Free energy of the gas of spin 1/2 fermions beyond the second order and the Stoner phase transition

Applying the previously developed systematic thermal (imaginary time) perturbative expansion to the relevant effective field theory we compute the free energy $F$ of the diluted gas of (nonrelativistic) spin $1\mathrm{/}2$ fermions interacting through a spin-independent repulsive two-body potential as a function of the numbers $N_{+}$ and $N_{-}$ of spin up and spin down fermions (i.e. as a function of the system’s polarization) and the temperature $T$. We give the complete order $(k_{\mathrm{F}}{a}_0{)}^3$ ($k_{\mathrm{F}}$ is the Fermi wave vector and $a_0$ is the $s$-wave scattering length characterizing the interaction potential) contribution to $F$. We also extend the computation beyond a fixed order by resumming to all orders in the parameter $k_{\mathrm{F}}{a}_0$ the contributions to $F$ of two infinite sets of Feynman diagrams: the so-called particle-particle rings and the particle-hole rings. We find that including the second one of these two contributions has a dramatic consequence for the transition of the system from the paramagnetic to the ferromagnetic phase (the so called Stoner phase transition): in this approximation the phase transition simply disappears. This result does not contradict the expectation that a transition to the magnetically ordered state should occur in truly repulsive systems. The $p$-wave and higher scattering lengths, as well as other parameters chacterizing the interaction potential, are in such systems generally of the same order of magnitude as $a_0$ and contributions depending on them should be, therefore, also included in $F$. Our results may, however, have implications for the search of the itinerant ferromagnetism of cold atomic gases in which large $a_0$, much larger than all other parameters, is artificially created by exploiting the physics of the Feshbach resonance.