Magnetism and Cooper pairing in one-dimensional large spin fermions with repulsive interactions

The recent experimental realization of ultracold large-spin fermionic systems provides a new opportunity to investigate exotic magnetism and Cooper pairing physics. By means of exact diagonalizaton and the density matrix renormalization group, we systematically study the magnetic properties of the Mott-insulating state of the simplest large-spin systems with hyperfine spin $F=3/2$ in one-dimension and at quarter filling. Such a system is characterized by an exact $SO(5)$ symmetry. The ground state shows various profiles at various $\theta=\tan^{-1}J_0/J_2$, where $J_0/J_2$ is the ratio of exchange strengths of the singlet ($S_T=0$) and quintet ($S_T=2$) channels. As $\theta> 45^{\circ}$ the ground state is a gapped state with dimerization patterns whereas as $\theta \le 45^{\circ}$ it is a gapless Luttinger liquid state. Furthermore, we found that in the Luttinger liquid phase the static correlation functions show power-law decays with a four-site periodicity, which is similar to an SU(4) chain. We also study the spin-$3/2$ model with doping. In the regime of $\theta> 45^{\circ}$ and at moderate doping, the singlet pairing correlations indicate power-law decays whereas the quintet pairing correlations have exponential decays. On the other hand, in the regime of $\theta \le 45^{\circ}$ the quintet pairing correlations are more robust than the singlet pairing correlations.