A Numerical Perspective on Hartree-Fock-Bogoliubov Theory
For researchers in computational quantum physics, this work establishes foundational numerical guarantees for HFB theory, which is crucial for simulating superconductivity and nuclear systems.
This paper provides the first numerical analysis of Hartree-Fock-Bogoliubov (HFB) theory, proving that the fixed-point algorithm either converges or oscillates, and adapting the Optimal Damping Algorithm to HFB. Numerical experiments on a gravitational system show pairing always occurs, and on a nucleon model demonstrate the necessity of damping.
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 Cancès, 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 Cancès 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.