Ground-State Static Correlation Functions of the Lieb–Liniger Model

  • Guillaume LangEmail author
Part of the Springer Theses book series (Springer Theses)


In this chapter, I characterize a strongly-correlated, ultracold one-dimensional Bose gas on a ring through its equilibrium, static correlation functions. The gas is described by the Lieb–Liniger model, that corresponds to contact interactions. This model is arguably the most conceptually simple in the class of continuum quantum field theories, and the most studied as well.


  1. 1.
    M.K. Panfil, Density fluctuations in the 1D Bose gas. Ph.D. thesis (2013)Google Scholar
  2. 2.
    J. Yu-Zhu, C. Yang-Yang, G. Xi-Wen, Understanding many-body physics in one dimension from the Lieb-Liniger model. Chin. Phys. B 24, 050311 (2015)CrossRefGoogle Scholar
  3. 3.
    F. Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems. Lecture notes in Physics, vol. 940 (2017)zbMATHCrossRefGoogle Scholar
  4. 4.
    E.H. Lieb, W. Liniger, Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605 (1963)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    J.B. McGuire, Study of exactly soluble one-dimensional N-body problems. J. Math. Phys. 5, 622 (1964)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    G.E. Astrakharchik, J. Boronat, J. Casulleras, S. Giorgini, Beyond the Tonks-Girardeau gas: strongly correlated regime in quasi-one-dimensional Bose gases. Phys. Rev. Lett. 95, 190407 (2005)ADSCrossRefGoogle Scholar
  7. 7.
    M.T. Batchelor, M. Bortz, X.W. Guan, N. Oelkers, Evidence for the super Tonks-Girardeau gas. J. Stat. Mech. L10001 (2005)Google Scholar
  8. 8.
    E. Haller, M. Gustavsson, M.J. Mark, J.G. Danzl, R. Hart, G. Pupillo, H.-C. Nägerl, Realization of an excited, strongly correlated quantum gas phase. Science 325, 1222 (2009)ADSCrossRefGoogle Scholar
  9. 9.
    S. Chen, X.-W. Guan, X. Yin, L. Guan, M.T. Batchelor, Realization of effective super Tonks-Girardeau gases via strongly attractive one-dimensional Fermi gases. Phys. Rev. A 81, 031608(R) (2010)ADSCrossRefGoogle Scholar
  10. 10.
    G.E. Astrakharchik, Y.E. Lozovik, Super-Tonks-Girardeau regime in trapped one-dimensional dipolar gases. Phys. Rev. A 77, 013404 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    M.D. Girardeau, G.E. Astrakharchik, Super-Tonks-Girardeau state in an attractive one-dimensional dipolar gas. Phys. Rev. Lett. 109, 235305 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    J.N. Fuchs, A. Recati, W. Zwerger, Exactly solvable model of the BCS-BEC crossover. Phys. Rev. Lett. 93, 090408 (2004)ADSCrossRefGoogle Scholar
  13. 13.
    M.T. Batchelor, X.-W. Guan, J.B. Mc Guire, Ground state of 1D bosons with delta interaction: link to the BCS model. J. Phys. A 37, 497 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    T. Iida, M. Wadati, Exact analysis of \(\delta \)-function attractive fermions and repulsive Bosons in one-dimension. J. Phys. Soc. Jpn. 74, 1724 (2005)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    P. Calabrese, P. Le Doussal, Interaction quench in a Lieb-Liniger model and the KPZ equation with flat initial conditions. J. Stat. Mech. P05004 (2014)Google Scholar
  16. 16.
    A. De Luca, P. Le Doussal, Crossing probability for directed polymers in random media. Phys. Rev. E 92, 040102(R) (2015)CrossRefGoogle Scholar
  17. 17.
    M. Panchenko, The Lieb-Liniger model at the critical point as toy model for Black Holes (2015). arXiv:1510.04535v1[hep-th]
  18. 18.
    D. Flassig, A. Franca, A. Pritzel, Large-N ground state of the Lieb-Liniger model and Yang-Mills theory on a two-sphere. Phys. Rev. A 93, 013627 (2016)ADSCrossRefGoogle Scholar
  19. 19.
    L. Piroli, P. Calabrese, Local correlations in the attractive one-dimensional Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation. Phys. Rev. A 94, 053620 (2016)ADSCrossRefGoogle Scholar
  20. 20.
