Third-order corrections to the ground-state energy of the gas of spin- s fermions with arbitrary densities of different spin projections

Abstract

Recently, we computed third-order corrections to the ground-state energy of an arbitrarily polarized diluted gas of spin-$\frac{1}{2}$ fermions interacting through a spin-independent repulsive two-body potential. Here we extend this result to the gas of spin-$s$ fermions [a system whose Hamiltonian has an accidental $\text{SU}(2s+1)$ symmetry] with arbitrary densities of fermions having different spin projections. The corrections are computed semianalytically using the effective-field-theory approach and are parametrized by the $s$- and $p$-wave scattering lengths ${a}_{0}$ and ${a}_{1}$ and the $s$-wave effective radius ${r}_{0}$, measurable in the low-energy fermion-fermion elastic scattering. The result is used to study the impact the higher-order corrections can have on the characteristics of the phase transition (at zero temperature) to the ordered phase (on the emergence of the itinerant ferromagnetism).