Dobrovolny B., Laanait L.
Temperature phase transitions associated with local
minima of energy in continous unbounded spins
(117K, LATeX)

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.