Journal of Statistical Physics

, Volume 25, Issue 4, pp 669–678 | Cite as

Absence of long-range order in one-dimensional spin systems

  • Jeffrey B. Rogers
  • Colin J. Thompson


For a one-dimensional Ising model with interaction energy
$$E\left\{ \mu \right\} = - \sum\limits_{1 \leqslant i< j \leqslant N} {J(j - i)} \mu _\iota \mu _j \left[ {J(k) \geqslant 0,\mu _\iota = \pm 1} \right]$$
it is proved that there is no long-range order at any temperature when
$$S_N = \sum\limits_{k = 1}^N {kJ\left( k \right) = o} \left( {[\log N]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)$$
The same result is shown to hold for the corresponding plane rotator model when
$$S_N = o\left( {\left[ {{{\log N} \mathord{\left/ {\vphantom {{\log N} {\log \log N}}} \right. \kern-\nulldelimiterspace} {\log \log N}}} \right]} \right)$$

Key words

Ising model plane rotator model inequalities long-range order 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. J. Dyson,Commun. Math. Phys. 12:212 (1969);21:269 (1971).Google Scholar
  2. 2.
    P. W. Anderson, G. Yuval, and D. R. Hamann,Phys. Rev. B 1:4464 (1970).Google Scholar
  3. 3.
    D. G. Kelly and S. Sherman,J. Math. Phys. 9:466 (1968); J. L. Monroe and P. A. Pearce,J. Stat. Phys. 21:615 (1979).Google Scholar
  4. 4.
    H. E. Stanley,Introduction to Phase Transitions and Critical Phenomena (Clarendon, Oxford, 1971).Google Scholar
  5. 5.
    Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1965).Google Scholar
  6. 6.
    C. J. Thompson,Phys. Lett. 43A; 259 (1973).Google Scholar
  7. 7.
    D. Szasz,J. Stat. Phys. 19:453 (1978).Google Scholar
  8. 8.
    J. Bricmont,Phys. Lett. 57A:411 (1976).Google Scholar
  9. 9.
    J. L. Lebowitz,J. Stat. Phys. 16:463 (1977).Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • Jeffrey B. Rogers
    • 1
  • Colin J. Thompson
    • 1
  1. 1.Mathematics DepartmentUniversity of MelbourneParkvilleAustralia

Personalised recommendations