Single particle energies in the many-particle green's function formalism

Viktor Borisovich Bobrov

Shohei Watabe

Abstract Strong connection between the single-particle excitation and the collective excitation stands out as one of the features of Bose–Einstein condensates (BECs). We discuss theoretically these single-particle and density excitations of BECs focusing on the exact properties of the one-body and two-body Green’s functions developed by Gavoret and Nozières. We also investigate these excitations by using the many-body approximation theory at nonzero temperatures. First, we revisited the earlier study presented by Gavoret and Nozières, involving the subsequent results given by Nepomnyashchii and Nepomnyashchii, in terms of the matrix formalism representation. This matrix formalism is an extension of the Nambu representation for the single-particle Green’s function of BECs to discuss the density and current response functions efficiently. We describe the exact low-energy properties of the correlation functions and the vertex functions, and discuss the correspondence of the spectra between the single-particle excitation and the density excitation in the low-energy and low-momentum limits at T = 0. After deriving the exact low-energy structures of the one-body and two-body Green’s functions, we develop a many-body approximation theory of BECs with making the use of the matrix formalism for describing the single-particle Green’s function and the density response function at nonzero temperatures. We show how the peaks of the single-particle spectral function and the density response function behave with an increasing temperature. Many-body effect on the single-particle spectral function and the density response function is included within a random phase approximation, where satellite structures emerge because of beyond-mean-field effects. Criticisms are also made on recent theories casting doubt upon the conventional wisdom of the BEC: the equivalence of the dispersion relations between the single-particle excitation and the collective excitation in the low-energy and low-momentum regime.

John C. Inkson

А. К. Власников, Artem Lunev, V. M. Mikhailov

The single-particle Green function formalism (Gor’kov’s method) is applied to calculate the energies of finite superconducting Fermi systems with a finite number of particles. It is shown that strong pairing leads to equations similar to those from the Bardeen-Cooper-Schrieffer theory.

Roman Smit

We show that the single-particle Green's functions used in many body theory have an elegant description in the form of hyperfunctions. We summarize the necessary hyperfunction concepts. We show that the analytical properties and the relations between different Green's functions are natural within this formulation. Important results from the standard formalism are recovered straightforwardly. We argue that hyperfunctions could possibly provide a powerful new tool for many body theory, if the formalism could be developed beyond single-particle Green's functions.