Approximation of Green's functions

Силантьев Анатолий Владимирович

Background. The Hubbard model is widely used for theoretical description of strongly correlated electronic systems. Various approximate methods are used for investigation of these systems. At the present time there are four approaches to calculation of the Green’s function in the approximation of static fluctuations. The purpose of this paper is to develop two more approaches to calculation of the Green’s function in the approximation of static fluctuations. Materials and methods. The basis for the approximation of static fluctuations is the method of the motion equations for creation operators. The approximation of static fluctuations allows to obtain a closed system of differential equations for finding creation operators. The Green’s functions, correlation functions and energy spectrum can be calculated using creation operators. Results. The developed approaches for finding the Green’s functions in the Hubbard model by the approximation of static fluctuations were used to obtain the Green’s functions of dimer, hexagon, pentagon and fullerene С 20. Conclusions. This work demonstrates that the Green’s functions of dimer, hexagon, pentagon and fullerene С 20 obtained by the new approaches coincide with the Green’s functions of these systems obtained by the earlier approaches.

Benno Fuchssteiner

Adrian Stan

In this chapter we introduce the fundamental concepts behind the Green Function Theory. We start from a description of the second quantization method and then, after introducing the ensemble average of an operator, we define the main object of the theory, the Green function. Further on, after obtaining the equation of motion for the Green function, we introduce the self-energy operator and we discuss the Hartree-Fock approximation, the second-Born approximation and the GW approximation to the self-energy.

Eleftherios N. Economou

Eleftherios N. Economou

Christian Brouder, Gianluca Panati, Gabriel Stoltz

A rigorous nonperturbative adiabatic approximation of the evolution operator in the many-body physics of degenerate systems is derived. This approximation is used to solve the long-standing problem of the choice of the initial states of H(0) leading to eigenstates of H(0) + V for degenerate systems. These initial states are eigenstates of P(0)VP(0), where P(0) is the projection onto a degenerate eigenspace of H(0). This result is used to give the proper definition of the Green function, the statistical Green function and the nonequilibrium Green function of degenerate systems. The convergence of these Green functions is established.