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arXiv:2303.09921 · Physical Review A

Third-order corrections to the ground-state energy of the gas of spin- s fermions with arbitrary densities of different spin projections

Piotr H. Chankowski, Jacek Wojtkiewicz, Szymon Augustynowicz

Published Jun 21, 2023• 2 citations
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Abstract

Recently, we computed third-order corrections to the ground-state energy of an arbitrarily polarized diluted gas of spin-$\frac{1}{2}$ fermions interacting through a spin-independent repulsive two-body potential. Here we extend this result to the gas of spin-$s$ fermions [a system whose Hamiltonian has an accidental $\text{SU}(2s+1)$ 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 $s$- and $p$-wave scattering lengths ${a}_{0}$ and ${a}_{1}$ and the $s$-wave effective radius ${r}_{0}$, 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).

DOI
10.1103/physreva.107.063311

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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).
Third order corrections to the ground state energy of the polarized diluted gas of spin 1/21/21/2 fermions
We present the results of the computation of the third order corrections to the ground state energy of the diluted polarized gas of nonrelativistic spin 1/21/21/2 fermions interacting through a spin-independent repulsive two-body potential. The corrections are computed within the effective field theory approach which does not require specifying the interaction potential explicitly but only to characterize it by only a few parameters - the scattering lengths a0a_0a0​, a1,…a_1,\dotsa1​,… and effective radii r0,…r_0,\dotsr0​,… - measurable in low energy fermion-fermion elastic scattering. The corrections are computed semi-analytically, that is are expressed in terms of two functions of the system's polarization. The functions are given by the integrals which can be easily evaluated using the Mathematica built-in routines for numerical integration.
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.
Applying the previously developed systematic thermal (imaginary time) perturbative expansion to the relevant effective field theory we compute the free energy FFF of the diluted gas of (nonrelativistic) spin 1/21/21/2 fermions interacting through a spin-independent repulsive two-body potential as a function of the numbers N+N_+N+​ and N−N_-N−​ of spin up and spin down fermions (i.e. as a function of the system's polarization) and the temperature TTT. We give the complete order (kFa0)3(k_{\rm F}a_0)^3(kF​a0​)3 (kFk_{\rm F}kF​ is the Fermi wave vector and a0a_0a0​ is the sss-wave scattering length characterizing the interaction potential) contribution to FFF. We also extend the computation beyond a fixed order by resumming to all orders in the parameter kFa0k_{\rm F}a_0kF​a0​ the contributions to FFF 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 ppp-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 a0a_0a0​ and contributions depending on them should be, therefore, also included in FFF. Our results may, however, have implications for the search of the itinerant ferromagnetism of cold atomic gases in which large a0a_0a0​, much larger than all other parameters, is artificially created by exploiting the physics of the Feshbach resonance.
We report the results of the computation of the order (kFa0)2(k_{\rm F}a_0)^2(kF​a0​)2 correction, where kF=3π2ρk_{\rm F}=3π^2ρkF​=3π2ρ is the Fermi wave vector and a0a_0a0​ the sss-wave scattering length of the repulsive interaction, to the ground-state energy of a mixture of oppositely polarized NaN_aNa​ spin 1/21/21/2 fermions aaa of masses mam_ama​ and NbN_bNb​ spin 1/21/21/2 fermions bbb of masses mbm_bmb​ (ρ=N/Vρ=N/Vρ=N/V, N=Na+NbN=N_a+N_bN=Na​+Nb​). It is shown that the results of the paper \cite{FraPil} in which the same correction has been computed entirely numerically, using a more traditional approach, can be easily and semianalytically reproduced using the effective field theory technique.
Itinerant ferromagnetism is one of the most-studied quantum phase transitions, the transition point and the nature of this phase transition being widely discussed. In dilute Fermi liquids, this analysis has been carried out up to second order in the gas parameter, where the results for any spin degeneracy are universal in terms of only the s -wave scattering length a 0 . We extend this analysis to third order, where energies depend not only on a 0 , but also on the s -wave effective range r 0 and the p -wave scattering length a 1 . The introduction in the theory of these new parameters changes the transition point, with respect to the second-order estimation, and also can modify the nature of the phase transition itself. We analyze these interaction effects on the phase transition for different spin values. The emerging phase diagram shows that the type of ferromagnetic transition changes dramatically as a function of r 0 and a 1 and, importantly, that this classification is not solely determined by the spin value, as happens at second order.
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.