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

Abstract

In the previous work we have developed a systematic thermal (imaginary time) perturbative expansion and applying it to the relevant effective field theory computed, up to the second order in the interaction, the free energy $F$ of the diluted gas of (nonrelativistic) spin $1/2$ fermions interacting through a spin-independent repulsive two-body potential. Here we extend this computations to higher orders: assuming that the only relevant parameter specifying the interaction potential is the $s$-wave scattering length $a_0$, we compute the complete order $(k_{\rm F}a_0)^3$ ($k_{\rm F}$ is the Fermi wave vector) contribution to the system's free energy as a function of the numbers $N_+$ and $N_-$ of spin up and spin down fermions (i.e. as a function of its polarization) and the temperature $T$. We also extend the computation beyond a fixed order by resumming 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.

Figures (9)

• Figure 1: The order $C_0^2$ diagram contributing to the thermodynamic potential $F$ of the gas of spin $1/2$ fermions and two "elementary" one-loop diagrams ($A$- and $B$-"blocks") out of which the second order and those higher order (in the $C_0$ coupling) contributions which are taken into account in this work are composed. Solid and dashed lines denote propagators of fermions with the spin projections $+$ and $-$, respectively.
• Figure 2: The particle-particle and the particle-hole diagrams contributing in the order $C^3_0$ to the thermodynamic potential $F$.
• Figure 3: The difference $(F(P)-F(0))^{pp}/N$ (in units $(3/5)\varepsilon_{\rm F}$) for $T=0$ and $T=0.2~\!T_{\rm F}\equiv0.2~\!\varepsilon_{\rm F}/k_{\rm B}$ as a function of the polarization $P=(N_+-N_-)/N$ for different values of the gas parameter $k_{\rm F}a_0$.
• Figure 4: As in Figure \ref{['fig:FresPPonlyT00and02']} but for $T=0.3~\!T_{\rm F}$ and $T=0.5~\!T_{\rm F}$.
• Figure 5: Dependence on the gas parameter $k_{\rm F}a_0$ of the resumed contributions of the particle-particle diagrams given by the expressions (\ref{['eqn:FppSummed']}) (dashed blue lines), of the particle-hole diagrams given by (\ref{['eqn:FphSummed']}) (dotted red lines) and of their sum (solid green lines) for zero temperature and two values of the polarization $P$.