A Numerical Perspective on Hartree-Fock-Bogoliubov Theory

TL;DR

This paper introduces a numerical analysis perspective to Hartree-Fock-Bogoliubov (HFB) theory, analyzing discretization, convergence of algorithms, and demonstrating pairing phenomena through numerical experiments in gravitational and nuclear models.

Contribution

It is the first to analyze HFB theory from a numerical standpoint, including convergence properties of fixed point algorithms and the adaptation of damping methods.

Findings

Fixed point algorithm either converges or oscillates without solutions.

Damping algorithms improve convergence stability.

Numerical experiments show pairing always occurs in studied models.

Abstract

The method of choice for describing attractive quantum systems is Hartree-Fock-Bogoliubov (HFB) theory. This is a nonlinear model which allows for the description of pairing effects, the main explanation for the superconductivity of certain materials at very low temperature. This paper is the first study of Hartree-Fock-Bogoliubov theory from the point of view of numerical analysis. We start by discussing its proper discretization and then analyze the convergence of the simple fixed point (Roothaan) algorithm. Following works by Canc\`es, Le Bris and Levitt for electrons in atoms and molecules, we show that this algorithm either converges to a solution of the equation, or oscillates between two states, none of them being a solution to the HFB equations. We also adapt the Optimal Damping Algorithm of Canc\`es and Le Bris to the HFB setting and we analyze it. The last part of the paper is devoted to numerical experiments. We consider a purely gravitational system and numerically discover that pairing always occurs. We then examine a simplified model for nucleons, with an effective interaction similar to what is often used in nuclear physics. In both cases we discuss the importance of using a damping algorithm.