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arXiv:cond-mat/9807339 · Journal of the Physical Society of Japan

Electrical Conductivity of Fermi Liquids. I. Many-Body Effect on the Drude Weight

Takuya Okabe

Published Aug 15, 1998• 21 citations
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Abstract

On the basis of the Fermi liquid theory, we investigate the many-body effect on the Drude weight. In a lattice system, the Drude weight $D$ is modified by electron-electron interaction due to Umklapp processes, while it is not renormalized in a Galilean invariant system. This is explained by showing that the effective mass $m'$ for $D\propto n/m'$ is defined through the current, not velocity, of quasiparticle. It is shown that the inequality $D>0$ is required for the stability against the uniform shift of the Fermi surface. The result of perturbation theory applied for the Hubbard model indicates that $D$ as a function of the density $n$ is qualitatively modified around half filling $n\sim 1$ by Umklapp processes.

DOI
10.1143/jpsj.67.2792

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On the basis of the Fermi liquid theory, we investigate the many-body effect on the Drude weight. In a lattice system, the Drude weight DDD is modified by electron-electron interaction due to Umklapp processes, while it is not renormalized in a Galilean invariant system. This is explained by showing that the effective mass m′m'm′ for D∝n/m′D\propto n/m'D∝n/m′ is defined through the current, not velocity, of quasiparticle. It is shown that the inequality D>0D>0D>0 is required for the stability against the uniform shift of the Fermi surface. The result of perturbation theory applied for the Hubbard model indicates that DDD as a function of the density nnn is qualitatively modified around half filling n∼1n\sim 1n∼1 by Umklapp processes.
The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}$, where \ensuremath{\Delta} is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.
The electrical resistivity due only to mutual Coulomb interaction of the d -dimensional lattice electron systems ( d ≥2) in the presence of Umklapp scattering processes is studied on the basis of the Fermi liquid theory. It is shown that normal scattering processes generally contribute to the resistivity if Umklapp scattering processes are present. This fact gives a natural explanation of the resistivity of the d =∞ Hubbard model which is determined only by the damping rate of quasi-particles, γ. In the case of d =2, however, it is shown that the resistivity is proportional to T 2 , T being the temperature, even when normal scattering processes, which give T 2 log T -contribution to γ, contribute to the resistivity. In the special case of the square lattice, where the transfer integrals are only between the nearest neighbors, the coefficient of the T 2 -term in the second order of the interaction is shown to diverge as (1/δ) log (1 / δ) when δ→0, δ being the doping rate away from half-filling.
The Drude weight D and the dc conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) of strongly correlated electrons are investigated theoretically. Analytic results are derived for the homogeneous phase of the Hubbard model at d=\ensuremath{\infty}, and for spinless fermions in this limit with 1/d corrections systematically included to lowest order. It is found that ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$(T) is finite for all T>0, displaying Fermi-liquid behavior, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{dc}}$\ensuremath{\propto}1/T2{\mathit{T}}^{2}T2, at low temperatures. The validity of this result for finite dimensions is examined by investigating the importance of umklapp scattering processes and vertex corrections. A finite dc conductivity for T>0 is argued to be a generic feature of correlated lattice electrons in not too low dimensions.
We study the temperature dependence of the electrical resistivity of interacting two-dimensional Fermi liquids. We perform a numerical simulation of the nonequilibrium state based on semiclassical Boltzmann transport theory. A two-dimensional system of quasiparticles on a square lattice serves as a model system. We use a single-orbital tight-binding model of the dispersion. For conceptual purposes, we choose a simple repulsive onsite interaction that leads to quasiparticle scattering and delta-potential scatterers as a model of nonmagnetic impurity scattering. Through our simulation, we demonstrate that deviations from the predictions of standard Fermi-liquid theory can arise due to the nontrivial scattering geometry of umklapp processes, in special cases even in the ultralow-temperature limit. We show through qualitative arguments how these unconventional scaling properties of the electrical resistivity, which are often interpreted as indication of a non-Fermi-liquid state, can arise due to special geometric conditions of the Fermi surface. The appearance of robust deviations from the predictions of Fermi-liquid theory within our simple model presents a viewpoint in order to interpret unconventional transport properties in electron-electron scattering dominated metallic systems.
The Landau parameters F 0 s , F 0 a , F 1 s and F 1 a in the two-dimensional Hubbard model are calculated by the second order perturbation method of Abrikosov-Khalatnikov. The following points are revealed as the half-filling is approached: (1) F 0 a changes its sign from negative to positive, if the Coulomb interaction U is strong enough, i.e., U/t≳2.0U/t\gtrsim 2.0 U/t≳2.0. This means that the spin density oscillations, zero spin-sound, can propagate there; (2) F 1 s changes its sign from positive to negative, so that we can understand, on the basis of the anisotropic Fermi liquid theory, the behavior of the Drude weight whose magnitude decreases as the half-filling is approached even if there were no mass enhancement; (3) From the results of F 0 s and F 0 a , it is shown that both the charge and spin susceptibility decrease rapidly close to the half-filling. The latter is regarded as the so-called spin-gap behavior.
We obtain an exact expression for the conductivity of a disordered, non-Galilean-invariant Fermi liquid by solving the kinetic equation with both screened Coulomb and z=3z=3z=3 Pomeranchuk critical interactions. While consistent with previous asymptotic results, our solution shows that electron-electron interactions enter the conductivity solely via the quasiparticle scattering time, τeeτ_\mathrm{ee}τee​. Accordingly, the crossovers between the collisionless and hydrodynamic regimes occur when 1/τee1/τ_\mathrm{ee}1/τee​ becomes comparable to the larger of the impurity scattering rate and the probe frequency, ΩΩΩ. In addition, the exact solution yields the optical response in the hydrodynamic regime, Ω≪1/τeeΩ\ll 1/τ_\mathrm{ee}Ω≪1/τee​, which is inaccessible within perturbation theory. Near a z=3z=3z=3 Pomeranchuk quantum critical point, consistency between the kinetic-equation and Kubo approaches requires proper inclusion of mass renormalization within the Eliashberg approximation, which also ensures that the crossover between the collisionless and hydrodynamic regimes in the optical conductivity occurs at the Planckian scale Ω∼TΩ\sim TΩ∼T.