The Quantum-Mechanical Many-Body Problem: The Bose Gas

  • Elliott H. Lieb
  • Robert Seiringer
  • Jan Philip Solovej
  • Jakob Yngvason
Chapter

Abstract

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4–5 decades ago, and to explore new regimes not treated before. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous—and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πћ 2aρ/m where a is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, several other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the ‘traps’ actually used in the experiments. For this system it is also possible to prove complete Bose condensation and superfluidity, as we have shown. On the frontier of experimental developments is the possibility that a dilute gas in an elongated trap will behave like a one-dimensional system; we have proved this mathematically. Another topic is a proof that Foldy’s 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy; using this we can also prove the N7/5 formula for the ground state energy of the two-component charged Bose gas proposed by Dyson in 1967. All of this is quite recent work and it is hoped that the mathematical methodology might be useful, ultimately, to solve more complex problems connected with these interesting systems.

Keywords

Vortex Coherence Weinstein Cylin Hines 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AG]
    G.E. Astrakharchik and S. Giorgini, Quantum Monte Carlo study of the three-to one-dimensional crossover for a trapped Bose gas, Phys. Rev. A 66, 0536141–6 (2002).Google Scholar
  2. [ABCG]
    G.E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Superfluidity versus Bose-Einstein condensation in a Bose gas with disorder, Phys. Rev. A 66, 023603 (2002).Google Scholar
  3. [Ha]
    B. Baumgartner, The Existence of Many-particle Bound States Despite a Pair Interaction with Positive Scattering Length, J. Phys. A 30 (1997), L741 — L747.ADSCrossRefGoogle Scholar
  4. Bm]G.Baym, in: Math. Methods in Solid State and Superfluid Theory, Scottish Univ Summer School of Physics, Oliver and Boyd, Edinburgh (1969).Google Scholar
  5. [Be] F.A. Berezin, Izv. Akad. Nauk, ser. mat., 36 (No. 5) (1972); English translation: USSR Izv. 6 (No. 5)
    ). F.A. Berezin, General concept of quantization, Commun. Math. Phys. 40, 153–174 (1975).Google Scholar
  6. [BI]
    D. Blume, Fermionization of a bosonic gas under highly elongated confinement: A diffusion quantum Monte Carlo study, Phys. Rev. A 66, 053613–1–8 (2002).Google Scholar
  7. [Bo]
    N.N. Bogolubov, J. Phys. (U.S.S.R.) 11, 23 (1947); N.N. Bogolubov, D.N. Zubarev, Soy. Phys.-JETP 1, 83 (1955).Google Scholar
  8. [BBD]
    K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Artl, W. Ertmer, and K. Sengstok, Waveguides for Bose-Einstein condensates, Phys. Rev. A, 63, 031602 (2001)Google Scholar
  9. [B]
    S.N. Bose, Plancks Gesetz and Lichtquantenhypothese, Z. Phys. 26, 178–181 (1924).MATHCrossRefGoogle Scholar
  10. [CS1]
    A.Y. Cherny, A.A. Shanenko, Dilute Bose gas in two dimensions: Density expansions and the Gross-Pitaevskii equation, Phys. Rev. E 64, 027105 (2001)Google Scholar
  11. [CS2]
    A.Y. Cherny, A.A. Shanenko, The kinetic and interaction energies of a trapped Bose gas: Beyond the mean field, Phys. Lett. A 293, 287 (2002).ADSMATHCrossRefGoogle Scholar
  12. [CLY]