    M. Gaudin, La fonction d’onde de Bethe (Masson, Paris, 1983)zbMATHGoogle Scholar
  21. 21.
    M.A. Cazalilla, Bosonizing one-dimensional cold atomic gases. J. Phys. B: Atomic, Mol. Opt. Phys. 37, 7 S1 (2004)ADSCrossRefGoogle Scholar
  22. 22.
    I.S. Eliëns, On quantum seas. Ph.D. thesis (2017)Google Scholar
  23. 23.
    T.C. Dorlas, Orthogonality and completeness of the Bethe Ansatz Eigenstates of the nonlinear Schrödinger model. Commun. Math. Phys. 154(2), 347–376 (1993)ADSzbMATHCrossRefGoogle Scholar
  24. 24.
    K. Sakmann, A.I. Streltsov, O.E. Alon, L.S. Cederbaum, Exact ground state of finite Bose-Einstein condensates on a ring. Phys. Rev. A 72, 033613 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    E.P. Gross, Structure of a quantized vortex in boson systems. Il Nuovo Cimento 20, 454 (1961)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    L. Pitaevskii, Vortex lines in an imperfect Bose gas. Sov. Phys. JETP. 13, 451 (1961)MathSciNetGoogle Scholar
  27. 27.
    C.N. Yang, C.P. Yang, Thermodynamics of a one dimensional system of Bosons with repulsive delta function interaction. J. Math. Phys. 10, 1115 (1969)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    V. Hutson, The circular plate condenser at small separations. Proc. Camb. Phil. Soc. 59, 211–225 (1963)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    M.L. Mehta, Random Matrices. Pure and Applied Mathematics Series, 3rd edn. (2004)CrossRefGoogle Scholar
  30. 30.
    V.N. Popov, Theory of one-dimensional Bose gas with point interaction. Theor. Math. Phys. 30, 222 (1977)MathSciNetCrossRefGoogle Scholar
  31. 31.
    M. Wadati, Solutions of the Lieb-Liniger integral equation. J. Phys. Soc. Jpn 71, 2657 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    N. Bogoliubov, On the theory of superfluidity. J. Phys. USSR 11, 23 (1947)MathSciNetGoogle Scholar
  33. 33.
    M. Gaudin, Boundary energy of a Bose gas in one dimension. Phys. Rev. A 4, 1 (1971)CrossRefGoogle Scholar
  34. 34.
    T. Iida, M. Wadati, Exact analysis of \(\delta \)-function attractive fermions and repulsive Bosons in one-dimension. J. Phys. Soc. Jpn 74(6), 1724–1736 (2005)ADSzbMATHCrossRefGoogle Scholar
  35. 35.
    M. Takahashi, On the validity of collective variable description of Bose systems. Prog. Theor. Phys. 53, 386 (1975)ADSCrossRefGoogle Scholar
  36. 36.
    D.K. Lee, Ground-state energy of a many-particle Boson system. Phys. Rev. A 4, 1670 (1971)ADSCrossRefGoogle Scholar
  37. 37.
    D.K. Lee, Ground-state energy of a one-dimensional many-Boson system. Phys. Rev. A 9, 1760 (1974)ADSCrossRefGoogle Scholar
  38. 38.
    C.A. Tracy, H. Widom, On the ground state energy of the \(\delta \)-function Bose gas. J. Phys. A 49, 294001 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    T. Kaminaka, M. Wadati, Higher order solutions of Lieb-Liniger integral equation. Phys. Lett. A 375, 2460 (2011)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    T. Emig, M. Kardar, Probability distributions of line lattices in random media from the 1D Bose gas. Nucl. Phys. B 604, 479 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    S. Prolhac, Ground state energy of the \(\delta \)-Bose and Fermi gas at weak coupling from double extrapolation. J. Phys. A 50, 144001 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    R. Apéry, Irrationalité de \(\zeta (2)\) et \(\zeta (3)\). Société Mathématique de France, Astérisque 61, 11–13 (1979)zbMATHGoogle Scholar
  43. 43.
    M. Zvonarev, Correlations in 1D Boson and fermion systems: exact results. Ph.D. Thesis, Copenhagen University, Denmark (2005)Google Scholar
  44. 44.
    Z. Ristivojevic, Excitation spectrum of the Lieb-Liniger model. Phys. Rev. Lett. 113, 015301 (2014)ADSCrossRefGoogle Scholar
  45. 45.
