\sect{Phase transition of Ising cluster (based on Exam2002)}

Consider a cluster of ${N}$ spins ${s_{i}=\pm 1}$.
The interaction between {\em any} two spins 
is ${-\epsilon s_{i} s_{j}}$, with ${\epsilon>0}$.
The interaction of each spin with the external magnetic field is ${-\mathsf{H} s_{i}}$. 
The total magnetization is defined as ${\mathsf{m}=\sum s_{i}}$. 
The inverse temperature is ${\beta}$.

\begin {itemize}

\item[(a)]
Show that the partition function can be written as
${Z\left(\beta,\mathsf{H}\right)=\sum_{\mathsf{m}}g\left(\mathsf{m}\right)\exp\left(\half \mathsf{B} m^{2} + h \mathsf{m}\right)}$.
Express ${g\left(\mathsf{m}\right)}$ and ${\mathsf{B}}$ and $h$ using ${\left(N, \epsilon, \beta, \mathsf{H} \right)}$.

\item[(b)]
Assume that ${\mathsf{B}=b/{N}}$, and define the magnetization as ${M={\mathsf{m}}/{N}}$. 
Write the partition function as ${Z\left(b,h\right)=\sum_{M} \exp\left(-N \mathcal{A}\left(M\right)\right)}$.
Write the expressions for ${\mathcal{A}\left(M\right)}$ and for its derivatives ${\mathcal{A}'\left(M\right)}$ and ${\mathcal{A}''\left(M\right)}$.

\item[(c)]
Determine the critical temperature ${T_{c}}$, and write an equation
for the mean field value of ${M}$. Make a qualitative plot of ${\mathcal{A}\left(M\right)}$ 
below and above the critical temperature.

\item[(d)]
Write an approximation for ${\mathcal{A}\left(M\right)}$ up to order ${M^{4}}$.
On the basis of this expression determine the temperature range
where mean filed theory cannot be trusted. Hint: you have to
estimate the variance ${\langle M^{2} \rangle}$ in the Gaussian
approximation. What happens with this condition in the thermodynamic
limit (${N\rightarrow\infty}$)?

\item[(e)]
Find an expression for the heat capacity in the mean field 
and in the Gaussian approximations. 

\end {itemize}