Variational Principle Directly on the Coherent Pair Condensate. I. the BCS Case

TL;DR

This paper introduces a variational method directly on the coherent pair condensate that conserves particle number and avoids projection, demonstrated with realistic interactions and large model spaces, achieving high-precision energy minimization.

Contribution

It develops a new variational algorithm on the coherent pair condensate that conserves particle number and simplifies physical interpretation, applicable to realistic nuclear interactions.

Findings

The algorithm achieves high-precision energy minimization.

Realistic $V_{low-k}$ interaction does not cause divergences in pairing.

Tiny occupation numbers contribute minimally to the energy.

Abstract

We propose a scheme to perform the variational principle directly on the coherent pair condensate (VDPC). The result is equivalent to that of the so-called variation after particle-number projection, but now the particle number is always conserved and the time-consuming projection is avoided. This work considers VDPC+BCS. We derive analytical expressions for the average energy and its gradient in terms of the coherent pair structure. In addition, we give analytically the pair structure at the energy minimum, and discuss its asymptotic behavior away from the Fermi surface, which looks quite simple and allows easy physical interpretations. The new algorithm iterates these pair-structure expressions to minimize energy. We demonstrate the new algorithm in a semi-realistic example using the realistic $V_{low-k}$ interaction and large model spaces (up to 15 harmonic-oscillator major shells). The energy can be minimized to practically arbitrary precision. The result shows that the realistic $V_{low-k}$ interaction does not cause divergences in the pairing channel, although tiny occupation numbers (for example smaller than $10^{-5}$) contribute to the energy (by a few tens of keV). We also give analytical expressions for the gradient of energy with respect to changes of the canonical single-particle basis, which will be necessary for the next work in this series: VDPC+HFB.