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.
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.
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.
There exists a regime of overlap between the equilibrium and non-equilibrium behavior of a system, the non-equilibrium behavior of weakly perturbed states. When a system is perturbed ever so slightly, its response will be linear in the perturbation, say the current of the conduction electrons in a metal will be proportional to the strength of the applied electric field. This regime is called the linear response regime, and though the system is in a non-equilibrium state all its characteristics can be inferred from the properties of its equilibrium state. In the next chapter we shall go beyond the linear regime by showing how to obtain quantum kinetic equations. The kinetic-equation approach to transport is a general method, and allows in principle nonlinear effects to be considered. However, in many practical situations one is interested only in the linear response of a system to an external force. The linear response limit is a tremendous simplification in comparison with general non-equilibrium conditions, and is the subject matter of this chapter. In particular the linear response of the density and current of an electron gas are discussed. The symmetry properties of response functions, and the fluctuation–dissipation theorem are established. Lastly we demonstrate how correlation functions can be measured in scattering experiments, as illustrated by considering neutron scattering from matter. Needless to say, in measurements of (say) the current in a macroscopic body, far less information in the current correlation function is probed. Linear response In this section we consider the response of an arbitrary property of a system to a general perturbation.
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.
Introduction The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory , which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7. In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.
We review the linear response theory in the horizontal quantum liquids framework spanning from Coulomb liquids to Atomic gases. There are several well known references about this subject, like the onset of the Kramers-Kronig relations and the fluctuation-dissipation theorem. For the Coulomb systems we show the connection between the linear response function and the dielectric function which settles a parallelism between statistical mechanics and electrostatic properties. For (very degenerate, dilute, trapped, two dimensional) Atomic (Bose) gases we will prove, in full generality, how the response properties of the gas depend from the frequency of the harmonic trap.
Abstract This chapter presents a universal definition of linear response, which at first is studied for the example of the damped oscillator. From the quantal time evolution operator lowest order transition rates are deduced, together with microscopic response functions. Their fundamental properties are studied, in particular, the meaning of their reactive and dissipative parts for energy transfer. Correlation functions are introduced and related to response functions by way of the fluctuation dissipation theorem. For a random matrix model, the latter is shown to also hold true for microcanonical ensembles. Connections to thermal Green functions are established and an extension of response theory to unstable modes is discussed. With the help of the Mori product, relations among the static response, the adiabatic and the isothermal susceptibilities are given. Time dependent irreversible processes are studied by introducing relaxation functions and by following the variation of the density operator and of entropy. Onsager relations are presented and Kubo's expressions for transport coefficients are given.