Weighted Hartree-Fock-Bogoliubov method for interacting fermions: An application to ultracold Fermi superfluids

For several decades it has been known that divergences arise in the ground-state energy and chemical potential of unitary superfluids, where the scattering length diverges, due to particle-hole scattering. Leading textbooks and research articles recognize that there are serious issues but ignore them due to the lack of an approach that can regularize these divergences. We find a solution to this difficulty by proposing a general method, called the weighted Hartree-Fock-Bogoliubov theory, to handle multiple decomposition channels originating from the same interaction. We distribute the interaction in weighted channels determined by minimization of the action, and we apply this idea to unpolarized Fermi superfluids. Using our method, we solve a long-standing difficulty in the partitioning of the interaction into Hartree, Fock, and Bogoliubov channels for Fermi superfluids, and we obtain a phase diagram at the saddle-point level, which contains multichannel nonperturbative corrections. In particular, we find a previously overlooked superfluid phase for weak interactions, which is dominated by particle-hole processes, in addition to the usual superfluid phase only containing particle-particle physics.