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

TL;DR

This paper calculates the free energy of a spin 1/2 fermion gas beyond second order, explores the effects of many-body interactions on phase transitions, and finds that certain resummations eliminate the ferromagnetic transition.

Contribution

It introduces a systematic perturbative expansion for the free energy of interacting fermions and resums infinite classes of diagrams, revealing the disappearance of the Stoner transition.

Findings

Including particle-hole ring diagrams removes the ferromagnetic phase transition.

Complete order $(k_F a_0)^3$ contribution to free energy is provided.

Results have implications for cold atomic gases and itinerant ferromagnetism.

Abstract

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 $1/2$ 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 $(k_{\rm F}a_0)^3$ ($k_{\rm F}$ 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 $k_{\rm F}a_0$ 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.