    J. Conlon, E.H. Lieb, H.-T. Yau, The N7í5 Law for Charged Bosons, Commun. Math. Phys. 116, 417–448 (1988).MathSciNetADSGoogle Scholar
  13. [CCRCW]
    S.L. Cornish, N.R. Claussen, J.L. Roberts, E.A. Cornell, C.E. Wieman, Stable 85 Ró Bose-Einstein Condensates with Widely Tunable Interactions, Phys. Rev. Lett. 85, 1795–98 (2000).ADSCrossRefGoogle Scholar
  14. [DGPS]
    F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. S.ringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463–512 (1999).ADSCrossRefGoogle Scholar
  15. [DOW]
    K.K. Das, M.D. Girardeau, and E.M. Wright, Crossover from One to Three Dimensions for a Gas of Hard–Core Bosons, Phys. Rev. Lett. 89, 110402–1–4 (2002).Google Scholar
  16. [DLO]
    V. Dunjko, V. Lorent, and M. Olshanii, Bosons in Cigar-Shaped Traps: Thomas-Fermi Regime, Tonks-Girardeau Regime, and In Between, Phys. Rev. Lett. 86, 5413–5316 (2001).ADSCrossRefGoogle Scholar
  17. [D1]
    F.J. Dyson, Ground-State Energy of a Hard-Sphere Gas, Phys. Rev. 106, 20–26 (1957).MATHCrossRefGoogle Scholar
  18. [D2]
    F.J. Dyson, Ground State Energy of a Finite System of Charged Particles, J. Math. Phys. 8, 1538–1545 (1967).MathSciNetADSCrossRefGoogle Scholar
  19. [DLS]
    F.J. Dyson, E.H. Lieb, B. Simon, Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions, J. Stat. Phys. 18, 335–383 (1978).MathSciNetADSCrossRefGoogle Scholar
  20. [E]
    A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzber. Kgl. Preuss. Akad. Wiss., 261–267 (1924), and 3–14 (1925).Google Scholar
  21. [FS]
    A.L. Fetter and A.A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein condensate, J. Phys.: Condens. Matter 13, R135 (2001).ADSCrossRefGoogle Scholar
  22. [FH]
    D.S. Fisher, P.C. Hohenberg, Dilute Bose gas in two dimensions, Phys. Rev. B 37, 4936–4943 (1988).Google Scholar
  23. [F]
    L.L. Foldy, Charged Boson Gas, Phys. Rev. 124, 649–651 (1961); Errata ibid 125, 2208 (1962).Google Scholar
  24. [Gi]
    M.D. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension, J. Math. Phys. 1, 516 (1960).ADSCrossRefGoogle Scholar
  25. [GW]
    M.D. Girardeau and E.M. Wright, Bose–Fermi variational Theory for the BEC–Tonks Crossover, Phys. Rev. Lett. 87, 210402–1–4 (2001).Google Scholar
  26. [GWT]
    M.D. Girardeau, E.M. Wright, and J.M. Triscari, Ground–state properties of a one–dimensional system of hard–core bosons in a harmonic trap, Phys. Rev. A 63, 033601–1–6 (2001).Google Scholar
  27. Go] A. Görlitz, et al., Realization of Bose–Einstein Condensates in Lower Dimen–sion,Phys. Rev. Lett. 87 130402–1–4 (2001).Google Scholar
  28. [GS]
    G.M. Graf and J.P. Solovej, A correlation estimate with applications to quantum systems with Coulomb interactions, Rev. Math. Phys. 6, 977–997 (1994).MathSciNetMATHCrossRefGoogle Scholar
  29. G] M. Greiner, et al., Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates,Phys. Rev. Lett. 87 160405 (2001).Google Scholar
  30. [Grl]
    E.P. Gross, Structure of a Quantized Vortex in Boson Systems, Nuovo Cimento 20, 454–466 (1961).MATHCrossRefGoogle Scholar
  31. [Gr2]
    E.P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys. 4, 195–207 (1963).ADSCrossRefGoogle Scholar
  32. [HFM]
    D.F. Hines, N.E. Frankel, D.J. Mitchell, Hard disc Bose gas, Phys. Lett. 68A, 12–14 (1978).CrossRefGoogle Scholar
  33. [Ho]
    P.C. Hohenberg, Existence of Long-range Order in One and Two Dimensions, Phys. Rev. 158, 383–386 (1966).Google Scholar
