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

Article de recherche

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

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.

Résumé
Abstract

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.

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.

Reçu le : 2022-02-28
Révisé le : 2022-04-11
Accepté le : 2022-05-17
Publié le : 2022-08-12

DOI : 10.5802/crphys.108

Mot 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

Affiliations des auteurs :

Shimpei Endo ¹ ; Yvan Castin ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRPHYS_2022__23_G1_41_0,
     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},
     volume = {23},
     year = {2022},
     doi = {10.5802/crphys.108},
     language = {fr},
}

TY  - JOUR
AU  - Shimpei Endo
AU  - Yvan Castin
TI  - Quatrièmes coefficients d’amas et du viriel d’un gaz unitaire de fermions pour un rapport de masse quelconque
JO  - Comptes Rendus. Physique
PY  - 2022
SP  - 41
EP  - 110
VL  - 23
PB  - Académie des sciences, Paris
DO  - 10.5802/crphys.108
LA  - fr
ID  - CRPHYS_2022__23_G1_41_0
ER  -

%0 Journal Article
%A Shimpei Endo
%A Yvan Castin
%T Quatrièmes coefficients d’amas et du viriel d’un gaz unitaire de fermions pour un rapport de masse quelconque
%J Comptes Rendus. Physique
%D 2022
%P 41-110
%V 23
%I Académie des sciences, Paris
%R 10.5802/crphys.108
%G fr
%F CRPHYS_2022__23_G1_41_0

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/

Bibliographie
Cité par

[1] Yvan Castin; Félix Werner The Unitary Gas and its Symmetry Properties, BCS-BEC Crossover and the Unitary Fermi gas (W. Zwerger, ed.) (Lecture Notes in Physics), Volume 836, Springer, 2011, pp. 127-191 | DOI

[2] Mohit Randeria; Edward Taylor Crossover from Bardeen–Cooper–Schrieffer to Bose–Einstein Condensation and the Unitary Fermi Gas, Ann. Rev. Cond. Matter Phys., Volume 5 (2014) no. 1, 209, pp. 209-232 | DOI

[3] Martin W. Zwierlein; J. R. Abo-Shaeer; A. Schirotzek; C. H. Schunck; W. Ketterle Vortices and superfluidity in a strongly interacting Fermi gas, Nature, Volume 435 (2005), pp. 1047-1051 | DOI

[4] Leonid A. Sidorenkov; Meng Khoon Tey; Rudolf Grimm; Yan-Hua Hou; Lev Pitaevskii; Sandro Stringari Second sound and the superfluid fraction in a Fermi gas with resonant interactions, Nature, Volume 498 (2013), pp. 78-81 | DOI

[5] S. Nascimbène; N. Navon; K. J. Jiang; F. Chevy; C. Salomon Exploring the thermodynamics of a universal Fermi gas, Nature, Volume 463 (2010), pp. 1057-1060 | DOI

[6] Munekazu Horikoshi; Shuta Nakajima; Masahito Ueda; Takashi Mukaiyama Measurement of universal thermodynamic functions for a unitary Fermi gas, Science, Volume 327 (2010) no. 5964, pp. 442-445 | DOI

[7] Mark J. Ku; Ariel T. Sommer; Lawrence W. Cheuk; Martin W. Zwierlein Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science, Volume 335 (2012) no. 6068, pp. 563-567 | DOI

[8] C. Kohstall; M. Zaccanti; M. Jag; A. Trenkwalder; P. Massignan; G. Bruun; F. Schreck; Rudolf Grimm Metastability and coherence of repulsive polarons in a strongly interacting Fermi mixture, Nature, Volume 485 (2012), pp. 615-618 | DOI

[9] C. Ravensbergen; E. Soave; V. Corre; M. Kreyer; B. Huang; E. Kirilov; Rudolf Grimm Resonantly Interacting Fermi-Fermi Mixture of $^{161}$ Dy and $^{40}$ K, Phys. Rev. Lett., Volume 124 (2020) no. 20, 203402 | DOI

[10] E. Neri; A. Ciamei; C. Simonelli; I. Goti; M. Inguscio; A. Trenkwalder; M. Zaccanti Realization of a cold mixture of fermionic chromium and lithium atoms, Phys. Rev. A, Volume 101 (2020) no. 6, 063602 | DOI

[11] K. Van Houcke; Félix Werner; E. Kozik; N. Prokof’ev; B. Svistunov; M. J. H. Ku; Ariel T. Sommer; Lawrence W. Cheuk; A. Schirotzek; Martin W. Zwierlein Feynman diagrams versus Fermi-gas Feynman emulator, Nat. Phys., Volume 8 (2012), pp. 366-370 | DOI

[12] Riccardo Rossi Contributions to unbiased diagrammatic methods for interacting fermions, Ph. D. Thesis, Université Paris sciences et lettres, Paris, France (2017) (thèse en ligne tel-01704724v2, https://tel.archives-ouvertes.fr/tel-01704724v2)

