Introduction to Green-Function Methods

Robert van Leeuwen, Nils Erik Dahlen

J. B. Zhang, Z. K. Qin

It is shown in the present paper that if operators of a boson Green's function involve arbitrary c numbers, then the corresponding correlation function obtained from the spectral theorem is independent of the c numbers, which is in contradiction with the definition of the said correlation function. In order to solve this problem we have introduced an undetermined-constant method and applied it to ferromagnetic and ferroelectric systems, respectively. It is pointed out that the introduction of the undetermined constant is an intrinsic characteristic of the boson Green's-function method and that contradictory results can often be obtained if it is not introduced.

Sergei Vladimirovich Tyablikov

Mariana M. Odashima, Beatriz G. Prado, E. Vernek

The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions. Despite its power and versatility, it is known as a laborious and sometimes cumbersome method. Here we introduce the equilibrium Green's functions and the equation-of-motion technique, exemplifying the method in discrete lattices of non-interacting electrons. We start with simple models, such as the two-site molecule, the infinite and semi-infinite one-dimensional chains, and the two-dimensional ladder. Numerical implementations are developed via the recursive Green's function, implemented in Julia, an open-source, efficient and easy-to-learn scientific language. We also present a new variation of the surface recursive Green's function method, which can be of interest when simulating simultaneously the properties of surface and bulk.

Eleftherios N. Economou

Mariana M. Odashima, Beatriz G. Prado, E. Vernek

The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions. Despite its power and versatility, it is known as a laborious and sometimes cumbersome method. Here we introduce the equilibrium Green's functions and the equation-of-motion technique, exemplifying the method in discrete lattices of non-interacting electrons. We start with simple models, such as the two-site molecule, the infinite and semi-infinite one-dimensional chains, and the two-dimensional ladder. Numerical implementations are developed via the recursive Green's function, implemented in Julia, an open-source, efficient and easy-to-learn scientific language. We also present a new variation of the surface recursive Green's function method, which can be of interest when simulating simultaneously the properties of surface and bulk.