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arXiv:2108.00793 · Physical review. B./Physical review. B

Ground-state energy of the polarized dilute gas of interacting spin- 12 fermions

Piotr H. Chankowski, Jacek Wojtkiewicz

Published Oct 27, 2021• 9 citations
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Abstract

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 ${({k}_{\mathrm{F}}{a}_{0})}^{2}$ order (where ${k}_{\mathrm{F}}$ is the system's Fermi momentum and ${a}_{0}$ the $s$-wave scattering length) and (with more labor) to extend it up to the order ${({k}_{\mathrm{F}}{a}_{0})}^{4}$. The analogous expansion of the ground-state energy of the polarized gas of spin $1/2$ fermions is known only up to the ${k}_{\mathrm{F}}{a}_{0}$ order (where ${k}_{\mathrm{F}}$ stands for ${k}_{\mathrm{F}\ensuremath{\uparrow}}$ or ${k}_{\mathrm{F}\ensuremath{\downarrow}}$); the order ${({k}_{\mathrm{F}}{a}_{0})}^{2}$ 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 ${({k}_{\mathrm{F}}{a}_{0})}^{2}$ correction to the ground state of the polarized gas in a way applicable to all repulsive interactions.

DOI
10.1103/physrevb.104.144425

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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.
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).
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.
We prove a lower bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions. This correction depends on the ppp-wave scattering length of the interaction and matches the corresponding upper bound in [J. Funct. Anal. 286.7 (2024), p. 110320].
Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin 1/2 fermions, this is E∼E0+(ℏ2/2m)2πNρaE \sim E^0 + (\hbar^2/2m) 2 πN ρaE∼E0+(ℏ2/2m)2πNρa, where E0E^0E0 is the energy of the non-interacting system and ρρρ is the density.
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.