    G. Lang, F. Hekking, A. Minguzzi, Ground-state energy and excitation spectrum of the Lieb-Liniger model : accurate analytical results and conjectures about the exact solution. SciPost Phys. 3, 003 (2017)ADSCrossRefGoogle Scholar
  46. 46.
    T.V. Rao, Capacity of the circular plate condenser: analytical solutions for large gaps between the plates. J. Phys. A 38, 10037–10056 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    G.E. Astrakharchik, J. Boronat, I.L. Kurbakov, Y.E. Lozovik, F. Mazzanti, Low-dimensional weakly interacting Bose gases: nonuniversal equations of state. Phys. Rev. A 81, 013612 (2010)ADSCrossRefGoogle Scholar
  48. 48.
    S. Prolhac, Private CommunicationGoogle Scholar
  49. 49.
    W. Xu, M. Rigol, Universal scaling of density and momentum distributions in Lieb-Liniger gases. Phys. Rev. A 92, 063623 (2015)ADSCrossRefGoogle Scholar
  50. 50.
    T. Cheon, T. Shigehara, Fermion-Boson duality of one-dimensional quantum particles with generalized contact interaction. Phys. Rev. Lett. 82, 2536–2539 (1999)ADSCrossRefGoogle Scholar
  51. 51.
    T. Kinoshita, T. Wenger, D.S. Weiss, Local pair correlations in one-dimensional Bose gases. Phys. Rev. Lett. 95, 190406 (2005)ADSCrossRefGoogle Scholar
  52. 52.
    B. Laburthe Tolra, K.M. O’Hara, J.H. Huckans, W.D. Phillips, S.L. Rolston, J.V. Porto, Observation of reduced three-body recombination in a correlated 1D degenerate Bose gas. Phys. Rev. Lett. 92, 190401 (2004)Google Scholar
  53. 53.
    J. Armijo, T. Jacqmin, K.V. Kheruntsyan, I. Bouchoule, Probing three-body correlations in a quantum gas using the measurement of the third moment of density fluctuations. Phys. Rev. Lett. 105, 230402 (2010)ADSCrossRefGoogle Scholar
  54. 54.
    E. Haller, M. Rabie, M.J. Mark, J.G. Danzl, R. Hart, K. Lauber, G. Pupillo, H.-C. Nägerl, Three-body correlation functions and recombination rates for Bosons in three dimensions and one dimension. Phys. Rev. Lett. 107, 230404 (2011)ADSCrossRefGoogle Scholar
  55. 55.
    D.M. Gangardt, G.V. Shlyapnikov, Stability and phase coherence of trapped 1D Bose gases. Phys. Rev. Lett. 90, 010401 (2003)ADSCrossRefGoogle Scholar
  56. 56.
    V.V. Cheianov, H. Smith, M.B. Zvonarev, Exact results for three-body correlations in a degenerate one-dimensional Bose gas. Phys. Rev. A 73, 051604(R) (2006)ADSCrossRefGoogle Scholar
  57. 57.
    V.V. Cheianov, H. Smith, M.B. Zvonarev, Three-body local correlation function in the Lieb-Liniger model: bosonization approach. J. Stat. Mech. P08015 (2006)Google Scholar
  58. 58.
    D.M. Gangardt, G.V. Shlyapnikov, Local correlations in a strongly interacting one-dimensional Bose gas. New J. Phys. 5, 79 (2003)ADSCrossRefGoogle Scholar
  59. 59.
    M. Kormos, G. Mussardo, A. Trombettoni, Expectation values in the Lieb-Liniger Bose gas. Phys. Rev. Lett. 103, 210404 (2009)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    E.J.K.P. Nandani, R.A. Römer, S. Tan, X.-W. Guan, Higher-order local and non-local correlations for 1D strongly interacting Bose gas. New J. Phys. 18, 055014 (2016)ADSCrossRefGoogle Scholar
  61. 61.
    B. Pozsgay, Local correlations in the 1D Bose gas from a scaling limit of the XXZ chain. J. Stat. Mech. P11017 (2011)Google Scholar
  62. 62.
    M. Jimbo, T. Miwa, Y. Môri, M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé Transcendent. Physica 1D, 80 (1980)ADSzbMATHGoogle Scholar
  63. 63.
