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Finite range and upper branch effects on itinerant ferromagnetism in repulsive Fermi gases: Bethe-Goldstone ladder resummation approach

We investigate the ferromagnetic transition in repulsive Fermi gases at zero temperature with upper branch and effective range effects. Based on a general effective Lagrangian that reproduces precisely the two-body $s$-wave scattering phase shift, we obtain a nonperturbative expression of the energy density as a function of the polarization by using the Bethe-Goldstone ladder resummation. For hard sphere potential, the predicted critical gas parameter $k_{\rm F}a=0.816$ and the spin susceptibility agree well with the results from fixed-node diffusion Monte Carlo calculations. In general, positive and negative effective ranges have opposite effects on the critical gas parameter $k_{\rm F}a$: While a positive effective range reduces the critical gas parameter, a negative effective range increases it. For attractive potential or Feshbach resonance model, the many-body upper branch exhibits an energy maximum at $k_{\rm F}a=α$ with $α=1.34$ from the Bethe-Goldstone ladder resummation, which is qualitatively consistent with experimental results. The many-body T-matrix has a positive-energy pole for $k_{\rm F}a>α$ and it becomes impossible to distinguish the bound state and the scattering state. These positive-energy bound states become occupied and therefore the upper branch reaches an energy maximum at $k_{\rm F}a=α$. In the zero range limit, there exists a narrow window ($0.86<k_{\rm F}a<1.56$) for the ferromagnetic phase. At sufficiently large negative effective range, the ferromagnetic phase disappears. On the other hand, the appearance of positive-energy bound state resonantly enhances the two-body decay rate around $k_{\rm F}a=α$ and may prevent the study of equilibrium phases and ferromagnetism of the upper branch Fermi gas.