04-34 Dobrovolny B., Laanait L.

Temperature phase transitions associated with local minima of energy in continous unbounded spins (117K, LATeX) Feb 10, 04

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Abstract. {\footnotesize In this work we develop an alternative version of the theory of contour models adapted to continuous spins, $\omega_{x}\in {\large{\bf {R}}}$, located in sites, $x$ of a $d\geq 2$ dimensional lattice ${\large{\bf Z^{d}}}$. \\ The spins interacting via nearest neighbors ferromagnetic interactions are embedded in a single spin potential $V$ similar to that, already, introduced by Dobrushin and Shlosman.\\ The potential $V$, has an ordered sequence $\left( \omega_{1}< ...< \omega_{n}\right) $ of $n$(finite) local minima and satisfy: \begin{itemize} \item The value of the potential, $V(\omega_{q})$, at the minimum $\omega_{q}$ verify: $V(\omega_{q})< V(\omega_{q^{^{\prime }}})$, $q< q^{^{\prime }}$. \item The "mass", $m_{q}$, related to the second derivative, $m_{q}=\frac{1}{2}\frac{\partial ^{2}V(w)}{\partial ^{2}w}|_{\omega_{q}}$ exists and strictly positive and satisfy, $m_{q}>m_{q^{^{\prime }}}$, $q< q^{^{\prime }}$. \item The distance between two successive minima is sufficiently great and the they are separated by a sufficiently heigh energy barrier. \end{itemize} For all finite reciprocal temperature $\beta$, satisfying $1\leq \beta <\infty$, and for the mass $m_{n}$ ( corresponding to the $n^{th}$ minimum) large enough, we prove the Peierls condition, and we derive the phase diagram by proving that there exist sequences ($\beta_{1},... ,\beta_{N(n)}$) , $N(n)<n$ , of first order phase transition points, depending on $m_{q}$ and $V(\omega_{q})$, $q=1, ..., n$ , where the first derivative of the "physical" free energy with respect to $\beta$ is discontinuous. The method we use, among others, is the Bricmont- Kuroda- Lebowitz extension of the Pirogov -Sinai Theory combined with the "Gaussian Transformation" proposed by Zahradnik.

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