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Small Mass Implies Uniqueness of Gibbs States of a Quantum Crystal

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Abstract

A model of interacting quantum particles performing one-dimensional anharmonic oscillations around their equilibrium positions which form a lattice ℤd is considered. For this model, it is proved that the set of tempered Euclidean Gibbs measures is a singleton provided the particle mass is less than a certain bound m *, which is independent of the temperature β−1. This settles a problem that was open for a long time and is an essential improvement of a similar result proved before by the same authors [5], where the bound m * depended on β in such a way that \({{m_{{\ast}} (\beta) \rightarrow 0}}\) as \({{\beta \rightarrow +\infty}}\).

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References

  1. Aksenov, V.L., Plakida, N.M., Stamenković, S.: Neutron Scattering by Ferroelectrics. Singapore: World Scientific, 1990

  2. Albeverio, S., Høegh–Krohn, R.: Homogeneous Random Fields and Quantum Statistical Mechanics. J. Funct. Anal. 19, 242–272 (1975)

    MATH  Google Scholar 

  3. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu.: Suppression of critical fluctuations by strong quantum effects in quantum lattice systems. Commun. Math. Phys. 194, 493–512 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu.: Classical limits of Euclidean Gibbs states of quantum lattice models. Lett. Math. Phys. 48, 221–233 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu., Röckner, M.: Uniqueness for Gibbs measures of quantum lattices in small mass regime. Ann. Inst. H. Poincaré, Probab. Statist. 37, 43–69 (2001)

    Google Scholar 

  6. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu., Röckner, M.: Euclidean Gibbs states of quantum lattice systems. Rev. Math. Phys. 14, 1335–1401 (2002)

    Article  Google Scholar 

  7. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu., Röckner, M.: Gibbs states of a quantum crystal: uniqueness by small particle mass. C.R. Acad. Sci. Paris, Ser. I 335, 693–698 (2002)

    Google Scholar 

  8. Albeverio, S., Kondratiev, Yu., Kozitsky, Yu., Röckner, M.: Quantum stabilization in anharmonic crystals. Phys. Rev. Lett. 90, 170603-1-4 (2003)

    Article  Google Scholar 

  9. Albeverio, S., Kondratiev, Yu., Pasurek, T., Röckner, M.: Gibbs states on loop latices: existence and a priori estimates. C. R. Acad. Sci. Paris 333, Série I, 1005–1009 (2001)

    Google Scholar 

  10. Albeverio, S., Kondratiev, Yu., Pasurek, T., Röckner, M.: Euclidean Gibbs states of quantum crystals. Moscow Math. J. 1, 1–7 (2001)

    Google Scholar 

  11. Albeverio, S., Kondratiev, Yu., Pasurek, T., Röckner, M.: Euclidean Gibbs measures on loop spaces: existence and a priori estimates. BiBiS Preprint Nr. 02-05-086, 2002. To appear in Ann. Probab.

  12. Albeverio, S., Kondratiev, Yu., Röckner, M., Tsikalenko, T.: Uniqueness of Gibbs states for quantum lattice systems. Probab. Theory. Relat. Fields 108, 193–218 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Albeverio, S., Kondratiev, Yu., Röckner, M., Tsikalenko, T.: Dobrushin's uniqueness for quantum lattice systems with nonlocal interaction. Commun. Math. Phys. 189, 621–630 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Albeverio, S., Kondratiev, Yu., Röckner, M., Tsikalenko, T.: Glauber dynamics for quantum lattice systems, Rev. Math. Phys. 13, 51–124 (2001)

    Article  MathSciNet  Google Scholar 

  15. Barbulyak, V.S., Kondratiev, Yu., G.: The quasiclassical limit for the Schrödinger operator and phase transitions in quantum statistical physics. Func. Anal. Appl. 26(2), 61–64 (1992)

    MATH  Google Scholar 

  16. Bellissard, J., Høegh-Krohn, R.: Compactness and the maximal Gibbs state for random Gibbs fields on a lattice. Commun. Math. Phys. 84, 297–327 (1982)

    MathSciNet  MATH  Google Scholar 

  17. Blinc, R., Žekš, B.: Soft Modes in Ferroelectrics and Antiferroelectrics. Amsterdam-Oxford-New York: North-Holland Publishing Company/American Elsevier, 1974

  18. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I, II. New York: Springer-Verlag, 1979, 1981

    Google Scholar 

  19. Bruce, A.D., Cowley, R.A.: Structural Phase Transitions. London: Taylor and Francis Ltd., 1981

  20. Driessler, W., Landau, L., Perez, J.F.: Estimates of critical lengths and critical temperatures for classical and quantum lattice systems. J. Stat. Phys. 20, 123–162 (1979)

    MathSciNet  Google Scholar 

  21. Freericks, J.K., Jarrell, M., Mahan, G.D.: The anharmonic electron-phonon problem. Phys. Rev. Lett. 77, 4588–4591 (1996)