[13] K. Huang Statistical Mechanics, John Wiley & Sons, New York, 1987

[14] Xia-Ji Liu Virial expansion for a strongly correlated Fermi system and its application to ultracold atomic Fermi gases, Phys. Rep., Volume 524 (2013) no. 2, pp. 37-83 | DOI | MR

[15] Xia-Ji Liu; Hui Hu Virial expansion for a strongly correlated Fermi gas with imbalanced spin populations, Phys. Rev. A, Volume 82 (2010) no. 4, 043626 | DOI

[16] Riccardo Rossi; T. Ohgoe; K. Van Houcke; Félix Werner Resummation of Diagrammatic Series with Zero Convergence Radius for Strongly Correlated Fermions, Phys. Rev. Lett., Volume 121 (2018) no. 13, 130405 | DOI

[17] Erich Beth; George E. Uhlenbeck The quantum theory of the non-ideal gas I. Deviations from the classical theory, Physica, Volume 3 (1936) no. 8, pp. 729-745 | DOI | Zbl

[18] Erich Beth; George E. Uhlenbeck The quantum theory of the non-ideal gas. II. Behaviour at low temperatures, Physica, Volume 4 (1937) no. 10, pp. 915-924 | DOI | Zbl

[19] L. Landau; E. Lifchitz Physique statistique - 1ère partie, Éditions Mir, Moscou, 1984

[20] Yvan Castin; Félix Werner Le troisième coefficient du viriel du gaz de Bose unitaire, Canadian Journal of Physics, Volume 91 (2013) no. 5, pp. 382-389 | DOI

[21] Chao Gao; Shimpei Endo; Yvan Castin The third virial coefficient of a two-component unitary Fermi gas across an Efimov-effect threshold, Eur. Phys. Lett., Volume 109 (2015) no. 1, 16003 | DOI

[22] P. C. Hemmer The hard core quantum gas at high temperatures, Phys. Lett., A, Volume 27 (1968) no. 6, pp. 377-378 | DOI

[23] B. Jancovici Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperature, Phys. Rev., Volume 178 (1969) no. 1, pp. 295-297 | DOI

[24] B. Jancovici Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperature. II, Phys. Rev., Volume 184 (1969) no. 1, pp. 119-123 | DOI

[25] B. Jancovici; S. P. Merkuriev Quantum-mechanical third virial coefficient of a hard-sphere gas at high temperature, Phys. Rev. A, Volume 12 (1975) no. 6, pp. 2610-2621 | DOI

[26] T. D. Lee; C. N. Yang Many-Body Problem in Quantum Statistical Mechanics. II. Virial Expansion for Hard-Sphere Gas, Phys. Rev., Volume 116 (1959) no. 1, pp. 25-31 | DOI | MR | Zbl

[27] A. Pais; George E. Uhlenbeck On the Quantum Theory of the Third Virial Coefficient, Phys. Rev., Volume 116 (1959) no. 2, pp. 250-269 | DOI | Zbl

[28] Sadhan K. Adhikari; R. D. Amado Low-Temperature Behavior of the Quantum Cluster Coefficients, Phys. Rev. Lett., Volume 27 (1971) no. 8, pp. 485-487 | DOI

[29] W. G. Gibson Low-Temperature Expansion of the Third-Cluster Coefficient of a Quantum Gas, Phys. Rev. A, Volume 6 (1972) no. 6, pp. 2469-2477 | DOI

[30] A. Comtet; Y. Georgelin; S. Ouvry Statistical aspects of the anyon model, J. Phys. A, Math. Gen., Volume 22 (1989) no. 18, pp. 3917-3925 | DOI | MR

[31] J. McCabe; S. Ouvry Perturbative three-body spectrum and the third virial coefficient in the anyon model, Phys. Lett. B, Volume 260 (1991), pp. 113-119 | DOI

[32] Xia-Ji Liu; Hui Hu; Peter D. Drummond Virial Expansion for a Strongly Correlated Fermi Gas, Phys. Rev. Lett., Volume 102 (2009) no. 16, 160401 | DOI

[33] Xia-Ji Liu; Hui Hu; Peter D. Drummond Three attractively interacting fermions in a harmonic trap : Exact solution, ferromagnetism, and high-temperature thermodynamics, Phys. Rev. A, Volume 82 (2010) no. 2, 023619 | DOI

[34] Félix Werner; Yvan Castin Unitary gas in an isotropic harmonic trap : Symmetry properties and applications, Phys. Rev. A, Volume 74 (2006) no. 5, 053604 | DOI

[35] V. Efimov Energy levels of three resonantly interacting particles, Nucl. Phys. A, Volume 210 (1973) no. 1, pp. 157-188 | DOI