  34. HoM] P.C. Hohenberg and P.C. Martin, Microscopic theory of helium, Ann. Phys. (NY) 34, 291 (1965).ADSCrossRefGoogle Scholar
  35. [H]
    K. Huang, in: Bose-Einstein Condensation, A. Griffin, D.W. Stroke, S. Stringari, eds., Cambridge University Press, 31–50 (1995).Google Scholar
  36. [HY]
    K. Huang, C.N. Yang, Phys. Rev. 105, 767–775 (1957); T.D. Lee, K. Huang, C. N. Yang, Phys. Rev. 106, 1135–1145 (1957); K.A. Brueckner, K. Sawada, Phys. Rev. 106, 1117–1127, 1128–1135 (1957); S.T. Beliaev, Soy. Phys.-JETP 7, 299–307 (1958); T.T. Wu, Phys. Rev. 115, 1390 (1959); N. Hugenholtz, D. Pines, Phys. Rev. 116, 489 (1959); M. Girardeau, R. Arnowitt, Phys. Rev. 113, 755 (1959); T.D. Lee, C.N. Yang, Phys. Rev. 117, 12 (1960).Google Scholar
  37. [JK]
    A.D. Jackson and G.M. Kavoulakis, Lieb Mode in a Quasi-One-Dimendional Bose-Einstein Condensate of Atoms, Phys. Rev. Lett. 89, 070403 (2002).Google Scholar
  38. [KLS]
    T. Kennedy, E.H. Lieb, S. S.astry, The XY Model has Long-Range Order for all Spins and all Dimensions Greater than One, Phys. Rev. Lett. 61, 25822584 (1988).Google Scholar
  39. [KD]
    W. Ketterle, N. J. van Druten, Evaporative Cooling of Trapped Atoms, in B. Bederson, H. Walther, eds., Advances in Atomic, Molecular and Optical Physics, 37, 181–236, Academic Press (1996).Google Scholar
  40. [KNSQ]
    E.B. Kolomeisky, T.J. Newman, J.P. Straley, X. Qi, Low-dimensional Bose liquids: beyond the Gross-Pitaevskii approximation, Phys. Rev. Lett. 85, 11461149 (2000).Google Scholar
  41. [KT]
    M. Kobayashi and M. Tsubota, Bose-Einstein condensation and superfluidity of a dilute Bose gas in a random potential, Phys. Rev. B 66, 174516 (2002).Google Scholar
  42. [KP]
    S. Komineas and N. Papanicolaou, Vortex Rings and Lieb Modes in a Cylin- drical Bose-Einstein Condensate, Phys. Rev. Lett. 89, 070402 (2002).Google Scholar
  43. [Le]
    A. Lenard, Momentum distribution in the ground stat of the one-dimensional system of impenetrable bosons, J. Math. Phys. 5, 930–943 (1964).ADSCrossRefGoogle Scholar
  44. L1] E.H. Lieb, Simplified Approach to the Ground State Energy of an Imperfect Bose Gas,Phys. Rev. 130 2518–2528 (1963). See also Phys. Rev. 133 (1964), A899–A906 (with A.Y. Sakakura) and Phys. Rev. 134 (1964), A312–A315 (with W. Liniger).Google Scholar
  45. [L2]
    E.H. Lieb, The Bose fluid, in W.E. Brittin, ed., Lecture Notes in Theoretical Physics VIIC, Univ. of Colorado Press, pp. 175–224 (1964).Google Scholar
  46. [L3]
    E.H. Lieb, The classical limit of quantum spin systems, Commun. Math. Phys. 31, 327–340 (1973).ADSGoogle Scholar
  47. [L4]
    E.H. Lieb, The Bose Gas: A Subtle Many-Body Problem, in Proceedings of the XIII International Congress on Mathematical Physics, London, A. Fokas, et al. eds. International Press, pp. 91–111, 2001.Google Scholar
  48. [LL]
    E.H. Lieb, W. Liniger, Exact Analysis of an Interacting Bose Gas.I . The Gen-eral Solution and the Ground State, Phys. Rev. 130, 1605–1616 (1963); E.H. Lieb, Exact Analysis of an Interacting Bose Gas. II . The Excitation Spectrum, Phys. Rev. 130, 1616–1624 (1963).MathSciNetMATHCrossRefGoogle Scholar
  49. [LLo]
    E.H. Lieb, M. Loss, Analysis, 2nd ed., Amer. Math. Society, Providence, R.I. (2001).MATHGoogle Scholar
  50. [LN] E.H. Lieb, H. Narnhofer, The Thermodynamic Limit for Jellium, J. Stat. Phys. 12, 291-310 (1975).
