Perturbation Theory of Response Functions

Gabriele F. Giuliani, Giovanni Vignale

Irina Bostan

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.

Paolo Amore, Francisco Marcelo Fernández

Lin Lin, Jianfeng Lu

Many properties of chemical or materials systems are characterized by how they respond to external perturbations. When the external perturbation is sufficiently small that only the leading-order perturbation is important, this is known as the linear response regime. It is in some sense analogous to the linear stability analysis of a dynamical system or that in fluid mechanics. This chapter will give an in-depth discussion of the linear response theory for both static and dynamic perturbations and their applications. The analysis will be from the perspective of the perturbation theory of Green's functions. Towards the end of the chapter, the perturbation theory of a many-body quantum system and its connection to DFT will also be briefly addressed.

Michele Cini

Ulf von Barth, Nils Erik Dahlen, Robert van Leeuwen +1 more

In the present work, we propose a theory for obtaining successively better approximations to the linear response functions of time-dependent density or current-density functional theory. The new technique is based on the variational approach to many-body perturbation theory (MBPT) as developed during the sixties and later expanded by us in the mid-nineties. Due to this feature, the resulting response functions obey a large number of conservation laws such as particle and momentum conservation and sum rules. The quality of the obtained results is governed by the physical processes built in through MBPT but also by the choice of variational expressions. We here present several conserving response functions of different sophistication to be used in the calculation of the optical response of solids and nanoscale systems.