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Acta Physica Polonica B

Third-0rder Corrections to the Ground-state Energy of the Polarized Diluted Gas of Spin \(1/2\) fermions

Piotr H. Chankowski, J. Wojtkiewicz, R. Bakhshizada

Published Jan 1, 2022• 2 citations
Publisher

Abstract

The purpose of this note is threefold: (i) to recall (with some points made more explicit) the mathematical Weyl algebra model formulation, given before, of the Staruszkiewicz theory of quantum Coulomb field; (ii) to add some new elements to the discussion of the representation of the Lorentz group within this model; (iii) to comment on some statements on the structure of the theory which appeared recently.

DOI
10.5506/aphyspolb.53.9-a4

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Third-0rder Corrections to the Ground-state Energy of the Polarized Diluted Gas of Spin 1/21/21/2 fermions
The purpose of this note is threefold: (i) to recall (with some points made more explicit) the mathematical Weyl algebra model formulation, given before, of the Staruszkiewicz theory of quantum Coulomb field; (ii) to add some new elements to the discussion of the representation of the Lorentz group within this model; (iii) to comment on some statements on the structure of the theory which appeared recently.
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.
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).
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.