    Errata J. Stat. Phys. 14, 465 (1976).Google Scholar
  51. [LSe]
    E.H. Lieb, R. Seiringer, Proof of Bose–Einstein Condensation for Dilute Trapped Gases, Phys. Rev. Lett. 88, 170409–1–4 (2002).Google Scholar
  52. [LSSY]
    E.H. Lieb, R. Seiringer, J.P. Solovej, and J. Yngvason, The ground state of the Bose gas, in: Current Developments in Mathematics, 2001, 131–178, International Press, Cambridge (2002).Google Scholar
  53. [LSeY1]
    E.H. Lieb, R. Seiringer, J. Yngvason, Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional, Phys. Rev A 61, 043602 (2000).Google Scholar
  54. LSeY2] EH. Lieb, R. Seiringer, J. Yngvason, A Rigorous Derivation of the GrossPitaevskii Energy Functional for a Two-dimensional Bose Gas, Commun. Math. Phys. 224, 17 (2001).ADSGoogle Scholar
  55. [LSeY3]
    E.H. Lieb, R. Seiringer, J. Yngvason, The Ground State Energy and Density of Interacting Bosons in a Trap, in Quantum Theory and Symmetries, Goslar, 1999, H.-D. Doebner, V.K. Dobrev, J.-D. Hennig and W. Luecke, eds., pp. 101–110, World Scientific (2000).Google Scholar
  56. [LSeY4]
    E.H. Lieb, R. Seiringer, J. Yngvason, Two-Dimensional Gross-Pitaevskii Theory, in: Progress in Nonlinear Science, Proceedings of the International Conference Dedicated to the 100th Anniversary of A.A. Andronov, Volume II, A.G. Litvak, ed., 582–590, Nizhny Novgorod, Institute of Applied Physics, University of Nizhny Novgorod (2002).Google Scholar
  57. [LSeY5]
    E.H. Lieb, R. Seiringer, J. Yngvason, Superfluidity in Dilute Trapped Bose Gases, Phys. Rev. B 66, 134529 (2002).Google Scholar
  58. [LSeY6]
    E.H. Lieb, R. Seiringer, J. Yngvason, One–Dimensional Behavior of Dilute, Trapped Bose Gases, Commun. Math. Phys. 244, 347–393 (2004). See also: One–Dimensional Bosons in Three–Dimensional Traps, Phys. Rev. Lett. 91, 150401–1–4 (2003).Google Scholar
  59. [LSeY7]
    E.H. Lieb, R. Seiringer, and J. Yngvason, Poincaré Inequalities in Punctured Domains, Ann. Math. 158, 1067–1080 (2003).MathSciNetMATHCrossRefGoogle Scholar
  60. [LSo]
    E.H. Lieb, J.P. Solovej, Ground State Energy of the One-Component Charged Bose Gas, Commun. Math. Phys. 217, 127–163 (2001). Errata 225, 219–221 (2002).MathSciNetGoogle Scholar
  61. LSo2] E.H. Lieb, J.P. Solovej, Ground State Energy of the Two-Component Charged Bose Gas,Commun. Math. Phys. (in press), arxiv:math-ph/0311010, mp..arc 03–490.Google Scholar
  62. [LY1]
    E.H. Lieb, J. Yngvason, Ground State Energy of the low density Bose Gas, Phys. Rev. Lett. 80, 2504–2507 (1998).ADSGoogle Scholar
  63. [LY2]
    E.H. Lieb, J. Yngvason, The Ground State Energy of a Dilute Two-dimensional Bose Gas, J. Stat. Phys. 103, 509 (2001).MathSciNetMATHGoogle Scholar
  64. [LY3]
    E.H. Lieb, J. Yngvason, The Ground State Energy of a Dilute Bose Gas, in Differential Equations and Mathematical Physics, University of Alabama, Birmingham, 1999, R. Weikard and G. Weinstein, eds., 271–282 Amer. Math. Soc./Internat. Press (2000).Google Scholar
  65. [MSKE]
    H. Moritz, T. Stöferle, M. Köhl and T. Esslinger, Exciting Collective Oscillations in a Trapped 1D Gas, Phys. Rev. Lett. 91, 250402 (2003).Google Scholar
  66. M] W.J. Mullin, Bose-Einstein Condensation in a Harmonic Potential,J. LowGoogle Scholar
  67. Temp. Phys. 106, 615–642 (1997).Google Scholar
  68. [01]
