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arXiv:cond-mat/0104229 · Physical review. B, Condensed matter

Dynamical local field, compressibility, and frequency sum rules for quasiparticles

Klaus Morawetz

Published Aug 30, 2002• 10 citations
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Abstract

The finite temperature dynamical response function including the dynamical local field is derived within a quasiparticle picture for interacting one-, two-, and three-dimensional Fermi systems. The correlations are assumed to be given by a density-dependent effective mass, quasiparticle energy shift, and relaxation time. The latter one describes disorder or collisional effects. This parametrization of correlations includes local-density functionals as a special case and is therefore applicable for density-functional theories. With a single static local field, the third-order frequency sum rule can be fulfilled simultaneously with the compressibility sum rule by relating the effective mass and quasiparticle energy shift to the structure function or pair-correlation function. Consequently, solely local-density functionals without taking into account effective masses cannot fulfill both sum rules simultaneously with a static local field. The comparison to the Monte Carlo data seems to support such a quasiparticle picture.

DOI
10.1103/physrevb.66.075125

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The finite temperature dynamical response function including the dynamical local field is derived within a quasiparticle picture for interacting one-, two-, and three-dimensional Fermi systems. The correlations are assumed to be given by a density-dependent effective mass, quasiparticle energy shift, and relaxation time. The latter one describes disorder or collisional effects. This parametrization of correlations includes local-density functionals as a special case and is therefore applicable for density-functional theories. With a single static local field, the third-order frequency sum rule can be fulfilled simultaneously with the compressibility sum rule by relating the effective mass and quasiparticle energy shift to the structure function or pair-correlation function. Consequently, solely local-density functionals without taking into account effective masses cannot fulfill both sum rules simultaneously with a static local field. The comparison to the Monte Carlo data seems to support such a quasiparticle picture.
The general possible form of mean-field parametrization in a running frame in terms of current, energy, and density functionals is examined under the restrictions of Galilean invariance. It is found that only two density-dependent parameters remain which are usually condensed in a position-dependent effective mass and the self-energy formed by current and mass. The position-dependent mass induces a position-dependent local current, which is identified for different nonlinear frames. In a second step the response to an external perturbation and relaxation towards a local equilibrium is investigated. The response function is found to be universal in the sense that the actual parametrization of the local equilibrium does not matter and is eliminated from the theory due to the conservation laws. The explicit form of the response with respect to density, momentum, and energy is derived. The compressibility sum rule as well as the sum rule by first- and third-order frequency moments are proved analytically to be fulfilled simultaneously. The results are presented for Bose or Fermi systems in one, two, and three dimensions.
Linearizing the appropriate kinetic equation we derive general response functions including self-consistent mean fields or density functionals and collisional dissipative contributions. The latter ones are considered in relaxation time approximation conserving successively different balance equations. The effect of collisions is represented by correlation functions which are possible to calculate with the help of the finite temperature Lindhard random-phase approximation expression. The presented results are applicable to the finite temperature response of interacting quantum systems if the quasiparticle or mean-field energy is parametrized within Skyrme-type functionals including density, current, and energy dependencies which can be considered alternatively as density functionals. In this way we allow to share correlations between density functional and collisional dissipative contributions appropriate for special treatment. We present results for collective modes such as the plasmon in plasma systems and the giant resonance in nuclei. The collisions lead in general to an enhanced damping of collective modes. If the collision frequency is close to the frequency of the collective mode, resonance occurs and the collective mode is enhanced showing a collisional narrowing.
We reanalyzed the recently derived response function for interacting systems in relaxation-time approximation respecting density, momentum, and energy conservation. We find that momentum conservation leads exactly to the local-field corrections for both cases respecting only density conservation and respecting density and energy conservation. This rewriting simplifies the former formulas dramatically. We discuss the small wave vector expansion and find that the response function shows a high-frequency dependence of omega(-5), which allows to fulfill higher-order sum rules. The momentum conservation also resolves a puzzle about the conductivity, which should only be finite in multicomponent systems.
The quadratic density response function and the quadratic dynamical structure function (the Fourier transform of the equilibrium three-point density correlations) contain important information about a many body system; they are also ingredients for an improved dynamical mean field theory for strongly coupled Fermi systems. We examine the analytic properties of the density response function and establish new single frequency and double frequency moment sum rules. We relate the sum rule coefficients to the high frequency expansion of the response function. Next we invoke the quadratic fluctuation-dissipation theorem to relate these frequency moments to weighted frequency moments of the dynamical structure function. These latter reduce to straight frequency moments in the high temperature classical and zero temperature degenerate limits. \textcopyright{} 1996 The American Physical Society.
Time-dependent quantal density-functional theory (Q-DFT) is a description of the s-system of noninteracting fermions with electronic density equivalent to that of Schr\"odinger theory, in terms of fields whose sources are quantum-mechanical expectations of Hermitian operators. The theory delineates and defines the contribution of each type of electron correlation to the local electron-interaction potential ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ of the s system. These correlations are due to the Pauli exclusion principle, Coulomb repulsion, correlation-kinetic, and correlation-current-density effects, the latter two resulting, respectively, from the difference in kinetic energy and current density between the interacting Schr\"odinger and noninteracting systems. We employ Q-DFT to prove the following sum rules and properties of the s system: (i) the components of the potential due to these correlations separately exert no net force on the system; (ii) the torque of the potential is finite and due solely to correlation-current-density effects; (iii) two sum rules involving the curl of the dynamic electron-interaction kernel defined as the functional derivative of ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ are derived and shown to depend on the frequency dependent correlation-current-density effect. Furthermore, via adiabatic coupling constant (\ensuremath{\lambda}) perturbation theory, we prove: (iv) the exchange potential ${\ensuremath{\nu}}_{x}(\mathbf{r},t)$ is the work done in a conservative field representative of Pauli correlations and lowest-order $O(\ensuremath{\lambda})$ correlation-kinetic and correlation-current-density effects; (v) the correlation potential ${\ensuremath{\nu}}_{c}(\mathbf{r},t)$ commences in $O({\ensuremath{\lambda}}^{2}),$ and, at each order, it is the work done in a conservative field representative of Coulomb correlations and correlation-kinetic and correlation-current-density effects; (vi) we derive the integral virial theorem relating ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ to the electron-interaction and correlation-kinetic energy for arbitrary coupling constant strength \ensuremath{\lambda}, and show there are no explicit correlation-current-density contributions to the energy. From this integral virial theorem we (vii) obtain the fully interacting $(\ensuremath{\lambda}=1)$ and exchange-only $(\ensuremath{\lambda}=0)$ integral virial theorems as special cases, the latter showing there is no explicit correlation-kinetic contribution to the exchange energy; and (viii) write expressions for the electron-interaction and correlation-kinetic actions for arbitrary coupling constant \ensuremath{\lambda} in terms of the corresponding fields.
We present an analytical study of the dynamical local-field factors associated with the response of a homogeneous two-dimensional interacting electron liquid as functions of momentum, frequency, and density. We derive sum rules that constrain their asymptotic forms (in momentum and frequency) for both the spin-symmetric and spin-antisymmetric cases. Parametrized expressions for the local-field factors are proposed, based on all available sum rules and on many-body perturbation theory, and these are found to be in good agreement with quantum Monte Carlo calculations. Finally, these expressions are used to evaluate the effective electron-electron interaction in a local approximation for two-dimensional systems. It is shown that both the quantitative and qualitative behaviors of the interaction are sensitive to the inclusion of dynamical correlations.