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Physical review. B, Condensed matter

Temperature-dependent low-frequency conductivity in marginal-Fermi-liquid theory

E. J. Nicol, J. P. Ćarbotte

Published Oct 1, 1991• 52 citations
Publisher

Abstract

We have calculated the temperature-dependent optical conductivity within phenomenological marginal-Fermi-liquid theory. A ``coherencelike'' peak occurs from a drop in the quasiparticle damping in the superconducting state. Results are given for varying impurity scattering rate.
DOI
10.1103/physrevb.44.7741

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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 $\mathrm{Re}\phantom{\rule{0.2em}{0ex}}\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}$.
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.
We calculate the optical conductivity $\ensuremath{\sigma}(T,\mathrm{\ensuremath{\Omega}})$ of a gas of Bogoliubov quasiparticles from their Green's function and the Kubo formula. We make comparisons with corresponding normal-state and superconducting-state results. The superconducting case includes the dynamic response of the condensate through additional contributions to the Kubo formula involving the Gor'kov anomalous Green's function. The differences in the optical scattering rate are largest just above the optical gap and become progressively smaller as the photon energy is increased or the temperature is raised. Our results are compared with those obtained using a recently advocated phenomenological procedure for eliminating the effect of the condensate. In this procedure the $\ensuremath{\delta}$-function contribution at zero photon energy, proportional to the superfluid density, is dropped in the real part of the conductivity $[{\ensuremath{\sigma}}_{1}(T,\mathrm{\ensuremath{\Omega}})]$ and its Kramers-Kronig transform is subtracted from the imaginary part ${\ensuremath{\sigma}}_{2}(T,\mathrm{\ensuremath{\Omega}})$. This results in deviations from our Bogoliubov quasiparticle and superconducting-state optical scattering rates even in the region where these have merged and are, in addition, not close to the normal-state result.
A Fermi surface coupled to a scalar field can be described in a 1/N1/N1/N expansion by choosing the fermion-scalar Yukawa coupling to be random in the NNN-dimensional flavor space, but invariant under translations. We compute the conductivity of such a theory in two spatial dimensions for a critical scalar. We find a Drude contribution, and verify that the proposed $1/{\ensuremath{\omega}}^{2/3}$ contribution to the optical conductivity at frequency $\ensuremath{\omega}$ has vanishing coefficient for a convex Fermi surface. We also describe the influence of impurity scattering of the fermions, and find that while the self-energy resembles a marginal Fermi liquid, the resistivity and optical conductivity behave like a Fermi liquid.
The two-dimensional Hubbard model exhibits superconductivity with d-wave symmetry even at half-filling in the presence of a next-nearest neighbor hopping. Using plaquette cluster dynamical mean-field theory with a continuous-time quantum Monte Carlo impurity solver, we reveal the non-Fermi liquid character of the metallic phase in proximity to the superconducting state. Specifically, the low-frequency scattering rate for momenta near (π, 0) varies nonmonotonically at low temperatures, and the dc conductivity is T linear at elevated temperatures with an upturn upon cooling. Evidence is provided that pairing fluctuations dominate the normal-conducting state even considerably above the superconducting transition temperature.
The Fermi-liquid transport theory of electrical conductivity formulated by Eliashberg [Zh. Eksp. Teor. Fiz. 41, 1241 (1961)] is further developed, using the analytic continuation of the finite-temperature Ward-Takahashi relations obtained by the present author [Ann. Phys. (N.Y.) 173, 226 (1987)] to maintain the theoretical consistency of approximation schemes for solving the Bethe-Salpeter equation. In particular, the finite-temperature Ward-Takahashi relations corresponding to the local conservation of the electron number density and to the conservation of the electron momentum are considered. A general expression for the electrical conductivity in terms of the proper vertex part is also obtained. In the formula, cancellation of the quasiparticle renormalization factors at the Fermi surface is explicitly shown in a general way without assuming a particular model. In the case that the electrons are scattered by phonons, the formula gives the Gr\"uneisen-Bloch formula for the on-shell phonons. For the impurity resistivity, the theory of Edwards [Philos. Mag. 3, 1020 (1958)] is developed to finite temperature by introducing a hypothetical boson field that replaces the impurity averaging.