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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
Articles

Abstract

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 

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Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

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

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