8 - Functionals in many-particle physics

Summary

To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved.

A. Klein, Phys. Rev. 121, 950 (1961)

Summary

The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H.

Many physical quantities can be expressed as functionals, i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction (r₁, r₂, …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions.

In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E[Q] of this quantity. The variational character of E allows one to derive equations that determine Q₀, the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F_m[Q] that can then be evaluated at Q = Q₀. This strategy is outlined in Fig. 8.1.

To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n(r) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E[n] leads to equations to determine the actual ground-state density n₀(r) and total energy E₀.