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.
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.
Consistency between reflection M-EELS and optical spectroscopy measurements of the long-wavelength density response of Bi2Sr2CaCu2O8+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 Bi2Sr2CaCu2O8+x (Bi-2212) at large momentum, q, exhibits a constant-in-frequency continuum [Mitrano, PNAS 115, 5392 (2018); Husain, PRX 9, 041062 (2019)] reminiscent of the marginal Fermi liquid (MFL) hypothesis of the late 1980s [Varma, PRL 63, 1996 (1989)]. However, reconciling this observation with infrared (IR) optics experiments, which show a well-defined plasmon excitation at q∼0, has been challenging. Here we report M-EELS measurements of Bi-2212 using 4× 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.
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 Sr2RuO4. 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 kBT. 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 Sr2RuO4, 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, Sr2RuO4 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 Sr2RuO4 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 strange metallic regime across a number of high-temperature superconducting materials presents numerous challenges to the classic theory of Fermi liquid metals. Recent measurements of the dynamical charge response of strange metals, including optimally doped cuprates, have revealed a broad, featureless continuum of excitations, extending over much of the Brillouin zone. The collective density oscillations of this strange metal decay into the continuum in a manner that is at odds with the expectations of Fermi liquid theory. Inspired by these observations, we investigate the phenomenology of bosonic collective modes and the particle-hole excitations in a class of strange metals by making an analogy to the phonons of classical lattices falling apart across an unconventional jamming-like transition associated with the onset of rigidity. By making comparisons to the experimentally measured dynamical response functions, we reproduce many of the qualitative features using the above framework. We conjecture that the dynamics of electronic charge density over an intermediate range of energy scales in a class of strongly correlated metals can be at the brink of a jamming-like transition.
The strange metal is a peculiar phase of matter in which the electron scattering rate, τ−1∼kBT/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). Here, we present momentum-resolved inelastic electron scattering measurements of the strange metal Bi2Sr2CaCu2O8+x showing that the susceptibility has the scale-invariant form χ′′(q,ω)=T−νf(ω/T), with exponent ν=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/T in imaginary time, characterized by conformal dimension Δ=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.
Understanding the T-linear normal-state resistivity of cuprates remains a central physics challenge. The associated momentum relaxation rate, ΓM, saturates near the conjectured ``Planckian" bound ΓM∼kT/ℏ, but the mechanism underlying the anomalous scattering remains unresolved. Here we employ nonlinear terahertz spectroscopy to systematically study La2−xSrxCuO4 across a broad temperature and doping range. We measure the normal-state third-order susceptibility, ∣χ(3)∣≈6×10−9 m2/V2, among the largest in the THz regime, enabling direct access to the rarely measured electronic energy relaxation rate, ΓE. Strikingly, ΓE is 10-40 times smaller than ΓM, revealing that the scatterings responsible for momentum loss and T-linear resistivity do not remove appreciable energy from the electrons. While 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). Our results exclude phonons as the source of T-linear resistivity and impose strong constraints on possible mechanisms.
Recent progress in experimental techniques has made it possible to extract detailed information on dynamics of carriers in a correlated electron material from its optical conductivity, [Formula: see text]. This review consists of three parts, addressing the following three aspects of optical response: (1) the role of momentum relaxation; (2) [Formula: see text] scaling of the optical conductivity of a Fermi-liquid metal, and (3) the optical conductivity of non-Fermi-liquid metals. In the first part (section 2), we analyze the interplay between the contributions to the conductivity from normal and umklapp electron-electron scattering. As a concrete example, we consider a two-band metal and show that although its optical conductivity is finite it does not obey the Drude formula. In the second part (sections 3 and 4), we re-visit the Gurzhi formula for the optical scattering rate, [Formula: see text], and show that a factor of [Formula: see text] is the manifestation of the 'first-Matsubara-frequency rule' for boson response, which states that [Formula: see text] must vanish upon analytic continuation to the first boson Matsubara frequency. However, recent experiments show that the coefficient b in the Gurzhi-like form, [Formula: see text], differs significantly from b = 4 in most of the cases. We suggest that the deviations from Gurzhi scaling may be due to the presence of elastic but energy-dependent scattering, which decreases the value of b below 4, with b = 1 corresponding to purely elastic scattering. In the third part (section 5), we consider the optical conductivity of metals near quantum phase transitions to nematic and spin-density-wave states. In the last case, we focus on 'composite' scattering processes, which give rise to a non-Fermi-liquid behavior of the optical conductivity at T = 0: [Formula: see text] at low frequencies and [Formula: see text] at higher frequencies. We also discuss [Formula: see text] scaling of the conductivity and show that [Formula: see text] in the same model scales in a non-Fermi-liquid way, as [Formula: see text].