\sect{Phase transition of Ising cluster (Exam2002 Q3)}
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 ${H}$ is ${-H s_{i}}$. 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
${Z\left(\beta,H\right)=\sum_{\mathsf{m}}g\left(\mathsf{m}\right)\exp\left(\half
B m^{2} + h \mathsf{m}\right)}$.
Express
${g\left(\mathsf{m}\right)}$ and ${B}$ and ${h}$ using
${\left(N,\epsilon, H, \beta\right)}$.
\item[(b)]
Assume that ${B=\frac{b}{N}}$, define the magnetization as
${M=\frac{\mathsf{m}}{N}}$, and write the partition function as
${Z\left(b,h\right)=\sum_{M} \exp\left(-N*A\left(M\right)\right)}$.
Write the expressions for ${A\left(M\right)}$ and for its
derivatives ${A'\left(M\right)}$ and ${A''\left(M\right)}$.
\item[(c)]
Determine the critical temperature ${T_{c}}$, and write an equation
for the mean field value of ${M}$. Make a qualitative plot of
${A\left(M\right)}$ below and above the critical temperature.
\item[(d)]
Write an approximation for ${A\left(M\right)}$ 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 (${N\rightarrow\infty}$)?
\end {itemize}
${\left(\frac{1}{2}\right)\ln\left(\left(1+x\right)\left(1-x\right)\right)
\ \approx \ x + \left(\frac{1}{3}\right)x^{3} + ...}$