Superfluid LDA (SLDA): Local Density Approximation for Systems with Superfluid Correlations

The theorem of Hohenberg and Kohn 1 concerning the existence of a universal Energy Density Functional (EDF), and its subsequent implementation as a Local Density Approximation (LDA), lead to a new qualitative approach to the study of electron systems, from atoms, to molecules, to condensed matter systems and macromolecules in particular and other fermionic systems in general. Even though neither Hohenberg and Kohn nor Kohn and Sham gave us recipes on how to construct the EDF, various approximation schemes with increasing level of sophistication have been created. Moreover, the EDF ideology has been extended to finite temperatures and finite excitation energies as well. However, essentially all of the implementations of the LDA and EDF have been limited so far to normal Fermi systems, namely, systems with no pairing correlations. There were two attempts to extend the LDA to superconducting systems , however, the pairing field in this approach was still a nonlocal object. One can present the argument that because electron superconductivity is phonon mediated, and since phonons have a spectrum limited by the Debye frequency, such a genuinely nonlocal character of the electron pairing field is natural. In our opinion this kind of argumentation is somewhat tenuous. The normal part of the EDF arises from Coulomb interaction, which in itself has an infinite range. Nevertheless, a coherent LDA approach to normal systems is possible. In recent works 3,4,5,6,7,8,9,10 we have been able to develop a genuinely local extention of the LDA to systems with superfluid correlations and apply it to a number of nuclear and atomic systems. Besides the fact that a genuine local approach to pairing correlations within an LDA à la Kohn and Sham is certainly possible, such a framework is physically meaningful. In nuclear physics for example, the so called coherence length, or in other words the size of the Cooper pair,