Skip to main content
SpringerLink
Log in

Peierls Argument and Long-Range Order Behavior of Quantum Lattice Systems with Unbounded Spins

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Results on long-range order behavior are obtained for systems in arbitrary dimension (v≥2) with a wide class of spin–spin long-range interactions, without assuming the reflection positivity property.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. J. Fröhlich, B. Simon, and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50:79 (1976).

    Google Scholar 

  2. F. Dyson, E. Lieb, and B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactions, J. Stat. Phys. 18:335 (1978).

    Google Scholar 

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

    Google Scholar 

  4. L. A. Pastur and B. A. Khoruzhenko, Phase transitions in quantum models of rotators and ferroelectrics, Theor. Math. Phys. 73:1094 (1987).

    Google Scholar 

  5. V. S. Barbulyak and Yu. G. Kondratiev, Functional integrals and quantum lattice systems: Phase transitions, Reports Nat. Acad. Sci. of Ukraine 10:19 (1991).

    Google Scholar 

  6. V. S. Barbulyak and Yu. G. Kondratiev, The semiclassical limit for the Schrödinger operator and phase transitions in quantum statistical mechanics, Funct. Anal. Appl. 26:124 (1992).

    Google Scholar 

  7. Yu. G. Kondratiev, Phase transitions in quantum models of ferroelectrics, in Stochastic Processes, Physics and Geometry II (World Scientific, Singapore/New Jersey, 1994), pp. 465-475.

    Google Scholar 

  8. A. Yu. Kondratiev, Phase transitions in quantum models of ferromagnetics, Approach with using the functional integrals, Reports Nat. Acad. Sci. of Ukraine 6 (1996).

  9. J. Glimm, A. Jaffe, and T. Spencer, Phase transitions for φ 42 quantum fields, Common. Math. Phys. 45:203 (1975).

    Google Scholar 

  10. R. Peierls, On Ising's model of ferromagnetics, Proc. Cambridge Phil. Soc. 32:477 (1936).

    Google Scholar 

  11. J. Fröhlich, Phase transitions, Goldstone bosons and topological superselections rules, Acta Physica Austriaca, Suppl. XV, 133-269 (1976).

    Google Scholar 

  12. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

    Google Scholar 

  13. S. Albeverio and R. Hoegh-Krohn, Homogeneous random fields and statistical mechanics, J. Funct. Anal. 19:242 (1975).

    Google Scholar 

  14. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer-Verlag, New York, 1982).

    Google Scholar 

  15. D. W. Ruelle, Superstable interaction in classical statistical mechanics, Common. Math. Phys. 18:127 (1970).

    Google Scholar 

  16. Y. M. Park, Quantum statistical mechanics of unbounded continuous spin systems, J. Korean Math. Soc. 22:43 (1985).

    Google Scholar 

  17. Y. M. Park and H. J. Yoo, Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems, J. Stat. Phys. 80:223 (1995).

    Google Scholar 

  18. B. Simon, The P(φ) 2 Euclidean (Quantum) Field Theory (Princeton, New Jersey, 1974).

    Google Scholar 

  19. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II (Springer-Verlag, New York/Heidelberg/Berlin, 1981).

    Google Scholar 

  20. V. S. Barbulyak and Yu. G. Kondratiev, Functional integrals and quantum lattice systems: Existence of Gibbs states, Reports Nat. Acad. Sci. of Ukraine 8:31 (1991).

    Google Scholar 

  21. S. Pirogov and Ya. G. Sinai, Phase diagrams of classical lattice systems, Theor. Math. Phys. 25:1185 (1975); 26:39 (1976).

    Google Scholar 

  22. R. Esposito, F. Nicoló, and M. Pulvirenti, Superstable interactions in quantum statistical mechanics: Maxwell-Boltzmann statistics, Ann. Inst. Henri Poincaré XXXIV:127 (1982).

    Google Scholar 

  23. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, Mathematical Foundations of Classical Statistical Mechanics: Continuous Systems (Gordon & Breach, London, 1989).

    Google Scholar 

  24. J. Ginibre, Some applications of functional integration in statistical mechanics, in Statistical Mechanics and Quantum Field Theory, C. De Witt and R. Stora, eds. (New York, Gordon & Breach).

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität, D 44780, Bochum, Germany;

    S. Albeverio & A. Yu. Kondratiev

  2. BiBoS Research Center, Bielefeld, Germany;

    S. Albeverio & A. Yu. Kondratiev

  3. SFB 237, Essen-Bochum-Düsseldorf, Germany;

    S. Albeverio & A. Yu. Kondratiev

  4. CERFIM, Locarno, Switzerland

    S. Albeverio & A. Yu. Kondratiev

  5. Institute of Mathematics, NAS of Ukraine, Kiev, Ukraine

    A. L. Rebenko

Authors
  1. S. Albeverio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Yu. Kondratiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. L. Rebenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Albeverio, S., Kondratiev, A.Y. & Rebenko, A.L. Peierls Argument and Long-Range Order Behavior of Quantum Lattice Systems with Unbounded Spins. Journal of Statistical Physics 92, 1137–1152 (1998). https://doi.org/10.1023/A:1023056913416

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023056913416

Search

Navigation