Analytical properties of the quadratic density response and quadratic dynamical structure functions: Conservation sum rules and frequency moments.

Abstract

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. @S1063-651X~96!09010-1# Sum rules for the linear response functions have played an important role throughout the development of electronliquid theory. The knowledge of the compressibility sum rule and the v 3 moment of the density-density response function x(q,v) has led to much improved local field corrections. Less explored are possible sum rules relating to quadratic response functions. With a little reflection it is easy to realize that the fundamental physical effects that operate in the generation of the linear sum rules—namely, symmetry and conservation laws—must also be responsible for creating sum rules for quadratic response functions. Some of these sum rules for the quadratic density-density response function x(q1 ,v 1 ;q2 ,v 2) have already been identified: Golden, Kalman, and Datta @1# have shown the existence of a static compressibility sum rule and established the high frequency behavior of x(q1 ,v 1 ;q2 ,v 2); more recently Tao and Kalman @2# have derived a frequency moment sum rule, analogous to the linear f -sum rule. In this paper we will systematically establish and analyze a number of sum rules for the quadratic density-density response and the quadratic dynamical structure function. The precise definition of the former is the relation