    M. Olshanii, Atomic Scattering in the Presence of an External Confinement Google Scholar
  69. and a Gas of Impenetrable Bosons,Phys. Rev. Lett. 81 938–941 (1998).Google Scholar
  70. [0]
    A.A. Ovchinnikov, On the description of a two-dimensional Bose gas at low densities, J. Phys. Condens. Matter 5, 8665–8676 (1993). See also JETP Letters 57, 477 (1993); Mod. Phys. Lett. 7, 1029 (1993).Google Scholar
  71. [PS]
    C. Pethick, H. Smith, Bose Einstein Condensation of Dilute Gases, Cambridge University Press, 2001.Google Scholar
  72. [PSW]
    D.S. Petrov, G.V. Shlyapnikov, and J.T.M. Walraven, Regimes of Quantum Degeneracy in Trapped ID Gases, Phys. Rev. Lett. 85, 3745–3749 (2000).ADSCrossRefGoogle Scholar
  73. [Pi]
    L.P. Pitaenskii,Vortes in an imperfect Bose gas, Sov. Phys.JETP. 13, 451–454 (1961).Google Scholar
  74. PiSt] L. Pitaevskii, S. Stringari, Uncertainty Principle, Quantum Fluctuations, and Broken Symmetries, J. Low Temp. Phys. 85, 377 (1991).ADSCrossRefGoogle Scholar
  75. [Po]
    V.N. Popov, On the theory of the superfluidity of two-and one-dimensional Bose systems, Theor. and Math. Phys. 11, 565–573 (1977).Google Scholar
  76. PrSv] N.V. Prokof’ev and B.V. Svistunov, Two definitions of superfluid density, Phys. Rev. B 61, 11282 (2000).CrossRefGoogle Scholar
  77. [S]
    M. Schick, Two-Dimensional System of Hard Core Bosons, Phys. Rev. A 3, 1067–1073 (1971).CrossRefGoogle Scholar
  78. Sc] F. Schreck, et al., Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea,Phys. Rev. Lett. 87 080403 (2001).Google Scholar
  79. [Sel]
    R. Seiringer, Diplom thesis, University of Vienna, 1999.Google Scholar
  80. [Se2]
    R. Seiringer, Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii Ground State Energy Formula, in: Partial Differential Equations and Spectral Theory, PDE2000 Conference in Clausthal, Germany, M. Demuth and B.-W. Schulze, eds., 307–314, Birkhäuser (2001).Google Scholar
  81. [Se3]
    R. Seiringer, Gross-Pitaevskii Theory of the Rotating Bose Gas, Commun. Math. Phys. 229, 491–509 (2002); Ground state asymptotics of a dilute, rotating gas, J. Phys. A: Math. Gen. 36, 9755–9778 (2003).Google Scholar
  82. [Sh]
    S.I. Shevchenko, On the theory of a Bose gas in a nonuniform field, Sov. J. Low Temp. Phys. 18, 223–230 (1992).MathSciNetGoogle Scholar
  83. [Si]
    B. Simon, 7’race ideals and their application, Cambridge University Press (1979).Google Scholar
  84. So] J.P. Solovej, Upper Bounds to the Ground State Energies of the One-and Two-Component Charged Bose gases,preprint, arxiv:math-ph/0406014.Google Scholar
  85. [T]
    G. Temple, The theory of Rayleigh’s Principle as Applied to Continuous Sys- tems, Proc. Roy. Soc. London A 119, 276–293 (1928).ADSMATHCrossRefGoogle Scholar
  86. [TT]
    D.R. Tilley and J. Tilley, Superfluidity and Superconductivity, third edition, Adam Bulger, Bristol and New York (1990).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Robert Seiringer
    • 2
  • Jan Philip Solovej
    • 3
    • 4
  • Jakob Yngvason
    • 5
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Department of PhysicsPrinceton UniversityPrincetonUSA
  3. 3.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  4. 4.On Leave From Dept. of MathUniversity of CopenhagenCopenhagenDenmark
  5. 5.Institut fur Theoretische PhysikUniversität WienViennaAustria

Personalised recommendations