Green's function approach to the quantum many-body theory of the solid state

M R Bhagavan

James K. Freericks

This is an introductory chapter on how to calculate nonequilibrium Green's functions via dynamical mean-field theory for the Autumn School on Correlated Electrons: Many-Body Methods for Real Materials, 16-20 September 2019, Forschungszentrum Juelich. It is appropriate for graduate students with a solid state physics and advanced quantum mechanics background.

N. E. Bickers

Ville J. Härkönen, Robert van Leeuwen, Eberhard K. U. Gross

The method of many-body Green's functions is developed for arbitrary systems of electrons and nuclei starting from the full (beyond Born-Oppenheimer) Hamiltonian of Coulomb interactions and kinetic energies. The theory presented here resolves the problems arising from the translational and rotational invariance of this Hamiltonian that afflict the existing many-body Green's function theories. We derive a coupled set of exact equations for the electronic and nuclear Green's functions and provide a systematic way to approximately compute the properties of arbitrary many-body systems of electrons and nuclei beyond the Born-Oppenheimer approximation. The case of crystalline solids is discussed in detail.

Charles Paul Enz

William J. Herrera, Herbert Vinck-Posada, Shirley Gómez Páez

The use of Green's functions is valuable when solving problems in electrodynamics, solid-state physics, and many-body physics. However, its role in quantum mechanics is often limited to the context of scattering by a central force. This work shows how Green's functions can be used in other examples in quantum mechanics courses. In particular, we introduce time-independent Green's functions and the Dyson equation to solve problems with an external potential. We calculate the reflection and transmission coefficients of scattering by a Dirac delta barrier and the energy levels and local density of states of the infinite square well potential.