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arXiv:1208.3485 · Physical Review B

First-Matsubara-frequency rule in a Fermi liquid. II. Optical conductivity and comparison to experiment

Dmitrii L. Maslov, Andrey V. Chubukov

Published Oct 19, 2012• 75 citations
arXivHTMLPDF

Abstract

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 $T$. 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, $b=1$. Our analysis implies that the value of $b\ensuremath{\approx}1$, reported for ${\mathrm{URu}}_{2}{\mathrm{Si}}_{2}$ 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 $5.6$, reported for underdoped cuprates and organics, correspondingly, imply that both elastic and inelastic mechanisms contribute to the optical conductivity.

DOI
10.1103/physrevb.86.155137

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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].

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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.

arXiv:0809.1104•Physical Review B•Jan 23, 2009•20 citations
Motivated by recent optical measurements on a number of strongly correlated electron systems, we revisit the dependence of the conductivity of a Fermi liquid \ensuremathσ(\ensuremathΩ,T)\ensuremath{\sigma}(\ensuremath{\Omega},T)\ensuremathσ(\ensuremathΩ,T) on the frequency \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ and temperature TTT. Using the Kubo formalism and taking full account of vertex corrections, we show that the Fermi-liquid form Re\ensuremathσ\ensuremath−1(\ensuremathΩ,T)\ensuremath∝\ensuremathΩ2+4\ensuremathπ2T2\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)\ensuremath{\propto}{\ensuremath{\Omega}}^{2}+4{\ensuremath{\pi}}^{2}{T}^{2}Re\ensuremathσ\ensuremath−1(\ensuremathΩ,T)\ensuremath∝\ensuremathΩ2+4\ensuremathπ2T2 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 Re\ensuremathσ\ensuremath−1(\ensuremathΩ,T)\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)Re\ensuremathσ\ensuremath−1(\ensuremathΩ,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 Re\ensuremathσ\ensuremath−1(\ensuremathΩ,T)\ensuremath∝\ensuremathΩ2+b\ensuremathπ2T2\mathrm{Re}{\ensuremath{\sigma}}^{\ensuremath{-}1}(\ensuremath{\Omega},T)\ensuremath{\propto}{\ensuremath{\Omega}}^{2}+b{\ensuremath{\pi}}^{2}{T}^{2}Re\ensuremathσ\ensuremath−1(\ensuremathΩ,T)\ensuremath∝\ensuremathΩ2+b\ensuremathπ2T2, with 1\ensuremath≤b<\ensuremath∞1\ensuremath{\le}b<\ensuremath{\infty}1\ensuremath≤b<\ensuremath∞. For purely elastic scattering, b=1b=1b=1. Our analysis implies that the value of b\ensuremath≈1b\ensuremath{\approx}1b\ensuremath≈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≈2.3b\ensuremath{\approx}2.3b\ensuremath≈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.
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].
We analyze in detail the fermionic self-energy $\ensuremath{\Sigma}(\ensuremath{\omega},T)$ in a Fermi liquid (FL) at finite temperature TTT and frequency $\ensuremath{\omega}$. We consider both canonical FLs, i.e., systems in spatial dimension D>2D>2D>2, where the leading term in the fermionic self-energy is analytic [the retarded $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)=C({\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2})]$, and noncanonical FLs in 1<D<21<D<21<D<2, where the leading term in $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ scales as TD{T}^{D}TD or ${\ensuremath{\omega}}^{D}$. We relate the ${\ensuremath{\omega}}^{2}+{\ensuremath{\pi}}^{2}{T}^{2}$ form to a special property of the self-energy, ``the first-Matsubara-frequency rule,'' which stipulates that ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ in a canonical FL contains an O(T)\mathcal{O}(T)O(T) but no T2{T}^{2}T2 term. We show that in any D>1D>1D>1, the next term after O(T)\mathcal{O}(T)O(T) in ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ is of order TD{T}^{D}TD (T3lnT{T}^{3}\mathrm{ln}TT3lnT in D=3D=3D=3). This TD{T}^{D}TD term comes from only forward and backward scattering, and is expressed in terms of fully renormalized amplitudes for these processes. The overall prefactor of the TD{T}^{D}TD term vanishes in the ``local approximation,'' when the interaction can be approximated by its value for the initial and final fermionic states right on the Fermi surface. The local approximation is justified near a Pomeranchuk instability, even if the vertex corrections are non-negligible. We show that the strength of the first-Matsubara-frequency rule is amplified in the local approximation, where it states that not only the TD{T}^{D}TD term vanishes, but also that ${\ensuremath{\Sigma}}^{R}(i\ensuremath{\pi}T,T)$ does not contain any terms beyond O(T)\mathcal{O}(T)O(T). This rule imposes two constraints on the scaling form of the self-energy: upon replacing $\ensuremath{\omega}$ by $i\ensuremath{\pi}T$, $\mathrm{Im}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ must vanish and $\mathrm{Re}{\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ must reduce to O(T)\mathcal{O}(T)O(T). These two constraints should be taken into consideration in extracting scaling forms of ${\ensuremath{\Sigma}}^{R}(\ensuremath{\omega},T)$ from experimental and numerical data.
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.
The Fermi liquid theory is a central concept in modern condensed matter physics used to describe conventional metals. A state of this universality class consists of well-defined quasiparticles, which occupy a finite number of states within a Fermi volume defined in momentum space. According to the Luttinger’s theorem, such a state encloses a Fermi volume proportional to the electron density modulo the filled bands if no symmetry is broken. In the past decades strong deviations from Fermi liquid theory have been observed in the scaling behavior of thermodynamic observables in different materials, among which are for instance cuprates and iron pnictides. This thesis is about two prototypical systems that can not be described by Fermi liquid theory. The first part of this work investigates two-dimensional effective models, which are in a Mott insulating state at half-filling and have a finite conductivity when doped with holes. In such a Mott insulating state, the charge carriers are strongly localized due to the Coulomb repulsion. Hence, strong correlations are assumed to have a strong impact on the formation of the ground state, also in the regime of small hole concentration. In a first project we have used a so-called spinon dopon mean-field theory to