Advertisement

Models of statistical mechanics in one dimension originating from quantum ground states

  • Herbert Spohn
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 257)

Abstract

We consider the ground state of a few quantum mechanical degrees of freedom coupled to a Bose field. Examples are the spin-boson Hamiltonian, the polaron, an electron in an external potential coupled to the radiation field, and, the recently popular, quantum coherence with dissipation. Feynman's method to integrate over the Bose field yields models of statistical mechanics in one dimension, possibly with several components. They are ferromagnetic and may undergo a phase transition as the coupling is varied. In simple cases we have a fairly complete understanding. Other cases are not yet covered and require a non-trivial extension of the methods available for one-dimensional systems. We summarize recent results and list nine challenging problems.

Keywords

Ising Model Brownian Particle Path Measure Free Measure Spin Impurity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T.Spencer, The Schrbdinger equation with a random potential — a mathematical review. In: Percolation and Random Media, Les Houches Summer School 1984.Google Scholar
  2. [2]
    L.A.Bunimovich and Ya.G.Sinai, Comm.Math.Phys. 78, 479 (1980)Google Scholar
  3. [3]
    D.W.Jespen, J.Math.Phys. 6, 405 (1965)Google Scholar
  4. [3a]
    F.Spitzer, J.Math.and Phys. 18, 973 (1969)Google Scholar
  5. [4]
    Ya.G.Sinai and M.R.Soloveichik, One-dimensional classical massive particle in the ideal gas, preprint.Google Scholar
  6. [4a]
    D. Szász and B.Tóth, Bounds for the limiting variance of the heavy particle in R1, preprint.Google Scholar
  7. [5]
    D.Dürr, S.Goldstein and J.L.Lebowitz, Comm.Math.Phys. 78, 507 (1981) and Z.Wahrscheinlichkeitstheorie verw.Gebiete 62, 427 (1983).Google Scholar
  8. [6]
    H.Kesten and G.C.Papanicalaou, Comm.Math.Phys. 78, 19 (1980) andGoogle Scholar
  9. [6a]
    D. Dürr, S.Goldstein and J.L.Lebowitz, preprintGoogle Scholar
  10. [7]
    H.Spohn, J.Stat.Phys. 17, 385 (1977).Google Scholar
  11. [8]
    H.Spohn, Rev.Mod.Phys. 53, 569 (1980) and J.L.Lebowitz and H.Spohn, J.Stat.Phys. 29, 39 (1982).Google Scholar
  12. [9]
    E.Presutti, Ya.G.Sinai and M.R.Soloveichik, preprint.Google Scholar
  13. [10]
    S.Goldstein, J.L.Lebowitz and K.Ravishankar, Comm.Math.Phys. 85, 419 (1982).Google Scholar
  14. [11]
    O.E.Lanford and D.W.Robinson, Comm.Math.Phys. 24, 193 (1972)Google Scholar
  15. [11a]
    H.Narnhofer,Acta Physica Austriaca, Suppl.XI, 527 (1973)Google Scholar
  16. [11b]
    D.W.Robinson, Comm.Math.Phys. 31, 171 (1973).Google Scholar
  17. [12]
    E.B.Davies, Quantum Theory of Open Systems. Academic Press, London (1976).Google Scholar
  18. [13]
    K.Hepp and E.H.Lieb, Helv.Phys.Acta 46, 573 (1973)Google Scholar
  19. [13a]
    A.Frigerio and V.Gorini, J.Math.Phys 17, 2123 (1976).Google Scholar
  20. [14]
    R. Dümcke, Comm.Math.Phys. 97, 331 (1985).Google Scholar
  21. [15]
    R.Feynman, Rev.Mod.Phys. 20, 367 (1948).Google Scholar
  22. [16]
    H.Spohn and R. Dümcke, J.Stat.Phys., to appear.Google Scholar
  23. [17]
    G.Yuval and P.W.Anderson, Phys.Rev. B9, 1522 (1970).Google Scholar
  24. [18]
    M.Blume, V.J.Emery and A.Luther, Phys.Rev.Lett. 25, 450 (1970).Google Scholar
  25. [19]
    V.J.Emery and A.Luther, Phys.Rev. B9, 215 (1974).Google Scholar
  26. [20]
    A.O.Barut, Ed., Foundations of Radiation Theory and Quantum Electrodynamics. Plenum Press New York, 1980.Google Scholar
  27. [21]
    P.Pfeifer, Chiral Molecules-a Superselection Rule Induced by the Radiation Field, P.h.D.Thesis, ETH Zürich (1980).Google Scholar
  28. [22]
    L.M.Sander and H.B.Shore, Phys.Rev. B3, 1472 (1979).Google Scholar
  29. [23]
