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Reduced Density Matrix Functional Theory for Superconductors

Authors: J. Schmidt

Ref.: Master thesis, Martin-Luther University of Halle-Wittenberg (2018)

Abstract: The first chapter will be concerned with giving an overview of different theories, concepts and formalisms that are required as a background for the thesis. We will give a short introduction to zero- and finite-temperature RDMFT (FT-RDMFT) as well as SC-DFT.
Based on the knowledge of these previous theories we will begin to build the theoretical foundations of SC-RDMFT in the form of a Gilbert theorem in the second chapter. The theorem will provide a 1-to-1 mapping between the equilibrium state, a pair of external potentials and the equilibrium pair of densities as well as a variational principle. We will also investigate a more more general minimization approach à la Lieb, discuss the properties of the universal functional and research the equilibrium-V-representability of the Nambu-Gorkov 1RDMs.
The third chapter will be devoted to the question of the existence of a Kohn-Sham system at finite and zero temperature in RDMFT as well as DFT. We know from SC-DFT that the full electron-nuclear system is practically impossible to solve but in Sc-DFT it is possible to circumvent this problem through an approximation known as decoupling approximation. In the fourth chapter we will research the feasibility of introducing this approximation to SC-RDMFT.
The fifth chapter will be concerned with the development of a first anomalous exchange- correlation functional for SC-RDMFT. For this purpose, we will introduce the Sham- Schlüter connection [16] to SC-RDMFT and compare the resulting functional and de- velopment process to the one in SC-DFT. Lastly, we will suggest a formulation of SC-RDMFT that corresponds more closely to the classical natural orbital formulation of RDMFT and try to reconcile the two approaches presented in the thesis.

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