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arXiv:2510.22719 · Physical Review A

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

Oskar Grocholski, Piotr H. Chankowski

Published Mar 5, 2026• 0 citations
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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.

DOI
10.1103/tcjx-q14m

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It is generally believed that a dilute spin-12\frac{1}{2}21​ Fermi gas with repulsive interactions can undergo a ferromagnetic phase transition to a spin-polarized state at a critical gas parameter (kFa)c{({k}_{\mathrm{F}}a)}_{c}(kF​a)c​. Previous theoretical predictions of the ferromagnetic phase transition have been based on the perturbation theory, which treats the gas parameter as a small number. On the other hand, Belitz, Kirkpatrick, and Vojta (BKV) have argued that the phase transition in clean itinerant ferromagnets is generically of first order at low temperatures, due to the correlation effects that lead to a nonanalytic term in the free energy. The second-order perturbation theory predicts a first-order phase transition at (kFa)c=1.054{({k}_{\mathrm{F}}a)}_{c}=1.054(kF​a)c​=1.054, consistent with the BKV argument. However, since the critical gas parameter is expected to be of order O(1)O(1)O(1), perturbative predictions may be unreliable. In this paper we study the nonperturbative effects on the ferromagnetic phase transition by summing the particle-particle ladder diagrams to all orders in the gas parameter. We consider a universal repulsive Fermi gas where the effective range effects can be neglected, which can be realized in a two-component Fermi gas of 6{}^{6}6Li atoms by using a nonadiabatic field switch to the upper branch of a Feshbach resonance with a positive sss-wave scattering length. Our theory predicts a second-order phase transition, which indicates that ferromagnetic transition in dilute Fermi gases is possibly a counterexample to the BKV argument. The predicted critical gas parameter (kFa)c=0.858{({k}_{\mathrm{F}}a)}_{c}=0.858(kF​a)c​=0.858 is in good agreement with the recent quantum Monte Carlo result (kFa)c=0.86{({k}_{\mathrm{F}}a)}_{c}=0.86(kF​a)c​=0.86 for a nearly zero-range potential [S. Pilati et al., Phys. Rev. Lett. 105, 030405 (2010)]. We also compare the spin susceptibility with the quantum Monte Carlo result and find good agreement.
Recently, we computed third-order corrections to the ground-state energy of an arbitrarily polarized diluted gas of spin-12\frac{1}{2}21​ fermions interacting through a spin-independent repulsive two-body potential. Here we extend this result to the gas of spin-sss fermions [a system whose Hamiltonian has an accidental SU(2s+1)\text{SU}(2s+1)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 sss- and ppp-wave scattering lengths a0{a}_{0}a0​ and a1{a}_{1}a1​ and the sss-wave effective radius r0{r}_{0}r0​, 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).
Itinerant ferromagnetism in dilute Fermi gases is predicted to emerge at values of the gas parameter where second-order perturbation theory is not accurate enough to properly describe the system. We have revisited perturbation theory for SU(N) fermions and derived its generalization up to third order both in terms of the gas parameter and the polarization. Our results agree satisfactorily with quantum Monte Carlo results for hard-sphere and soft-sphere potentials for S = 1/2 S = 1 / 2 . Although the nature of the phase transition depends on the interaction potential, we find that for a hard-sphere potential a phase transition is guaranteed to occur. While for S= 1/2 S = 1 / 2 we observe a quasi-continuous transition, for spins 3/2 3 / 2 and 5/2 5 / 2 , a first-order phase transition is found. For larger spins, a double transition (combination of continuous and discontinuous) occurs. The critical density reduces drastically when the spin increases, making the phase transition more accessible to experiments with ultracold dilute Fermi gases. Estimations for Fermi gases of Yb and Sr with spin 5/2 5 / 2 and 9/2 9 / 2 , respectively, are reported.
Third order corrections to the ground state energy of the polarized diluted gas of spin 1/21/21/2 fermions
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The effective field theory approach simplifies the perturbative computation of the ground-state energy of the diluted gas of repulsively interacting fermions allowing in the case of the unpolarized system to easily rederive the classic results up to the (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 order (where kF{k}_{\mathrm{F}}kF​ is the system's Fermi momentum and a0{a}_{0}a0​ the sss-wave scattering length) and (with more labor) to extend it up to the order (kFa0)4{({k}_{\mathrm{F}}{a}_{0})}^{4}(kF​a0​)4. The analogous expansion of the ground-state energy of the polarized gas of spin 1/21/21/2 fermions is known only up to the kFa0{k}_{\mathrm{F}}{a}_{0}kF​a0​ order (where kF{k}_{\mathrm{F}}kF​ stands for ${k}_{\mathrm{F}\ensuremath{\uparrow}}$ or ${k}_{\mathrm{F}\ensuremath{\downarrow}}$); the order (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 contribution has been computed (analytically) only using a specific (hard-core type) interaction potential. Here we show that the effective field theory method also allows to easily obtain the order (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 correction to the ground state of the polarized gas in a way applicable to all repulsive interactions.