Dark Energy from Eternal Pair-production of Fermions

We study a toy model in which Majorana or Dirac fermions behave as the source for a small vacuum energy in the present Universe. In the model, a self-interacting scalar boson coupled with fermions induces attractive and repulsive interactions between the fermions simultaneously. These interactions allow for the existence of a metastable state with positive energy density comprised of fermions degenerate inside a Fermi surface. The energy density of the metastable state remains constant as the Universe expands. This is because pair-productions of fermions from the vacuum continuously take place at no energy cost and keep supplying fermions uniformly to the Universe. The observed vacuum energy density $\sim 10^{-47}~\textrm{GeV}^4$ is reproduced for the fermion and scalar mass of the order $10^{-3}~\textrm{eV}$.