    D.M. Gangardt, Universal correlations of trapped one-dimensional impenetrable Bosons. J. Phys. A 37, 9335–9356 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    H.G. Vaidya, C.A. Tracy, One-particle reduced density matrix of impenetrable Bosons in one dimension at zero temperature. Phys. Rev. Lett. 43, 1540 (1979)ADSCrossRefGoogle Scholar
  65. 65.
    P.J. Forrester, N.E. Frankel, T.M. Garoni, N.S. Witte, Finite one-dimensional impenetrable Bose systems: occupation numbers. Phys. Rev. A 67, 043607 (2003)ADSCrossRefGoogle Scholar
  66. 66.
    A. Lenard, Momentum distribution in the ground state of the one-dimensional system of impenetrable Bosons. J. Math. Phys. 5, 930 (1964)ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    F.D.M. Haldane, ‘Luttinger liquid theory’ of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas. J. Phys. C: Solid State Phys. 14, 2585 (1981)ADSCrossRefGoogle Scholar
  68. 68.
    N. Didier, A. Minguzzi, F.W.J. Hekking, Complete series for the off-diagonal correlations of a one-dimensional interacting Bose gas. Phys. Rev. A 80, 033608 (2009)ADSCrossRefGoogle Scholar
  69. 69.
    A. Shashi, M. Panfil, J.-S. Caux, A. Imambekov, Exact prefactors in static and dynamic correlation functions of one-dimensional quantum integrable models: applications to the Calogero-Sutherland, Lieb-Liniger, and XXZ models. Phys. Rev. B 85, 155136 (2012)ADSCrossRefGoogle Scholar
  70. 70.
    M. Olshanii, V. Dunjko, A. Minguzzi, G. Lang, Connection between nonlocal one-body and local three-body correlations of the Lieb-Liniger model. Phys. Rev. A 96, 033624 (2017)ADSCrossRefGoogle Scholar
  71. 71.
    E. Gutkin, Conservation laws for the nonlinear Schrödinger equation. Annales de l’I.H.P., section C, tome 2(1), 67–74 (1985)ADSzbMATHCrossRefGoogle Scholar
  72. 72.
    B. Davies, Higher conservation laws from the quantum non-linear Schrödinger equation. Phys. A 167, 433–456 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    B. Davies, V. Korepin, Higher conservation laws from the quantum non-linear Schrödinger equation (2011), arXiv:1109.6604v1
  74. 74.
    M. Olshanii, V. Dunjko, Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases. Phys. Rev. Lett. 91, 090401 (2003)ADSCrossRefGoogle Scholar
  75. 75.
    V. Dunjko, M. Olshanii, A Hermite-Padé perspective on the renormalization group, with an application to the correlation function of Lieb-Liniger gas. Jour. Phys. A 44, 055206 (2011)ADSzbMATHCrossRefGoogle Scholar
  76. 76.
    J.-S. Caux, P. Calabrese, N.A. Slavnov, One-particle dynamical correlations in the one-dimensional Bose gas. J. Stat. Mech. P01008 (2007)Google Scholar
  77. 77.
    C.A. Regal, M. Greiner, S. Giorgini, M. Holland, D.S. Jin, Momentum distribution of a Fermi gas of atoms in the BCS-BEC crossover. Phys. Rev. Lett. 95, 250404 (2005)ADSCrossRefGoogle Scholar
  78. 78.
    J.T. Stewart, J.P. Gaebler, T.E. Drake, D.S. Jin, Verification of universal relations in a strongly interacting Fermi gas. Phys. Rev. Lett. 104, 235301 (2010)ADSCrossRefGoogle Scholar
  79. 79.
    R.J. Wild, P. Makotyn, J.M. Pino, E.A. Cornell, D.S. Jin, Measurements of Tan’s contact in an atomic Bose-Einstein condensate. Phys. Rev. Lett. 108, 145305 (2012)ADSCrossRefGoogle Scholar
  80. 80.
    R. Chang, Q. Bouton, H. Cayla, C. Qu, A. Aspect, C.I. Westbrook, D. Clément, Momentum-resolved observation of thermal and quantum depletion in a Bose gas. Phys. Rev. Lett. 117, 235303 (2016)ADSCrossRefGoogle Scholar
  81. 81.
