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arXiv:1608.02514 · Reports on Progress in Physics

Optical response of correlated electron systems

Dmitrii L Maslov, Andrey V Chubukov

Published Dec 21, 2016• 97 citations
arXivHTMLPDF

Abstract

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].

DOI
10.1088/1361-6633/80/2/026503

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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].
Motivated by recent optical measurements on a number of strongly correlated electron systems, we revisit the dependence of the conductivity of a Fermi liquid $\ensuremath{\sigma}(\ensuremath{\Omega},T)$ on the frequency $\ensuremath{\Omega}$ and temperature TTT. Using the Kubo formalism and taking full account of vertex corrections, we show that the Fermi-liquid form $\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)\ensuremath{\propto}{\ensuremath{\Omega}}^{2}+4{\ensuremath{\pi}}^{2}{T}^{2}$ holds under very general conditions, namely, in any dimensionality above one, for a Fermi surface of an arbitrary shape (but away from nesting and van Hove singularities), and to any order in the electron-electron interaction. We also show that the scaling form of $\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)$ is determined by the analytic properties of the conductivity along the Matsubara axis. If a system contains not only itinerant electrons but also localized degrees of freedom which scatter electrons elastically, e.g., magnetic moments or resonant levels, the scaling form changes to $\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)\ensuremath{\propto}{\ensuremath{\Omega}}^{2}+b{\ensuremath{\pi}}^{2}{T}^{2}$, with $1\ensuremath{\le}b<\ensuremath{\infty}$. For purely elastic scattering, b=1b=1b=1. Our analysis implies that the value of $b\ensuremath{\approx}1$, reported for URu2Si2{\mathrm{URu}}_{2}{\mathrm{Si}}_{2}URu2​Si2​ and some rare-earth-based doped Mott insulators, indicates that the optical conductivity in these materials is controlled by an elastic scattering mechanism, whereas the values of $b\ensuremath{\approx}2.3$ and 5.65.65.6, reported for underdoped cuprates and organics, correspondingly, imply that both elastic and inelastic mechanisms contribute to the optical conductivity.
We calculate the low-frequency optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ of clean metals and superconductors at zero temperature neglecting the effects of impurities and phonons. In general, the frequency and temperature dependences of $\ensuremath{\sigma}$ have very little in common. For small Fermi surfaces in three dimensions (but not in two dimensions) we find, for example, that $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega}>0)\ensuremath{\approx}\mathrm{const}$ which corresponds to a scattering rate $\ensuremath{\Gamma}\ensuremath{\propto}{\ensuremath{\omega}}^{2}$ even in the absence of umklapp scattering when there is no T2{T}^{2}T2 contribution to $\ensuremath{\Gamma}$. In the main part of the paper we discuss in detail the optical conductivity of ddd-wave superconductors in two dimensions where $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega}>0)\ensuremath{\propto}{\ensuremath{\omega}}^{4}$ for the smallest frequencies and the umklapp processes typically set in smoothly above a finite threshold ${\ensuremath{\omega}}_{0}$ smaller than twice the maximal gap $\ensuremath{\Delta}$. In cases where the nodes are located at $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}∕2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}∕2)$, such that direct umklapp scattering among them is possible, one obtains $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}$.
We study the frequency dependence of the electron self-energy and the optical conductivity in a recently developed field theory of the spin-density-wave quantum phase transition in two-dimensional metals. We focus on the interplay between the Fermi surface ``hot spots'' and the remainder of the ``cold'' Fermi surface. Scattering of electrons off the fluctuations of the spin-density-wave order parameter, $\ensuremath{\varphi}$, is strongest at the hot spots; we compute the conductivity due to this scattering in a rainbow approximation. We point out the importance of composite operators, built of products of the primary electron or $\ensuremath{\varphi}$ fields: These have important effects also away from the hot spots. The simplest composite operator, ${\ensuremath{\varphi}}^{2}$, leads to deviations from Landau Fermi-liquid behavior on the entire Fermi surface. We also find an intermediate frequency window in which the cold electrons lose their quasiparticle form due to effectively one-dimensional scattering processes. The latter processes are part of umklapp scattering, which leads to singular contributions to the optical conductivity at the lowest frequencies at zero temperature.
Understanding the origin of electron incoherence is believed to be the first step toward the resolution of the mysteries of the high-Tc_{c}c​ cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude form of the optical absorption spectrum of the system. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system. However, a theoretical study of the optical conductivity of a strongly correlated model is a formidable task. Here we present an exact formula for the optical conductivity of the 2D Hubbard model from the low energy effective theory perspective. We show that the optical conductivity in Matsubara frequency of the 2D Hubbard model can be represented as the ensemble average of the optical conductivity of non-interacting systems in the background of fluctuating local moment. We find that such an ensemble average can be done exactly with a sign-problem-free Monte Carlo simulation if we assume the widely adopted Millis-Monien-Pines spin susceptibility for the fluctuating local moment. For thermal fluctuation of the local moment, our formula can be used to calculate directly the optical conductivity in real frequency which can be compared with the result of optical measurements in the cuprate superconductors.
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We investigate the frequency and temperature dependence of the low-energy electron dynamics in a Landau Fermi liquid with a local self-energy. We show that the frequency and temperature dependencies of the optical conductivity obey universal scaling forms, for which explicit analytical expressions are obtained. For the optical conductivity and the associated memory function, we obtain a number of surprising features that differ qualitatively from the Drude model and are universal characteristics of a Fermi liquid. Different physical regimes of scaling are identified, with marked non-Drude features in the regime where $\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}{k}_{\text{B}}T$. These analytical results for the optical conductivity are compared to numerical calculations for the doped Hubbard model within dynamical mean-field theory. For the ``universal'' low-energy electrodynamics, we obtain perfect agreement between numerical calculations and analytical scaling laws. Both results show that the optical conductivity displays a non-Drude ``foot,'' which could be easily mistaken as a signature of breakdown of the Fermi liquid, while it actually is a striking signature of its applicability. The aforementioned scaling laws provide a quantitative tool for the experimental identification and analysis of the Fermi-liquid state using optical spectroscopy, and a powerful method for the identification of alternative states of matter, when applicable.