Quatrièmes coefficients d’amas et du viriel d’un gaz unitaire de fermions pour un rapport de masse quelconque

Shimpei Endo; Yvan Castin

doi:10.5802/crphys.108

Research article

Quatrièmes coefficients d’amas et du viriel d’un gaz unitaire de fermions pour un rapport de masse quelconque
[Fourth cluster and virial coefficients of a unitary Fermi gas for an arbitrary mass ratio]

Shimpei Endo ¹; Yvan Castin ²

¹ Département de physique, Université de Tohoku, Sendai, Japon
² Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Université de la Sorbonne et Collège de France, 24 rue Lhomond, 75231 Paris, France

Comptes Rendus. Physique, Volume 23 (2022), pp. 41-110.

Abstract
Résumé

We calculate the fourth cluster coefficients of the homogeneous unitary spin 1/2 Fermi gas as functions of the internal-state mass ratio, over intervals constrained by the 3- or 4-body Efimov effect. For this we use our 2016 conjecture (validated for equal masses by Hou and Drut in 2020) in a numerically efficient formulation making the sum over angular momentum converge faster, which is crucial at large mass ratio. The mean cluster coefficient, relevant for equal chemical potentials, is not of constant sign and increases rapidly close to the Efimovian thresholds. We also get the fourth virial coefficients, which we find to be very poor indicators of interaction-induced 4-body correlations. We obtain analytically for all $n$ the cluster coefficients of order $n + 1$ for an infinity-mass impurity fermion, matching the conjecture for $n = 3$ . Finally, in a harmonic potential, we predict a non-monotonic behavior of the $3 + 1$ cluster coefficient with trapping frequency, near mass ratios where this coefficient vanishes in the homogeneous case. A multilingual version is available in separate files on the open archive HAL at https://hal.archives-ouvertes.fr/hal-03592961.

Nous calculons les quatrièmes coefficients d’amas du gaz unitaire homogène de fermions de spin 1/2 en fonction du rapport de masse entre les deux états de spin $↑$ et $↓$ , sur des intervalles limités par les seuils de l’effet Efimov à trois ou à quatre corps. Nous utilisons pour cela notre conjecture de 2016 (validée dans le cas de masses égales par le calcul direct de Hou et Drut de 2020) dans une formulation numériquement très efficace à base d’accélération de convergence de la somme sur le moment cinétique, un atout précieux à grand rapport de masse. Le coefficient d’amas moyen, défini pour des potentiels chimiques égaux, n’est pas de signe constant et s’accroît rapidement près des seuils. Nous déterminons aussi les quatrièmes coefficients du viriel, souvent évoqués mais jamais calculés, et que nous trouvons être de très mauvais indicateurs des corrélations à quatre corps induites par les interactions. En passant, nous calculons analytiquement pour tout $n$ les coefficients d’amas d’ordre $n + 1$ dans la limite où la masse du fermion seul dans son état de spin tend vers l’infini, et trouvons pour $n = 3$ qu’il y a accord avec la conjecture. Enfin, dans un potentiel harmonique, nous prédisons un comportement inattendu, non monotone, du coefficient d’amas d’ordre $3 + 1$ avec la raideur du piège, près des rapports de masse annulant ce coefficient dans le cas homogène. Une version multilingue est disponible en fichiers séparés sur l’archive ouverte HAL à l’adresse https://hal.archives-ouvertes.fr/hal-03592961.

Received: 2022-02-28
Revised: 2022-04-11
Accepted: 2022-05-17
Published online: 2022-08-12

DOI: 10.5802/crphys.108

Mots-clés : gaz de fermions, limite unitaire, invariance d’échelle, développement du viriel, développement en amas
Keywords: Fermi gases, unitary limit, scale invariance, virial expansion, cluster expansion

Author's affiliations:

Shimpei Endo ¹; Yvan Castin ²

¹ Département de physique, Université de Tohoku, Sendai, Japon
² Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Université de la Sorbonne et Collège de France, 24 rue Lhomond, 75231 Paris, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Shimpei Endo and Yvan Castin},
     title = {Quatri\`emes coefficients d{\textquoteright}amas et du viriel d{\textquoteright}un gaz unitaire de fermions pour un rapport de masse quelconque},
     journal = {Comptes Rendus. Physique},
     pages = {41--110},
     publisher = {Acad\'emie des sciences, Paris},
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     doi = {10.5802/crphys.108},
     language = {fr},
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Shimpei Endo; Yvan Castin. Quatrièmes coefficients d’amas et du viriel d’un gaz unitaire de fermions pour un rapport de masse quelconque. Comptes Rendus. Physique, Volume 23 (2022), pp. 41-110. doi : 10.5802/crphys.108. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.108/

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