The Kubo formula for the electrical conductivity is rewritten in terms of a sum of Drude-like contributions associated to the exact eigenstates of the interacting system, each characterized by its own frequency-dependent relaxation time. The structure of the novel and equivalent formulation, weighting the contribution from each eigenstate by its Boltzmann occupation factor, simplifies considerably the access to the static properties (dc conductivity) and resolves the long standing difficulties to recover the Boltzmann result for dc conductivity from the Kubo formula. It is shown that the Boltzmann result, containing the correct transport scattering time instead of the electron lifetime determined by the Green function, can be recovered in problems with elastic and inelastic scattering at the lowest order of interaction.
We derive a Kubo-type formula that describes proper non-equilibrium stationary states for finite-size systems. We first argue that the usual Kubo formula considers only equilibrium states of the perturbed system, which are inappropriate to describe transport properties. Moreover, we show that the standard Kubo formula for the dc conductance is only appropriate in the thermodynamic limit. We demonstrate that taking into consideration explicitly the coupling to the leads/baths solves both problems. This approach results in well-behaved response functions, without the singular contributions from degenerate states encountered when Kubo formulae for infinite-size systems are inappropriately used for finite-size systems. We also derive a second, more efficient formulation which is valid only for a set of special physical quantities, which includes the charge current density operator.
\nthe quantum calculation of the d.c. electrical conductivity for an electron system is reconsidered by using an improved form of the long-time solution of the density matrix. The starting time-evolution equation which is written as a linear system in the Liouville space of discrete quantum states, is first solved in terms of Cramer's determinants and next suitably adapted to tranport problems. The calculation of the statistical average of the many-body current density, in the presence of scattering potential, is carried out in the thermodynamic limit, using a characteristic property of collision matrices introduced earlier by van Hove as the so-called
delta-singularity. In contrast to previous methods based on kinetic equations solved by perturbations, Kubo's theory, or Mori type memory functions, the resolution of an integral equation, initially included in Cramer's formulae, is no longer required. Among its major advantages, the method succeeds in overcoming the well known zero frequency divergent behaviour of the conductivity. It also yields an explicit expression of an effective relaxation time which is just constructed out of those diagonal transition sequences responsible for self-energy effects in van Hove's theory. This expression which only assumes the randomization of phases of initial states can be used to get tractable expansions in powers of the collision potential. Its range of validity is limited by the effect of collisions on equilibrium statistics, which is shown to be negligible within the Peierls criterion,
hbar/taullmu,
tau being the relaxation time and
mu the Fermi energy. Finally, the comparison of the present approach with the elementary transport theory reveals that the relaxation time is close to the standard expression, even though the latter would be exactly recovered, in lowest order of collisions, only on disregarding divergent terms arising beyond second order.\n
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
URu2Si2 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.
Abstract The optical conductivity is the basic defining property of materials characterizing the current response toward time-dependent electric fields. In this work, following the approach of Kubo’s response theory, we study the general properties of the nonlinear optical conductivities of quantum many-body systems both in equilibrium and non-equilibrium. We obtain an expression of the second- and the third-order optical conductivity in terms of correlation functions and present a perturbative proof of the generalized Kohn formula proposed recently. We also discuss a generalization of the f-sum rule to a non-equilibrium setting by focusing on the instantaneous response.
We consider a system of weakly interacting Fermions which satisfies all those requirements necessary for the derivation of Van Hove’s weak coupling master equation. We show how the Boltzmann equation can be derived with the help of a Factorisation Theorem which we then prove, in the limit of an infinite system. We then use this Factorisation Theorem to show that the electrical conductivity derived from the Kubo formula is equivalent to the conductivity derived from the Boltzmann equation, for the same system of Fermions. Finally, we illustrate how the theory must be modified for a weakly interacting system of electrons and phonons.