Geometric and Probabilistic Aspects of Boson Lattice Models

  • Daniel Ueltschi
Part of the Progress in Probability book series (PRPR, volume 51)


This review describes quantum systems of bosonic particles moving on a lattice. These models are relevant in statistical physics, and have natural ties with probability theory. The general setting is recalled and the main questions about phase transitions are addressed. A lattice model with Lennard-Jones potential is studied as an example of a system where first-order phase transitions occur.

A major interest of bosonic systems is the possibility of displaying a Bose-Einstein condensation. This is discussed in the light of the main existing rigorous result, namely its occurrence in the hard-core boson model. Finally, we consider another approach that involves the lengths of the cycles formed by the particles in the space-time representation; Bose-Einstein condensation should be related to positive probability of infinite cycles.


Partition Function Probabilistic Aspect Quantum Lattice System Unique Tangent Infinite Cycle 
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  1. 1.
    M. Aizenman and B. Nachtergaele, Geometric aspects of quantum spin statesCommun. Math. Phys.164 (1994), 17–63.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vaporScience269 (1995), 198–202.CrossRefGoogle Scholar
  3. 3.
    C. Borgs, R. Koteck’, and D. Ueltschi, Low temperature phase diagrams for quantum perturbations of classical spin systemsCommun. Math. Phys.181 (1996), 409–446.MATHCrossRefGoogle Scholar
  4. 4.
    C. Borgs, R. Koteck’, and D. Ueltschi, Incompressible phase in lattice systems of interacting bosons, unpublished, 1997.Google Scholar
  5. 5.
    A. Bovier and M. Zahradník, A simple inductive approach to the problem of convergence of cluster expansions of polymer modelsJ. Stat. Phys.100 (2000), 765–778.MATHCrossRefGoogle Scholar
  6. 6.
    M. Cassandro and P. Picco, Existence of a phase transition in a continuous quantum systemsJ. Stat. Phys.103 (2001), 841–856.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J. Conlon and J.P. Solovej, Random walk representations of the Heisenberg modelJ. Stat. Phys.64 (1991), 251–270.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Conlon and J.P. Solovej, Upper bound on the free energy of the spin 1/2 Heisenberg ferromagnetLett. Math. Phys.23 (1991), 223–231.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    N. Datta, R. Fernández, and J. Fröhlich, Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground statesJ. Stat. Phys.84 (1996), 455–534.MATHCrossRefGoogle Scholar
  10. 10.
    N. Datta, R. Fernández, J. Fröhlich, and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracyHell). Phys. Acta69 (1996), 752–820.MATHGoogle Scholar
  11. 11.
    R.L. Dobrushin, Estimates of semi-invariants for the Ising model at low temperatures. InTopics of Statistical and Theoretical PhysicsAmerican Mathematical Society Transi. Ser. 2, 177, pp. 59–81, 1996.MathSciNetGoogle Scholar
  12. 12.
    F.J. Dyson, E.H. Lieb, and B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactionsJ. Stat. Phys.18 (1978), 335–383.CrossRefMathSciNetGoogle Scholar
  13. 13.
    R.P. Feynman, Atomic theory of the A transition in HeliumPhys. Rev.91 (1953), 1291–1301.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M.P.A. Fisher, P.B. Weichman, G. Grinstein, and D. Fisher, Boson localization and the superfluid-insulator transitionPhys. Rev. B40 (1989), 546–570.CrossRefGoogle Scholar
  15. 15.
    J. Fröhlich, L. Rey-Bellet, and D. Ueltschi, Quantum lattice models at intermediate temperatures, math-ph/0012011Commun. Math. Phys.224 (2001), 33–63.MATHCrossRefGoogle Scholar
  16. 16.
    J. Fröhlich, B. Simon, and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breakingCommun. Math. Phys.50 (1976), 7995.CrossRefGoogle Scholar
  17. 17.
    J. Ginibre, Some applications of functional integration in statistical mechanics. InStatistical Mechanics and Field Theory(C. De Witt and R. Stora, eds.), Gordon and Breach, 1971.Google Scholar
  18. 18.
    D. Ioffe, A note on the quantum Widom-Rowlinson modelJ. Stat. Phys.106 (2002), 375–384.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    R.B. Israel, Convexity in the Theory of Lattice Gases, Princeton University Press, 1979.Google Scholar
  20. 20.
    T. Kennedy, E.H. Lieb, and B.S. Shastry, The X-Y model has long-range order for all spins and all dimensions greater than onePhys. Rev. Lett.61 (1988), 2582–2584.CrossRefGoogle Scholar
  21. 21.
    R. KoteckÿPhase transitions of lattice models Rennes Lectures (1996).Google Scholar
  22. 22.
    R. Koteckÿ and D. Preiss, Cluster expansion for abstract polymer modelsCommun. Math. Phys.103 (1986), 491–498.MATHCrossRefGoogle Scholar
  23. 23.
    R. Koteckÿ and D. Ueltschi, Effective interactions due to quantum fluctuationsCommun. Math. Phys.206 (1999), 289–335.MATHCrossRefGoogle Scholar
  24. 24.
    J.L. Lebowitz, M. Lenci, and H. Spohn, Large deviations for ideal quantum systemsJ. Math. Phys.41 (2000), 1224–1243.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    E.H. Lieb, The Bose fluid. InLectures in Theoretical PhysicsVol.VII C (W.E. Brittin ed.), Univ. of Colorado Press, pp. 175–224, 1965.Google Scholar
  26. 26.
    E.H. Lieb, The Bose gas: A subtle many-body problem. InProceedings of the XIII Internat. Congress on Math. Physics, International Press, London, 2001.Google Scholar
  27. 27.
    E.H. Lieb and J. Yngvason, Ground state energy of the low density Bose gasPhys. Rev. Lett.80 (1998), 2504–2507.CrossRefGoogle Scholar
  28. 28.
    O. Penrose and L. Onsager, Bose-Einstein condensation and liquid HeliumPhys. Rev.104 (1956), 576–584.MATHCrossRefGoogle Scholar
  29. 29.
    Ch.-E. Pfister, Thermodynamical aspects of classical lattice systems, this volume, pp. 393–472.Google Scholar
  30. 30.
    S.A. Pirogov and Ya.G. Sinai, Phase diagrams of classical lattice systemsTheoretical and Mathematical Physics25 (1975), 1185–1192; 26 (1976), 39–49.CrossRefMathSciNetGoogle Scholar
  31. 31.
    B. SimonThe Statistical Mechanics of Lattice GasesPrinceton University Press, 1993.Google Scholar
  32. 32.
    Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon Press, 1982.Google Scholar
  33. 33.
    A. Sütô, Percolation transition in the Bose gasJ. Phys. A26 (1993), 4689–4710.CrossRefMathSciNetGoogle Scholar
  34. 34.
    A. Süt¨®, Non-uniform ground state for the Bose gasJ. Phys. A34 (1993), 37–55.CrossRefGoogle Scholar
  35. 35.
    B. T¨®th, Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnetLett. Math. Phys.28 (1993), 75–84.CrossRefMathSciNetGoogle Scholar
  36. 36.
    D. Ueltschi, Analyticity in Hubbard modelsJ. Stat. Phys.95 (1999), 693–717.MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    V. Zagrebnov and J.-B. Bru, The Bogoliubov model of weakly imperfect Bose gasPhys. Rep.350 (2001), 291–434.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Daniel Ueltschi
    • 1
  1. 1.Department of PhysicsPrinceton UniversityPrinceton

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