    T. Jacqmin, B. Fang, T. Berrada, T. Roscilde, I. Bouchoule, Momentum distribution of one-dimensional Bose gases at the quasicondensation crossover: theoretical and experimental investigation. Phys. Rev. A 86, 043626 (2012)ADSCrossRefGoogle Scholar
  82. 82.
    G.E. Astrakharchik, S. Giorgini, Correlation functions and momentum distribution of one-dimensional Bose systems. Phys. Rev. A 68, 031602(R) (2003)ADSCrossRefGoogle Scholar
  83. 83.
    N. Bleistein, R. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986)zbMATHGoogle Scholar
  84. 84.
    A. Minguzzi, P. Vignolo, M. Tosi, High-momentum tail in the Tonks gas under harmonic confinement. Phys. Lett. A 294, 222 (2002)ADSzbMATHCrossRefGoogle Scholar
  85. 85.
    S. Tan, Energetics of a strongly correlated Fermi gas. Ann. Phys. 323, 2952 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    S. Tan, Large momentum part of a strongly correlated Fermi gas. Ann. Phys. 323, 2971 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    S. Tan, Generalized virial theorem and pressure relation for a strongly correlated Fermi gas. Ann. Phys. 323, 2987 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    E. Braaten, D. Kang, L. Platter, Universal relation for identical Bosons from three-body physics. Phys. Rev. Lett. 106, 153006 (2011)ADSCrossRefGoogle Scholar
  89. 89.
    T. Busch, B.-G. Englert, K. Rza̧żewski, M. Wilkens, Two cold atoms in a harmonic trap. Found. Phys. 28(4) (1998)Google Scholar
  90. 90.
    I. Brouzos, P. Schmelcher, Construction of analytical many-body wave functions for correlated Bosons in a harmonic trap. Phys. Rev. Lett. 108, 045301 (2012)ADSCrossRefGoogle Scholar
  91. 91.
    B. Wilson, A. Foerster, C.C.N. Kuhn, I. Roditi, D. Rubeni, A geometric wave function for a few interacting bosons in a harmonic trap. Phys. Lett. A 378, 1065–1070 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    M.D. Girardeau, E.M. Wright, J.M. Triscari, Ground-state properties of a one-dimensional system of hard-core Bosons in a harmonic trap. Phys. Rev. A 63, 033601 (2001)ADSCrossRefGoogle Scholar
  93. 93.
    P. Vignolo, A. Minguzzi, One-dimensional non-interacting fermions in harmonic confinement: equilibrium and dynamical properties. J. Phys. B: At. Mol. Opt. Phys. 34, 4653–4662 (2001)ADSCrossRefGoogle Scholar
  94. 94.
    V.I. Yukalov, M.D. Girardeau, Fermi-Bose mapping for one-dimensional Bose gases. Laser Phys. Lett. 2(8), 375–382 (2005)ADSCrossRefGoogle Scholar
  95. 95.
    G. Lang, P. Vignolo, A. Minguzzi, Tan’s contact of a harmonically trapped one-dimensional Bose gas: strong-coupling expansion and conjectural approach at arbitrary interactions. Eur. Phys. J. Spec. Top. 226, 1583–1591 (2017)CrossRefGoogle Scholar
  96. 96.
    A.G. Volosniev, D.V. Fedorov, A.S. Jensen, N.T. Zinner, M. Valiente, Strongly interacting confined quantum systems in one dimension. Nat. Commun. 5, 5300 (2014)CrossRefGoogle Scholar
  97. 97.
    J. Decamp, J. Jünemann, M. Albert, M. Rizzi, A. Minguzzi, P. Vignolo, High-momentum tails as magnetic-structure probes for strongly correlated \(SU(\kappa )\) fermionic mixtures in one-dimensional traps. Phys. Rev. A 94, 053614 (2016)ADSCrossRefGoogle Scholar
  98. 98.
    P. Vignolo, A. Minguzzi, Universal contact for a Tonks-Girardeau gas at finite temperature. Phys. Rev. Lett. 110, 020403 (2013)ADSCrossRefGoogle Scholar
  99. 99.
    G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Schäfer, H. Hu, X.-J. Liu, J. Catani, C. Sias, M. Inguscio, L. Fallani, A one-dimensional liquid of fermions with tunable spin. Nat. Phys. 10, 198–201 (2014)CrossRefGoogle Scholar
  100. 100.
