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  • 1
    Electronic Resource
    Electronic Resource
    Absence of long-range order in one-dimensional spin systems (1981)
    Springer
    Journal of statistical physics 25 (1981), S. 669-678 
    Details
    ISSN: 1572-9613
    Keywords: Ising model ; plane rotator model ; inequalities ; long-range order
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: 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 $$S_N = o\left( {\left[ {{{\log N} \mathord{\left/ {\vphantom {{\log N} {\log \log N}}} \right. \kern-\nulldelimiterspace} {\log \log N}}} \right]} \right)$$
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01022361
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  • 2
    Electronic Resource
    Electronic Resource
    Local properties of an Ising model on a Cayley tree (1982)
    Springer
    Journal of statistical physics 27 (1982), S. 441-456 
    Details
    ISSN: 1572-9613
    Keywords: Ising model ; Cayley tree ; phase transition ; iteration ; fixed point ; bifurcation ; ferromagnetic ; antiferromagnetic.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The Ising model on a Cayley tree displays a peculiar (continuous order) phase transition with zero long-range order at all finite temperatures. When one studies expection values of spins far removed from the surface (which contains a finite fraction of the total number of spins in the thermodynamic limit), however, one obtains the so-called Bethe approximation. Here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface. In the ferromagnetic case the local magnetization is a fixed point of the iterative transformation, while in the antiferromagnetic case the fixed point bifurcates to a two-cycle of the transformation (for low temperatures and fields) giving rise to local sublattice magnetizations. In both cases, local thermodynamical properties are obtained by integration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01011085
    Library Location Call Number Volume/Issue/Year Availability
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    Paper (German National Licenses)
    Fulltext
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