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Physical Review

Long-Range Order in One-Dimensional Ising Systems

D. J. Thouless

Published Nov 10, 1969• 285 citations
Publisher

Abstract

The argument of Landau and Lifshitz for the absence of long-range order in one-dimensional systems is used to show that order is absent if the interaction energy falls off faster than ${n}^{\ensuremath{-}2}$. When the interaction falls off as ${n}^{\ensuremath{-}2}$, the order cannot go continuously to zero.

DOI
10.1103/physrev.187.732

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The argument of Landau and Lifshitz for the absence of long-range order in one-dimensional systems is used to show that order is absent if the interaction energy falls off faster than n\ensuremath−2{n}^{\ensuremath{-}2}n\ensuremath−2. When the interaction falls off as n\ensuremath−2{n}^{\ensuremath{-}2}n\ensuremath−2, the order cannot go continuously to zero.
The Mermin-Wagner theorem is strengthened so as to rule out magnetic long-range order at T>0T>0T>0 in one- or two-dimensional Heisenberg and XY systems with long-range interactions decreasing as ${R}^{\ensuremath{-}\ensuremath{\alpha}}$ with a sufficiently large exponent $\ensuremath{\alpha}$. For oscillatory interactions, ferromagnetic long-range order at T>0T>0T>0 is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}1(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)$ or $\ensuremath{\alpha}>5/2(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$. For systems with monotonically decreasing interactions, ferro- or antiferromagnetic long-range order at T>0T>0T>0 is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}2D$.
The authors present the results of the coherent anomaly method when applied to Ising models in one dimension with long-range interactions. This class of systems acts as an interesting and challenging test for the method because the critical exponents as well as the critical temperature values of which are given by the method, depend on the rate of fall-off of the interactions. Thus one can see how accurately the method gives results which correctly reflect this dependency. The results obtained from this method are compared with results obtained by a variety of other methods.
Abstract As the spatial dimensionality ddd decreases, fluctuations become larger and the stability of the low-temperature ordered state deteriorates. The dimensionality where long-range order disappears is known as lower critical dimension. For instance, the Ising model in one dimension does not display long-range order at finite temperatures, however in two dimensions Peierls argument explains why the same model has an ordered phase below a certain critical temperature. If the basic variables and symmetries are continuous as in the XYXYXY and Heisenberg models, the (long-range) ordered state at any finite temperature disappears already in two dimensions. This is the result of Mermin-Wagner's theorem. The XYXYXY model nevertheless undergoes an unusual phase transition without an onset of long-range order in two dimensions, which is known as the Kosterlitz-Thouless transition. Gauge or local symmetries cannot spontaneously be broken as elucidated by Elitzur's theorem when applied to lattice gauge theories.
Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.