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Theoretical and Mathematical Physics

A method of solving infinite systems of equations for two-time thermal Green's functions

Yu. A. Tserkovnikov

Published Nov 1, 1981• 73 citations
Publisher
DOI
10.1007/bf01028994

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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.

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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.
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.
The effect of boundary scattering of magnons in Heisenberg chains is studied by use of the thermal Green's function technique. In contrast with the usual approach, the periodic boundary condition is not imposed and the space Fourier transform is not used. The decoupled equation of motion for the two‐spin Green's functions is solved exactly as a system of simultaneous difference equations subject to very general boundary conditions. The solutions contain terms representing all the higher order reflections to infinite order of magnons from the two ends. The effect of chain length on the magnon frequency spectrum and its influence on the specific heat are calculated and discussed.