represent the two-dimensional Fermi-Hubbard model with strong on-site repulsive interaction in effective degrees of freedom, in which the holes can be embedded into a quantum spin liquid. The corresponding SU(2) invariant ground state belongs to the class of fractionalized Fermi liquids. In a second project we investigate a quantum dimer model, an effective model based on a Hilbert space spanned by short range singlets and bound states of holes and spins. The focus here is on the calculation of the hole-part of the electron spectral function by using exact diagonalization and its comparison with two analytic methods, a diagrammatic computation based on the Bethe-Salpeter equation and a so-called two-mode approximation. The electron spectral function shows a similar analytic form in momentum space between nodal and antinodal point when compared to results from photoemission spectroscopy experiments on cuprates. Furthermore, in a subsequent work we calculate the exact ground state of the quantum dimer model along a certain parameter line. In order to analyze the behavior of the elctron spectral function when increasing the density of holes, we investigate the Fermi-Hubbard model in a current project using a dynamical mean-field approach. To solve the 4-site cluster impurity problem, we use a numerical renormalization group approach. However, numerical limitations force us to restrict the analysis to spin-polarized baths. According to the spectral data, the system is similar to the SU(2) invariant case at half-filling in a Mott insulating state and posseses a momentum-selective energy gap at finite doping. The topology of the Fermi surface shows a Lifshitz transition when increasing the hole concentration. Here, the curvature of the Fermi surface changes from electron- to hole-like. For comparison we apply the dynamical mean-field theory also to the quantum dimer model and observe that the electron spectral functions at finite doping are qualitatively similar to that of the two-dimensional Fermi-Hubbard model. The second half of the work is about Tomonaga-Luttinger liquid theory, which is used to describe the low-energy effective degrees of freedom of one-dimensional systems. Here, we first discuss a conceptional extension of the operator-based bosonization theory for one-dimensional systems. This extension is especially suited for inhomogeneous one-dimensional systems. First, we investigate a one-dimensional system with a local interation potential and compute an exact solution of the single particle propagator at T = 0. The critical exponent of the single particle propagator has an unconventional form as a function of the microscopic Tomonaga-Luttinger parameters, which is not covered by the original Luttinger paradigm postulated by F. Duncan M. Haldane. In a second project on one-dimensional systems, we study the impact of scattering processes among bosonic low-energy excitations on the thermalization process. Such scattering processes are irrelevant on large length scales, however strongly affect the dynamics. In our analytic analysis we focus on a experimental setup, where a one-dimensional Bose gas is instantaneously splitted in two identical, however strongly correlated, halves of one-dimensional electronic systems. In the following, the corresponding non-equilibrium state runs through multiple regimes in time, such as a metastable prethermalization regime. However, above a certain threshold in time such scattering processes cause an effective thermalization of the system. In order to demonstrate this, we compute the kinetic equation in the Keldysh field integral formalism from a diagrammatic expansion based on a self-consistent Born approximation.
"This thesis consists of three Chapters. In the first Chapter, based on a planar unconventional Fermi liquid model, we present several results on the optimally doped and overdoped cuprate superconductors. For the normal state, we have analytically derived a linear in temperature and energy scattering rate for the carriers. This scattering rate yields necessarily a linear in temperature resistivity and a linear in 1/energy optical conductivity. The linearity of the scattering rate requires that the interacting Fermi liquid has strong peaks in its density of states (van-Hove singularities in 2 dimensions) near the chemical potential fl· Our results are backed by self-consistent Baym-Kadanoff (BK) numerical calculations, which also show that the interactions tend to pin the van- Hove singularities close to fl· We show that the low energy dependence of the Millis-Monien- Pines susceptibility XMMP can have a fermionic origin. We obtain particularly high transition temperatures Tc from our BK-Eliashberg scheme by introducing an ansatz for the susceptibility of the carriers. We postulate that the latter is enhanced in an additive manner due to the weak antiferromagnetic order of the Cu02 planes. Thus we have obtained a dx2_y2 gap with Tc > 120°K for nearest neighbor hopping t = 250me V. In the second Chapter we propose a novel mechanism for giant corrections to the transport quantities, including positive giant magnetoresistance, due to the presence of paramagnons in a weakly disordered metal. At low temperature and for finite impurity spin scattering, for a certain value, predicted from our theory, of the material-dependent paramagnon interaction, the total conductivity becomes highly sensitive to the orbital effects of a finite magnetic field. This is attributed to certain microscopic processes, otherwise negligibly small, which can be enhanced by a resonance factor, emanating from· the spin-density channel. We obtain very good agreement between this theory and four (4) different positive giant magnetoresistance experiments, including very recent ones, while making specific material-dependent predictions. We emphasize that our theory broadens the 'conventional wisdom' viewpoint that weak disorder would only generate small corrections to transport quantities. In the third Chapter we examine the effects of interface roughness and/or planar impurity doping in a superlattice, under the assumption that a weak disorder description is adequate. We find that these two types of disorder are equivalent, and that they can be viewed as effective ""bulk"" disorder, with anisotropic diffusion coefficients. This study can offer quantitative insight to transport properties of multilayers and devices, which contain inadvertently structural disorder at the interfaces."
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.