Establish quasiparticle orthonormalization for neural-network HFB or SLDA

Establish a method for imposing the quasiparticle orthonormality conditions on neural-network outputs representing the quasiparticle amplitudes $U$ and $V$ in Hartree--Fock--Bogoliubov theory or superfluid local density approximation within Kohn--Sham nuclear density functional theory.

Background

The paper identifies the extension of the neural-network variational framework from Kohn--Sham calculations without pairing to pairing-based approaches, specifically Hartree--Fock--Bogoliubov theory and the superfluid local density approximation, as a natural next step.

Such an extension requires neural networks to represent quasiparticle amplitudes UU and VV over a sufficiently large quasiparticle space, including continuum states. The required quasiparticle orthonormality constraints are more complicated than the orbital orthonormality constraint treated in the paper, and the appropriate enforcement mechanism remains unresolved.

References

The principal technical difficulty is to impose the quasiparticle orthonormality conditions on the network output, which are more involved than the orthonormality constraint treated here; how best to enforce them within the NN ansatz remains to be established.

— Neural-Network-Based Variational Method in Nuclear Density Functional Theory: Application to the Kohn--Sham method  (2609.00836 - Yoshimura et al., 1 Sep 2026) in Section 4, Summary and Outlook