Electronic Structure Reading Group: Connecting DFT with Green's function methods through the effective action formalism

Both density functional theory (DFT) and Green's function (GF) methods aim to reproduce the electronic structure of materials without directly computing the many-body wavefunction, instead focusing on simpler quantities: the ground-state electron density and the one-particle Green's function. Although they share similarities, the mathematical formalisms underlying DFT and GF methods are completely different. DFT is formulated within the framework of first quantization, whereas GF methods are rooted in quantum field theory. The effective action formalism, based on Feynman's path-integral and the Legendre transformation, offers a unified framework that can bridge this gap. By applying this technique and strategically choosing which operator to couple with a constraining field, it is possible to derive key functionals, such as the Kohn-Sham functional of DFT or the Luttinger-Ward functional of many-body perturbation theory.

References:
- Fukuda, R., Kotani, T., Suzuki, Y., & Yokojima, S. (1994). Density Functional Theory through Legendre Transformation. In Progress of Theoretical Physics (Vol. 92, Issue 4, pp. 833–862). Oxford University Press (OUP). https://doi.org/10.1143/ptp/92.4.833
- Georges, A., Kotliar, G., Krauth, W., & Rozenberg, M. J. (1996). Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. In Reviews of Modern Physics (Vol. 68, Issue 1, pp. 13–125). American Physical Society (APS). https://doi.org/10.1103/revmodphys.68.13
- Martin, R. M., Reining, L., & Ceperley, D. M. (2016). Functionals in many-particle physics. In Interacting Electrons: Theory and Computational Approaches (pp. 169–192). chapter 8, Cambridge: Cambridge University Press.


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The electronic structure reading group brings together researchers and students interested in mathematical aspects of electronic structure problems and adjacent topics, including:

• Density Functional Theory
• Many-body Schrödinger equation for electrons
• Born-Oppenheimer Molecular Dynamics
• Numerical analysis and error control