Irreducible Green functions method and many-particle interacting systems on a lattice

Abstract

The Green-function technique, termed the irreducible Green functions (IGF) method, that is a certain reformulation of the equation-of motion method for double-time temperature dependent Green functions (GFs) is presented. This method was developed to overcome some ambiguities in terminating the hierarchy of the equations of motion of double-time Green functions and to give a workable technique to systematic way of decoupling. The approach provides a practical method for description of the many-body quasi-particle dynamics of correlated systems on a lattice with complex spectra. Moreover, it provides a very compact and self-consistent way of taking into account the damping effects and finite lifetimes of quasi-particles due to inelastic collisions. In addition, it correctly defines the Generalized Mean Fields (GMF), that determine elastic scattering renormalizations and , in general, are not functionals of the mean particle densities only. The purpose of this article is to present the foundations of the IGF method. The technical details and examples are given as well. Although some space is devoted to the formal structure of the method, the emphasis is on its utility. Applications to the lattice fermion models such as Hubbard/Anderson models and to the Heisenberg ferro-and antiferromagnet, which manifest the operational ability of the method are given. It is shown that the IGF method provides a powerful tool for the construction of essentially new dynamical solutions for strongly interacting many-particle systems with complex spectra.

Equation of motion solutions to Hubbard model retaining Kondo effect

“…Instead of eq. ( 6) we will use the method with differentiating over the second time ( ' t ), which gives the EOM in the following form [34]:…”

Low-temperature asymptotic of the transverse dynamical structure factor for a magnetically polarized XX chain

“…At the same time for the real two-time Green function (19) the representation (26) may be proved analytically within the approach suggested and developed by N. M. Plakida and Yu. A. Tserkovnikov [19,20,21,22]. Moreover as it is shown in the paper an evaluation of Σ 1 (ω, q, T ) in this framework is rather simple and does not need a preliminary knowledge of χ 1 (ω, q, T ) (so that the resummation does not occur).…”

Variational principle of Bogoliubov and generalized mean fields in many-particle interacting systems

“…Moreover, the mean-field sign there alternates in the "chessboard" (staggered) order. The question of the true antiferromagnetic ground state is not completely clarified up to the present time [17,5,71,72,35,36,37]. This is related to the fact that, in contrast to ferromagnets, which have a unique ground state, antiferromagnets can have several different optimal states with the lowest energy.…”

“…We see that the mean-field approximation provides only a rough description of the real situation and overestimates the interaction between particles. Attempts to improve the homogeneous mean-field approximation were undertaken along different directions [17,5,71,72,35,36,37]. An extremely successful and quite nontrivial approach was developed by L. Neel [17,5,71,72], who essentially formulated the concept of local mean fields (1932).…”

“…The considerable progress in studying the spectra of elementary excitations and thermodynamic properties of many-body systems has been for the most part due to the development of the temperature-dependent Green functions methods [15,5,17,18]. The very important concept of the whole method is the concept of the generalized mean field [17,35,36,37]. These generalized mean fields have a complicated structure for the strongly correlated case and are not reduced to the functional of the mean densities of the electrons.…”