Direct and inverse photoemission spectroscopies are powerful experimental techniques to probe the electronic structure of matter. The resulting one-electron excitation spectra are most often extremely rich, as they exhibit plenty of interesting features which stem from the interaction of electrons in the material . Analyzing, interpreting and predicting such spectra is a major challenge of theoretical physics. The current state-of-the-art realistic calculations rely usually on many-body Green’s functions and complex, non-local and frequency-dependent self-energies, which are evaluated specifically for each material . Even though the calculated spectra are often in very good agreement with experiments, the computational cost is very large. Indeed, much superfluous information is produced that is not needed for the interpretation of experimental data.
In this thesis we propose two shortcuts with respect to the standard method. The first one is the introduction of an auxiliary system that, in principle, exactly targets the excitation spectrum of the real system. The prototypical example for an auxiliary system is density functional theory, in which the auxiliary system is the Kohn-Sham system : it exactly reproduces the density of the real system via a real and static potential, the Kohn-Sham potential. However, the Kohn-Sham system does not correctly describe excited-state properties: an example is the famous band-gap problem. The potential that we propose (named "spectral potential"), which is frequency dependent, yet local and real, can be viewed as a dynamical generalization of the Kohn-Sham potential that yields in principle the exact excitation spectrum.
The second shortcut is the idea of calculating this potential just once and forever in a model system, the homogeneous electron gas (HEG), and tabulating it. To study real materials, we design a connector that prescribes the use of the HEG results for calculating electronic spectra. All the non-trivial interplay between electron interaction and inhomogeneity of the real system enters the form of the connector. Thus the great challenge of the approach is finding an accurate expression for the connector. We propose an approximation based on local properties of the system, which we call the "dynamical local connector approximation”.
We have applied this procedure to model Hamiltonians and to four different prototypical materials: sodium, an almost homogeneous metal; aluminum, still a metal but less homogeneous; silicon, a semiconductor; argon, an inhomogeneous insulator. The spectra that we obtain agree to an impressive extent with the ones evaluated via the computationally expensive self-energy, demonstrating the potential of this approach .
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 R. M. Martin, L. Reining, D. M. Ceperley, Interacting Electrons (Cambridge University Press, 2016).
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 Marco Vanzini, Lucia Reining, Matteo Gatti, arXiv:1708.02450.