    R.Beck, W.Götze and P.Prelovsek, Phys.Rev. A20, 1140 (1979).Google Scholar
  30. [24]
    S.Chakravarty, Phys.Rev.Lett. 49, 681 (1982).Google Scholar
  31. [25]
    A.J.Bray and M.A.Moore, Phys.Rev.Lett. 49, 1545 (1982).Google Scholar
  32. [26]
    S.Chakravarty and S.Kivelson, Phys.Rev.Lett. 50, 1811 (1983).Google Scholar
  33. [27]
    W.Zwerger, Z.Physik B 53, 53 (1983).Google Scholar
  34. [28]
    B.Simon and A.D.Sokal, J.Stat.Phys. 25, 679 (1981).Google Scholar
  35. [29]
    F.J.Dyson, Comm.Math.Phys. 12, 91 (1969).Google Scholar
  36. [30]
    G.Gallavotti and H.Knops, Rev.Nuov.Cim. 5, 341 (1975).Google Scholar
  37. [31]
    P.Collet and J.P.Eckmann, A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics. Lecture Notes in Physics 74, Springer, Berlin, 1978.Google Scholar
  38. [32]
    J.Fröhlich and T.Spencer, Comm.Math.Phys. 84, 87 (1982).Google Scholar
  39. [33]
    Appendix in J.Bricmont, J.L.Lebowitz and C.E.Pfister, J.Stat. Phys. 24, 269 (1981).Google Scholar
  40. [34]
    S.Chakravarty and A.J.Leggett, Phys.Rev.Lett. 52, 5 (1984).Google Scholar
  41. [35]
    A.J.Leggett and A.Gary, Phys.Rev.Lett. 54, 857 (1985).Google Scholar
  42. [36]
    A.O.Caldeira and A.J.Leggett, Ann.Phys.(N.Y.) 149, 374 (1983).Google Scholar
  43. [37]
    E.Nelson, Schrödinger Particles Interacting with a Quantized Scalar Field. In: W.T.Martin and I. Sega., Analysis in Function Space. M.I.T.Press, 1964.Google Scholar
  44. [38]
    V.Hakim, A.Muramatsu and F.Guinea, Phys.Rev. B30, 464 (1984).Google Scholar
  45. [39]
    J. Ginibre, in: Statistical Mechanics and Quantum Field Theory. Eds. C.DeWitt and R.Stora, Gordon and Breach, New York (1971).Google Scholar
  46. [40]
    M.Aizenman, private communication.Google Scholar
  47. [41]
    J.L.Lebowitz and E.Presutti, Comm.Math.Phys. 50, 195 (1976).Google Scholar
  48. [42]
    J. Fröhlich, Thesis, ETH Zürich, 1972.Google Scholar
  49. [43]
    J.Frbhlich, Ann.Inst. Henri Poincaré, A19, 1 (1973).Google Scholar
  50. [44]
    J.Fröhlich, Fortschritte der Physik 22, 159 (1974).Google Scholar
  51. [45]
    E.Nelson, J.Math.Phys. 5, 1190 (1964).Google Scholar
  52. [46]
    J.P.Eckmann, Comm.Math.Phys. 18, 247 (1970).Google Scholar
  53. [47]
    S.Albeverio, Helv.Phys.Acta 45, 303 (1972).Google Scholar
  54. [48]
    N.Tokuda, H.Shoji and K.Yoneya, J.Phys. C 14, 4281 (1981).Google Scholar
  55. [49]
    H.Spohn, J.Phys. A, to appear.Google Scholar
  56. [50]
    J. Fröhlich and T.Spencer, Comm.Math.Phys. 81, 527 (1981).Google Scholar
  57. [51]
    A.Schmid, Phys.Rev.Lett. 51, 1506 (1983).Google Scholar
  58. [52]
    F.Guinea, V.Hakim and A.Muramatsu, Phys.Rev.Lett. 54, 263 (1985).Google Scholar
  59. [53]
    S.A. Bulgadaev, JETP Lett. 39, 315 (1984).Google Scholar
  60. [54]
    M.P.A.Fisher and W.Zwerger, Quantum Brownian motion in a periodic potential, preprint.Google Scholar
  61. [55]
    K.H.Kjaer and H.J.Hilhorst, J.Stat.Phys. 28, 621 (1982).Google Scholar
  62. [56]
    J.Slurink and H.J.Hilhorst, Physica 120A, 627 (1983).Google Scholar
  63. [57]
    J.T.Devreese, Path Integrals and Continuum Fröhlich Polarons. In: Path Integrals, ed. by G.J.Papadopoulos and J.R.Devreese. Plenum Press, New York, 1978.Google Scholar
  64. [58]
    R.P.Feynman, Phys.Rev. 97, 660 (1955).Google Scholar
  65. [59]
    R.L.Dobrushin, Matth.USSR Sbornik 23, 13 (1974).Google Scholar
  66. [60]
    M.D.Donsker and S.R.S.Varadhan, Comm. Pure and Appl.Math. 36, 505 (1983).Google Scholar
  67. [61]
    E.Lieb, Stud.Appl.Math. 57, 93 (1977).Google Scholar
  68. [62]
    J.Adamowski, B.Gerlach and H.Leschke, Physics Letters 79A, 249 (1980).Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Herbert Spohn
    • 1
  1. 1.Theoretische PhysikUniversität MünchenMünchen 2Germany

Personalised recommendations