Theoretical foundation of density functional theory in terms of field theoretical Legendre transformation is presented. The ground state energy is first written as a functional of local probe coupled to the density operator. The density functional is then defined by the functional Legendre transformation, which leads to a systematic formulation of density functional theory. Excitation spectrum is also determined within the same formalism in a unified way. The diagrammatic evaluation is most conveniently done by using auxiliary field method. Several generalizations of the formalism and extension to the case other than the density operator are also discussed.
Theoretical foundation of density functional theory in terms of field theoretical Legendre transformation is presented. The ground state energy is first written as a functional of local probe coupled to the density operator. The density functional is then defined by the functional Legendre transformation, which leads to a systematic formulation of density functional theory. Excitation spectrum is also determined within the same formalism in a unified way. The diagrammatic evaluation is most conveniently done by using auxiliary field method. Several generalizations of the formalism and extension to the case other than the density operator are also discussed.
The basic concern of density functional theory is the groundstate energy of an inhomogeneous interacting many-particle system as a function of the external potential acting on its density (and possibly as a function of the particle number, allowing for the consideration of ionization energies, affinities, and excitation gaps). This groundstate energy function is replaced by its Legendre transform as a function of the particle density, and the Legendre back transformation leads via insertion of an interaction-free reference system to an effective one-particle equation whose solution finds the ground-state energy and density to any given external potential. As was seen in Part I, this theory provides rather a reformulation than a solution of the many-body problem for the groundstate, because it does not explicitly provide the Legendre transform H [ n ] to E [ v, N ] (cf. (6.35, 6.31, and 6.36). The original many-body problem is now contained in H [ n ] which only can be guessed, although not without success. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Dynamical mean-field theory has become a method of choice for calculating spectral properties of strongly correlated systems. In this chapter the concept and the formalism of dynamical mean-field theory are described using a one-band Hubbard model as an illustration. The Anderson impurity model, which forms the auxiliary system, is used as a starting point to develop the formal dynamical mean-field equations. Functional integral formulation, well-suited for quantum Monte Carlo simulations, is also presented and analogy with density functional theory is made via the Legendre transform formulation. The success of the theory is illustrated by some generic examples.
The Legendre transformation is one of the ways to rewrite the theory by small number of selected variables. It leads naturally to the generalized action functional called the effective action and everything starts from this generating functional. It is applied to various problems of physics ranging from the relativistic field theory to the condensed matter physics. The main techniques used here are the on-shell expansion and the inversion method. The scheme presented in this article is thus a generating functional formalism of the many particle systems.
We present a rigorous formulation of generalized Kohn-Sham density-functional theory. This provides a straightforward Kohn-Sham description of many-body systems based not only on particle-density but also on any other observable. We illustrate the formalism for the case of a particle-density based description of a nonrelativistic many-electron system. We obtain a simple diagrammatic expansion of the exchange-correlation functional in terms of Kohn-Sham single-particle orbitals and energies; develop systematic Kohn-Sham formulation for one-electron propagators and many-body excitation energies. This work is ideally suited for practical applications and provides a rigorous basis for a systematic development of the existing body of first-principles calculations in a controllable fashion.