The Ground State Properties of a spin-3/2 Fermi gas

Large spin (f ≥ 3/2) fermions have recently become a focus of a rapidly growing interest in the context of cold atom physics. Some pioneering theoretical studies had been performed before the experimental realization of large spin Fermi atomic gases. Ho and Yip studied normal Fermi liquid behaviors and Cooper pairing structures of fermions with arbitrary large spins early in 1999. Wu et al. proved that there exists a hidden SO(5) symmetry in the spin-3/2 Fermi atomic gas . A series of experimental progresses are achieved in recent years  and realization of degenerate large spin Fermi gases in Yb , Li and Sr atoms is reported. In contrast to spin-1/2 system, large spin Fermi systems have more spin components and interacting channels, including a spin-mixing interaction . Consequently the systems exhibit more intriguing properties in novel quantum phases , quantum magnetism and magnetic impurity, Cooper pairing and spin-mixing dynamics .Though present research interests are mainly laid on correlated properties of large spin fermions, the Fermi-liquid theory of normal-state Fermi gas is of most fundamental. Actually, this issue has already attracted many attentions. Theoretically, Yip et al. investigated properties of the SU(N) Fermi gas at zero and finite temperatures and found that the Fermi-liquid parameters can be enhanced by the number of components . Ramires developed Fermi liquid theory for a Fermi gas with SP(N) symmetry. On the other hand, Fermi parameters of ultracold Fermi atoms can be experimentally checked. Scazza et al. examined the effective mass, the residue, and the decay rate of quasiparticles of a polarized ultracold ⁶Li mixture.

In this paper, we study ground state properties of a generalized spin-3/2 Fermi model in which interactions are set to be spin-dependent and emphasize contributions of the spin-mixing interaction. The system with SU(N) or SP(N) symmetry is just the special case of the generalized model by supposing 

some interaction parameters to be related. Our treatment is different from but complementary to the methods used previously for the SU(N) and SP(N) systems. We calculate the effective interaction via Galitskii's equation and integrate the self-energy with the tricky method that was originally used to treat the uniform electron gas by Onsager  and Ziesche . The obtained results can be compared to previous studies. 

After the calculation , we find that the ground state energy and effective mass always increase with the spin-dependent interaction strengths in the spin-3/2 Fermi system with lower symmetry which is consistent with the conclusion of SP(N) and SU(N) Fermi liquid. We derive the effective interaction iterately by using the  Green's function and ladder diagrams. These techniques make it easier to get the higher-order approximation terms of each spin channel and avoid some redundant cross terms. We can get the contribution of spin-mixing term g_0 separately.  The numerical integration results of spin-1/2 system is approximately equal to the result calculated by Galitskii in 1958 , but the scattering length is spin-dependent because of the contact interaction. Otherwise, there are more interaction parameters especially the spin-mixing term that always strengthens the ground state energy and effective mass in spin-3/2 Fermi gas. Taking the value of scattering length as ( g_1=g_2=g_3=g_4=1/2 g), we obtain the result of effective mass for spin-3/2 system, which can recover the SU(2) and SU(N) results but add a spin-mixing term.The chemical potential, ground state energy and effective mass are calculated after elaborately dealing with the effective interaction and the self-energy. Contributions of each interaction term are derived within the second-order perturbation. It is found that the spin-mixing term always strengthens the ground state energy and effective mass.