A novel homogeneous superconducting state to compete with the inhomogeneous FFLO states in conductors with exchange or Zeeman split Fermi surfaces

The Fulde-Ferrell-Larkin-Ovchinikov (FFLO) states have been proposed as the states to replace the usual BCS state when the Fermi surfaces of spin-up and down electrons are sufficiently different due to exchange or Zeeman splitting. The FF state has a space-varying phase and the LO state has an oscillating order-parameter magnitude. Thus both are inhomogeneous states. Here I propose a novel homogeneous state to compete with the FFLO state: Instead of forming Cooper pairing of one spin-up electron of momentum \textbf{k }and one spin-down electron on momentum --\textbf{k}, this new state involves a generalized Cooper pair which is actually a triplet: One spin-up electron of momentum (1+$\delta )$\textbf{k}, one spin-down electron of momentum --(1--$\delta )$\textbf{k}, AND one boson of momentum --2$\delta $\textbf{k}, which can be a phonon, or a magnon, etc. (This state might also form in Fermion-Boson atomic mixtures, and in quark-gluon plasma.) The value of $\delta $ is such that (1+$\delta )$k$_{F}$ is the spin-up Fermi momentum, and (1--$\delta )$k$_{F}$ is the spin-down Fermi momentum. Since the boson involved does not have a fixed momentum \textbf{q}, this state is not a density wave state of any sort coexisting with superconductivity. A mean field theory of this state will be presented.