[36] Félix Werner; Yvan Castin Unitary Quantum Three-Body Problem in a Harmonic Trap, Phys. Rev. Lett., Volume 97 (2006) no. 15, 150401 | DOI

[37] Seth T. Rittenhouse; N. P. Mehta; Chris H. Greene Green’s functions and the adiabatic hyperspherical method, Phys. Rev. A, Volume 82 (2010) no. 2, 022706 | DOI

[38] Yvan Castin; Edoardo Tignone Trimers in the resonant ( $2 + 1$ )-fermion problem on a narrow Feshbach resonance : Crossover from Efimovian to hydrogenoid spectrum, Phys. Rev. A, Volume 84 (2011) no. 6, 062704 | DOI

[39] Yvan Castin; Christophe Mora; Ludovic Pricoupenko Four-Body Efimov Effect for Three Fermions and a Lighter Particle, Phys. Rev. Lett., Volume 105 (2010) no. 22, 223201 | DOI

[40] Shimpei Endo; Yvan Castin Absence of a four-body Efimov effect in the $2 + 2$ fermionic problem, Phys. Rev. A, Volume 92 (2015) no. 5, 053624 | DOI

[41] Shimpei Endo; Yvan Castin The interaction-sensitive states of a trapped two-component ideal Fermi gas and application to the virial expansion of the unitary Fermi gas, J. Phys. A, Math. Gen., Volume 49 (2016) no. 26, 265301 | DOI | MR | Zbl

[42] Yangqian Yan; D. Blume Path integral Monte Carlo Determination of the Fourth-Order Virial Coefficient for Unitary Two-Component Fermi Gas with Zero-Range Interactions, Phys. Rev. Lett., Volume 116 (2016) no. 23, 230401 | DOI

[43] Yaqi Hou; Kaitlyn J. Morrell; Aleks J. Czejdo; J. E. Drut Fourth- and fifth-order virial expansion of harmonically trapped fermions at unitarity, Phys. Rev. Research, Volume 3 (2021) no. 3, 033099 | DOI

[44] Aleks J. Czejdo; Joaquim E. Drut; Yaqi Hou; Kaitlyn J. Morrell Toward an automated-algebra framework for high orders in the virial expansion of quantum matter, Condens. Matter, Volume 7 (2022) no. 1, 13 | DOI

[45] Yaqi Hou; J. E. Drut Fourth- and Fifth-Order Virial Coefficients from Weak Coupling to Unitarity, Phys. Rev. Lett., Volume 125 (2020) no. 5, 050403 | DOI | MR

[46] D. S. Petrov Three-body problem in Fermi gases with short-range interparticle interaction, Phys. Rev. A, Volume 67 (2003) no. 1, 010703 | DOI

[47] Shimpei Endo; Yvan Castin The interaction-sensitive states of a trapped two-component ideal Fermi gas and application to the virial expansion of the unitary Fermi gas (2021) (post-publication hal-01246611v5, https://hal.archives-ouvertes.fr/hal-01246611)

[48] Alexander L. Gaunt; Tobias F. Schmidutz; Igor Gotlibovych; Robert P. Smith; Zoran Hadzibabic Bose–Einstein condensation of atoms in a uniform potential, Phys. Rev. Lett., Volume 110 (2013) no. 20, 200406 | DOI

[49] Biswaroop Mukherjee; Zhenjie Yan; Parth B. Patel; Zoran Hadzibabic; Tarik Yefsah; Julian Struck; Martin W. Zwierlein Homogeneous Atomic Fermi Gases, Phys. Rev. Lett., Volume 118 (2017) no. 2, 123401 | DOI

[50] Tin-Lun Ho; Qi Zhou Obtaining the phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases, Nat. Phys., Volume 6 (2010) no. 2, pp. 131-134 | DOI

[51] R. Dum; M. Olshanii Gauge Structures in Atom-Laser Interaction : Bloch Oscillations in a Dark Lattice, Phys. Rev. Lett., Volume 76 (1996) no. 11, pp. 1788-1797 | DOI

[52] Ludovic Pricoupenko Isotropic contact forces in arbitrary representation : Heterogeneous few-body problems and low dimensions, Phys. Rev. A, Volume 83 (2011) no. 6, 062711 | DOI

[53] Wu-Ki Tung Group Theory in Physics, World Scientific, Philadelphie, 1985 | DOI | MR | Zbl

[54] Christophe Mora; Yvan Castin; Ludovic Pricoupenko Integral equations for the four-body problem, C. R. Physique, Volume 12 (2011) no. 1, pp. 71-85 | DOI

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Guest Editors

C. R. Phys (2023)

Three-body contact for fermions. I. General relations

Félix Werner; Xavier Leyronas

C. R. Phys (2024)

Integral equations for the four-body problem

Christophe Mora; Yvan Castin; Ludovic Pricoupenko

C. R. Phys (2011)