    C.N. Yang, Y. Yi-Zhuang, One-dimensional w-component fermions and Bosons with repulsive delta function interaction. Chin. Phys. Lett. 28, 020503 (2011)ADSCrossRefGoogle Scholar
  101. 101.
    X.-W. Guan, Z.-Q. Ma, B. Wilson, One-dimensional multicomponent fermions with \(\delta \)-function interaction in strong- and weak-coupling limits: \(\kappa \)-component Fermi gas. Phys. Rev. A 85, 033633 (2012)ADSCrossRefGoogle Scholar
  102. 102.
    N. Matveeva, G.E. Astrakharchik, One-dimensional multicomponent Fermi gas in a trap: quantum Monte Carlo study. New J. Phys. 18, 065009 (2016)ADSCrossRefGoogle Scholar
  103. 103.
    D.S. Petrov, D.M. Gangardt, G.V. Shlyapnikov, Low-dimensional trapped gases. J. Phys. IV France 116, 3–44 (2004)CrossRefGoogle Scholar
  104. 104.
    A.I. Gudyma, G.E. Astrakharchik, M. Zvonarev, Reentrant behavior of the breathing-mode-oscillation frequency in a one-dimensional Bose gas. Phys. Rev. A 92, 021601 (2014)ADSCrossRefGoogle Scholar
  105. 105.
    S. Choi, V. Dunjko, Z.D. Zhang, M. Olshanii, Monopole excitations of a harmonically trapped one-dimensional Bose gas from the ideal gas to the Tonks-Girardeau regime. Phys. Rev. Lett. 115, 115302 (2015)ADSCrossRefGoogle Scholar
  106. 106.
    J. De Nardis, M. Panfil, Exact correlations in the Lieb-Liniger model and detailed balance out of equilibrium. SciPost Phys. 1, 015 (2016)ADSCrossRefGoogle Scholar
  107. 107.
    H.E. Boos, V.E. Korepin, Quantum spin chains and Riemann zeta function with odd arguments. J. Phys. A: Math. Gen. 34, 5311–5316 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    T. Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris, t. 331, Série I, 267–270, Théorie des nombres/Number Theory (2000)Google Scholar
  109. 109.
    C.A. Tracy, H. Widom, On the ground state energy of the delta-function Fermi gas II: Further asymptotics (2016), arXiv:1609.07793v1 [math-ph]
  110. 110.
    Y. Brun, J. Dubail, One-particle density matrix of trapped one-dimensional impenetrable Bosons from conformal invariance. SciPost Phys. 2, 012 (2017)ADSCrossRefGoogle Scholar
  111. 111.
    M.-S. Wang, J.-H. Huang, C.-H. Lee, X.-G. Yin, X.-W. Guan, M.T. Batchelor, Universal local pair correlations of Lieb-Liniger Bosons at quantum criticality. Phys. Rev. A 87, 043634 (2013)ADSCrossRefGoogle Scholar
  112. 112.
    G. Lang, F. Hekking, A. Minguzzi, Dynamic structure factor and drag force in a one-dimensional Bose gas at finite temperature. Phys. Rev. A 91, 063619 (2015)ADSCrossRefGoogle Scholar
  113. 113.
    G. De Rosi, G.E. Astrakharchik, S. Stringari, Thermodynamic behavior of a one-dimensional Bose gas at low temperature. Phys. Rev. A 96, 013613 (2017)ADSCrossRefGoogle Scholar
  114. 114.
    K.V. Kheruntsyan, D.M. Gangardt, P.D. Drummond, G.V. Shlyapnikov, Pair correlations in a finite-temperature 1D Bose gas. Phys. Rev. Lett. 91, 040403 (2003)ADSCrossRefGoogle Scholar
  115. 115.
    A.G. Sykes, D.M. Gangardt, M.J. Davis, K. Viering, M.G. Raizen, K.V. Kheruntsyan, Spatial nonlocal pair correlations in a repulsive 1D Bose gas. Phys. Rev. Lett. 100, 160406 (2008)ADSCrossRefGoogle Scholar
  116. 116.
    P. Deuar, A.G. Sykes, D.M. Gangardt, M.J. Davis, P.D. Drummond, K.V. Kheruntsyan, Nonlocal pair correlations in the one-dimensional Bose gas at finite temperature. Phys. Rev. A 79, 043619 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Laboratoire de Physique et de Modélisation des Milieux Condensés (LPMMC)GrenobleFrance

Personalised recommendations