Competition between spin imbalance and mass imbalance in the 1D asymmetric Hubbard model

In this talk, I will discuss the spin imbalance in the 1D asymmetric Hubbard model in the negative U region by the Bosonization method. A ground-state phase diagram has been obtained. We find that, unlike the $N_\downarrow=N_\uparrow$ case, there is no other phase transition in the ground state (always Singlet Superconducting) before it enters into the phase separation region, and the pairing correlation function is founding to oscillate in real space (FFLO state). The maximum mode is only determined by difference of Fermi momenta, and the correlation exponent is determined by both the mass difference and spin polarization.