Reduced Density Matrix Functional Theory for Superconductors

We present a new ab-initio theory for superconducting systems with nonlocal external potentials based on the one-particle reduced density matrix $\rho$(r,r${}^{\prime }$), the anomalous density $\chi$(r,r${}^{\prime }$) and the nuclear density D(r,r${}^{\prime }$). We prove that all the system's equilibrium properties are determined uniquely by these three quantities and deduce a variational principle for the grand canonical potential in terms of them. By replacing the local electronic density, which is used in density functional theory for superconductors (SC-DFT), with the non-local one-particle reduced density matrix, our theory is able to solve difficulties which arise in SC-DFT through the combination of local and non-local quantities. Analogue to the Kohn-Sham system in DFT, we prove the existence of a Kohn-Sham system that is able to reproduce $\rho$, $\chi$ and D of an interacting system at finite temperature. On the basis of the Kohn-Sham system, we obtain a set of Bogoliubov-De Gennes-like single particle equations. Finally, we derive a first exchange-correlation functional through the Sham-Schlüter connection.