωωω, and nonzero qqq, some attribute of this Planckian metal randomizes the probe electron, causing it to lose information about its own momentum.
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.
Momentum-resolved electron energy-loss spectroscopy (EELS) on strange-metal BiSrCaCuO(Bi-2212) observes a broad charge continuum that is nearly momentum independent over an extended finite-momentum regime, whereas optical spectroscopy determines a metallic local conductivity. We analyze the long-wavelength response function that connects these regimes: the proper homogeneous bulk charge-density polarization. For any homogeneous-conserving metal whose longitudinal matter conductivity remains finite in thelimit at fixed nonzero frequency, the Ward identity gives. We then prove that any nonnegative normalized momentum-resolution kernel with a self-similar-scaled profile, uniformly bounded second moment, and tight second-moment tails preserves thisasymptotic behavior, changing only the prefactor. Consequently, a nearly local finite-continuum in Bi-2212, if continuously connected to a regular metallic optical limit of the same proper bulk response, is naturally described by a crossover surface. The theorem is a constraint on the proper bulk density response, or on a positive-scaled convolution of it; screened loss functions and surface EELS observables require the corresponding electrodynamic conversion before the constraint is applied.
The strange metal is a peculiar phase of matter in which the electron scattering rate, τ−1∼kBT/ℏτ^{-1} \sim k_B T/\hbarτ−1∼kB​T/ℏ, which determines the electrical resistance, is universal across a wide family of materials and determined only by fundamental constants. In 1989, theorists hypothesized that this universality would manifest as scale-invariant behavior in the dynamic charge susceptibility, χ′′(q,ω)χ''(q,ω)χ′′(q,ω). Here, we present momentum-resolved inelastic electron scattering measurements of the strange metal Bi2_22​Sr2_22​CaCu2_22​O8+x_{8+x}8+x​ showing that the susceptibility has the scale-invariant form χ′′(q,ω)=T−νf(ω/T)χ''(q,ω) = T^{-ν} f(ω/T)χ′′(q,ω)=T−νf(ω/T), with exponent ν=0.93ν= 0.93ν=0.93. We find the response is consistent with conformal invariance, meaning the dynamics may be thought of as occurring on a circle of radius 1/T1/T1/T in imaginary time, characterized by conformal dimension Δ=0.05Δ= 0.05Δ=0.05. Our study indicates that the strange metal is a universal phenomenon whose properties are not determined by microscopic properties of a particular material.
A hallmark of a Fermi liquid is a collective excitation associated with fluctuations of the Fermi surface, which evolves into the well-known plasmon in the presence of Coulomb interactions. This paper addresses the fate of the plasmon excitation in non-Fermi liquid metals with a sharply defined Fermi surface, but where long-lived single-electron excitations are absent. The authors use two complementary theoretical techniques to analyze the collective density fluctuations and their decay into the two-particle continuum for three distinct solvable examples of non-Fermi liquids.
Marginal Fermi liquid was originally introduced as a phenomenological description of the cuprates in a part of the metallic doping range which appears to be governed by fluctuations due to a quantum-critical point. An essential result due to the form of the assumed fluctuation spectra is that the large inelastic quasiparticle relaxation rate near the Fermi surface is proportional to the energy measured from the chemical potential, ${\ensuremath{\tau}}_{i}^{\ensuremath{-}1}\ensuremath{\propto}\ensuremath{\epsilon}$. We present a microscopic long-wavelength derivation of the hydrodynamic properties in such a situation by an extension of the procedure that Eliashberg used for the derivation of the hydrodynamic properties of a Landau-Fermi liquid. In particular, the density-density and the current-current correlations and the relation between the two are derived, and the connection to microscopic calculations of the frequency dependence of the optical conductivity with an additional Fermi-liquid correction factor shown to follow. The method used here may be necessary, quite generally, for the correct hydrodynamic theory for any problem of quantum-critical fluctuations in fermions.
Using two approaches to strongly correlated systems, the extremely correlated Fermi liquid theory and the dynamical mean field theory, we compute the transverse transport coefficients, namely, the Hall constants RH{R}_{H}RH​ and Hall angles ${\ensuremath{\theta}}_{H}$, and the longitudinal and transverse optical response of the $U=\ensuremath{\infty}$ Hubbard model in the limit of infinite dimensions. We focus on two successive low-temperature regimes, the Gutzwiller-correlated Fermi liquid (GCFL) and the Gutzwiller-correlated strange metal (GCSM). We find that the Hall angle $cot{\ensuremath{\theta}}_{H}$ is proportional to T2{T}^{2}T2 in the GCFL regime, while upon warming into the GCSM regime it first passes through a downward bend and then continues as T2{T}^{2}T2. Equivalently, RH{R}_{H}RH​ is weakly temperature dependent in the GCFL regime, but becomes strongly temperature dependent in the GCSM regime. Drude peaks are found for both the longitudinal optical conductivity ${\ensuremath{\sigma}}_{xx}(\ensuremath{\omega})$ and the optical Hall angles $tan{\ensuremath{\theta}}_{H}(\ensuremath{\omega})$ below certain characteristic energy scales. By comparing the relaxation rates extracted from fitting to the Drude formula, we find that in the GCFL regime there is a single relaxation rate controlling both longitudinal and transverse transport, while in the GCSM regime two different relaxation rates emerge. We trace the origin of this behavior to the dynamical particle-hole asymmetry of the Dyson self-energy, arguably a generic feature of doped Mott insulators.