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Physical Review A

Conservation sum rules for the quadratic density response function

Z. C. Tao, Gabor J. Kalman

Published Nov 1, 1990• 2 citations
Publisher

Abstract

The quadratic density response function has been shown to be determined by the three-point density correlations at equilibrium. Consequently, the conservation laws enjoyed by the density fluctuation are expected to have their manifestations in the quadratic density response function. New frequency-moment sum rules are established along these lines. We show that the lowest-order quadratic response function exhausts the quadratic equivalent of the f-sum rule.

DOI
10.1103/physreva.42.6201

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The quadratic density response function has been shown to be determined by the three-point density correlations at equilibrium. Consequently, the conservation laws enjoyed by the density fluctuation are expected to have their manifestations in the quadratic density response function. New frequency-moment sum rules are established along these lines. We show that the lowest-order quadratic response function exhausts the quadratic equivalent of the f-sum rule.
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.
A derivation of a general relationship between the frequency- and wave-number-dependent longitudinal quadratic response functions and the three-point dynamical structure function (the Fourier transform of the equilibrium three-point density-density correlations) is presented. This relationship is the full quadratic equivalent, valid for quantum systems of arbitrary degeneracy, of the customary (linear) fluctuation-dissipation theorem. The high-temperature limit reproduces earlier classical results, while its zero-temperature limit is characterized by a piecewise analytic behavior in the (${\ensuremath{\omega}}_{1}$${\ensuremath{\omega}}_{2}$) plane, generated by the respective dominance of the differently ordered triplets of density operators. There are two main versions of the theorem. In the first the completely symmetrized combination of the three-point functions is expressed in terms on a ``triangle-symmetrized'' [in the (k\ensuremath{\omega}) space] combination of the response functions. In the second ``inverted'' form, each of the six differently ordered three-point functions is expressed in terms of combinations of response functions, where the triangle symmetry is broken.
While the theory of linear response functions in many body theory has developed into a substantial lore over the past decades, rather little is known about the properties and behavior of higher order (quadratic, etc.) response functions of many body systems. In this poster we first review essential elements of quadratic response theory such as the quadratic fluctuation dissipation theorems and sum rules for the quadratic density response function. Next we show how the correlational local field G(k,ω) in the linear response is fundamentally linked to the quadratic response function which opens a way to a non-perturbational analysis of the former. We proceed to present the results of the explicit calculation of the quadratic density response of 2D and 3D electron gas. Both, real and imaginary part of the latter reveal an extremely rich behavior. Of particular interest is the comparison between the two- and the three-dimensional case.
Sum rules for response functions represent some of the few exact results in many-body physics, relating the dynamical response functions to equilibrium ground state properties. To date, however, sum rules have been examined primarily for linear response functions. In this work, the authors introduce a spectral density representation for second-order response functions, using it to systematically derive families of sum rules, including a nonlinear extension of the well-known fff-sum rule. This work places important constraints on future nonlinear spectroscopic experiments.
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.
In this work the basic formalism of non-equilibrium Green’s functions is presented and then applied to study a Ward identity in linear response theory, namely the frequency sum-rule. It can be proven that the frequency sum-rule is satisfied when the quantities involved are calculated using perturbation theory within a conserving approximation for the self-energy. To illustrate this equality along with other properties of the response function, a numerical application that solves the Kadanoff-Baym equations for systems of Hubbard chains was used. The results showed that the frequency sum-rule was satisfied to the same extent by all the conserving approximations used as by the exact diagonalization numerical results. The density response function was analyzed diagrammatically for a series of conserving approximations for the self-energy and this demonstrated that even for a first order in perturbation theory approximation for the self-energy, the response function has a corresponding complex, third order in the perturbation diagrammatic structure.