Abstract
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.
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References
M. Aizenman and B. Nachtergaele, Geometric aspects of quantum spin statesCommun. Math. Phys.164 (1994), 17–63.
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.
C. Borgs, R. Koteck’, and D. Ueltschi, Low temperature phase diagrams for quantum perturbations of classical spin systemsCommun. Math. Phys.181 (1996), 409–446.
C. Borgs, R. Koteck’, and D. Ueltschi, Incompressible phase in lattice systems of interacting bosons, unpublished, 1997.
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.
M. Cassandro and P. Picco, Existence of a phase transition in a continuous quantum systemsJ. Stat. Phys.103 (2001), 841–856.
J. Conlon and J.P. Solovej, Random walk representations of the Heisenberg modelJ. Stat. Phys.64 (1991), 251–270.
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.
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.
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.
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.
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.
R.P. Feynman, Atomic theory of the A transition in HeliumPhys. Rev.91 (1953), 1291–1301.
M.P.A. Fisher, P.B. Weichman, G. Grinstein, and D. Fisher, Boson localization and the superfluid-insulator transitionPhys. Rev. B40 (1989), 546–570.
J. Fröhlich, L. Rey-Bellet, and D. Ueltschi, Quantum lattice models at intermediate temperatures, math-ph/0012011Commun. Math. Phys.224 (2001), 33–63.
J. Fröhlich, B. Simon, and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breakingCommun. Math. Phys.50 (1976), 7995.
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.
D. Ioffe, A note on the quantum Widom-Rowlinson modelJ. Stat. Phys.106 (2002), 375–384.
R.B. Israel, Convexity in the Theory of Lattice Gases, Princeton University Press, 1979.
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.
R. KoteckÿPhase transitions of lattice models Rennes Lectures (1996).
R. Koteckÿ and D. Preiss, Cluster expansion for abstract polymer modelsCommun. Math. Phys.103 (1986), 491–498.
R. Koteckÿ and D. Ueltschi, Effective interactions due to quantum fluctuationsCommun. Math. Phys.206 (1999), 289–335.
J.L. Lebowitz, M. Lenci, and H. Spohn, Large deviations for ideal quantum systemsJ. Math. Phys.41 (2000), 1224–1243.
E.H. Lieb, The Bose fluid. InLectures in Theoretical PhysicsVol.VII C (W.E. Brittin ed.), Univ. of Colorado Press, pp. 175–224, 1965.
E.H. Lieb, The Bose gas: A subtle many-body problem. InProceedings of the XIII Internat. Congress on Math. Physics, International Press, London, 2001.
E.H. Lieb and J. Yngvason, Ground state energy of the low density Bose gasPhys. Rev. Lett.80 (1998), 2504–2507.
O. Penrose and L. Onsager, Bose-Einstein condensation and liquid HeliumPhys. Rev.104 (1956), 576–584.
Ch.-E. Pfister, Thermodynamical aspects of classical lattice systems, this volume, pp. 393–472.
S.A. Pirogov and Ya.G. Sinai, Phase diagrams of classical lattice systemsTheoretical and Mathematical Physics25 (1975), 1185–1192; 26 (1976), 39–49.
B. SimonThe Statistical Mechanics of Lattice GasesPrinceton University Press, 1993.
Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon Press, 1982.
A. Sütô, Percolation transition in the Bose gasJ. Phys. A26 (1993), 4689–4710.
A. Süt¨®, Non-uniform ground state for the Bose gasJ. Phys. A34 (1993), 37–55.
B. T¨®th, Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnetLett. Math. Phys.28 (1993), 75–84.
D. Ueltschi, Analyticity in Hubbard modelsJ. Stat. Phys.95 (1999), 693–717.
V. Zagrebnov and J.-B. Bru, The Bogoliubov model of weakly imperfect Bose gasPhys. Rep.350 (2001), 291–434.
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Ueltschi, D. (2002). Geometric and Probabilistic Aspects of Boson Lattice Models. In: Sidoravicius, V. (eds) In and Out of Equilibrium. Progress in Probability, vol 51. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0063-5_17
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DOI: https://doi.org/10.1007/978-1-4612-0063-5_17
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