    Article  Google Scholar 

  22. Georgii, H.O.: Gibbs Measures and Phase Transitions, Berlin: Walter de Gruyter, Springer, 1988

  23. Inoue, A.: Tomita-Takesaki Theory in Algebras of Unbounded Operators. Lecture Notes in Math. 1699, Berlin-Heidelberg-New York: Springer-Verlag, 1998

  24. Kondratiev, Ju. G.: Phase Transitions in Quantum Models of Ferroelectrics, In: Stochastic Processes, Physics, and Geometry II, Singapore, New Jersey: World Scientific, 1994, pp. 465–475

  25. Kozitsky, Yu.: Quantum effects in a lattice model of anharmonic vector oscillators. Lett. Math. Phys. 51, 71–81 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lebowitz, J.L., Presutti, E.: Statistical mechanics of systems of unbounded spins. Commun. Math. Phys. 50, 195–218 (1976)

    Google Scholar 

  27. Minlos, R.A., Verbeure, A., Zagrebnov, V.A.: A quantum crystal model in the light-mass limit: Gibbs states. Rev. Math. Phys. 12, 981–1032 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Park, Y.M., Yoo, H.H.: Characterization of Gibbs states of lattice boson systems. J. Stat. Phys. 75, 215–239 (1994)

    MathSciNet  MATH  Google Scholar 

  29. Parthasarathy, K.R.: Probability Measures on Metric Spaces. New York-London: Academic Press, 1967

  30. Pastur, L.A., Khoruzhenko, B.A.: Phase transitions in quantum models of rotators and ferroelectrics. Theor. Math. Phys 73, 111–124 (1987)

    Google Scholar 

  31. Plakida, N.M., Tonchev, M.S.: Quantum effects in a d-dimensional exactly solvable model for a structural phase transition. Phys. A 136, 176–188 (1986)

    Article  MathSciNet  Google Scholar 

  32. Ruelle, D.: Probability estimates for continuous spin systems. Commun. Math. Phys. 50, 189–194 (1976)

    Google Scholar 

  33. Schneider, T., Beck, H., Stoll, E.: Quantum effects in an n-component vector model for structural phase transitions, Phys. Rev. B13, 1123–1130 (1976)

    Google Scholar 

  34. Stamenković, S.: Unified model description of order-disorder and displacive structural phase transitions. Condensed Matter Physics (Lviv) 1(14), 257–309 (1998)

    Google Scholar 

  35. Stasyuk, I.V.: Local anharmonic effects in high-T c superconductors. Pseudospin-electron model. Condensed Matter Physics (Lviv) 2(19), 435–446 (1999)

    Google Scholar 

  36. Stasyuk, I.V.: Approximate analitical dynamical mean-field approach to strongly correlated electron systems. Condensed Matter Physics (Lviv) 3(22), 437–456 (2000)

    Google Scholar 

  37. Tibballs, J.E., Nelmes, R.J., McIntyre, G.J.: The crystal structure of tetragonal KH2 PO4 and KD2 PO4 as a function of temperature and pressure. J. Phys. C: Solid State Phys. 15, 37–58 (1982)

    Article  Google Scholar 

  38. Vaks, V.G.: Introduction to the Macroscopic Theory of Ferroelectrics. Moscow: Nauka, 1973 (in Russian)

  39. Verbeure, A., Zagrebnov, V.A.: Phase transitions and algebra of fluctuation operators in exactly soluble model of a quantum anharmonic crystal. J. Stat. Phys. 69, 37–55 (1992)

    Google Scholar 

  40. Verbeure, A., Zagrebnov, V.A.: No–go theorem for quantum structural phase transition. J. Phys. A: Math. Gen. 28, 5415–5421 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Walter, W.: Differential and Integral Inequalities. Berlin-Heidelberg-New York: Springer-Verlag, 1970

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Authors and Affiliations

  1. Abteilung für Stochastik, Universität Bonn, 53115, Bonn, Germany

    Sergio Albeverio

  2. BiBoS Research Centre, Bielefeld, Germany

    Sergio Albeverio & Yuri Kondratiev

  3. CERFIM, Locarno and Acc. Arch, USI, Switzerland

    Sergio Albeverio

  4. Dip. Matematica, Università di Trento, Italy

    Sergio Albeverio

  5. Fakultät für Mathematik, Universität Bielefeld, Germany

    Yuri Kondratiev & Michael Röckner

  6. Institute of Mathematics, Kiev, Ukraine

    Yuri Kondratiev

  7. Instytut Matematyki, Uniwersytet Marii Curie-Sklodowskiej, 20-031, Lublin, Poland

    Yuri Kozitsky

Authors
  1. Sergio Albeverio
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  2. Yuri Kondratiev
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  3. Yuri Kozitsky
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  4. Michael Röckner
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Correspondence to Sergio Albeverio.

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Communicated by H. Spohn

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Albeverio, S., Kondratiev, Y., Kozitsky, Y. et al. Small Mass Implies Uniqueness of Gibbs States of a Quantum Crystal. Commun. Math. Phys. 241, 69–90 (2003). https://doi.org/10.1007/s00220-003-0923-4

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  • DOI: https://doi.org/10.1007/s00220-003-0923-4

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