
\sect{Baruch's  C21.}

Consider the Ising model of magnetism with long range
interaction: the energy of a spin configuration $\{s_i\}$ with
${s_{i}=\pm 1}$ on an arbitrary lattice is given by,
\[\ E = -\left (J/2N\right)\sum_{ i,j} s_{i}s_{j}-    h\sum_{ i} s_{i}\]\\
where ${J>0}$ and the sum is on all ${i}$ and ${j}$ (in the usual
Ising model the sum is restricted to nearest neighbors) and
${h=\mu _{B}H, H}$ is the magnetic field.

\begin{itemize}
\item [(a)]
Write E in terms of $\,\,{m= \sum_{i}s_{i}/N}$ i.e.
${E\left(m,h\right)= -\left (1/2\right)JNm^{2}-   hNm}$; why is
${N}$ included in the definition of the coupling ${J/N}$?
\item [(b)]
Evaluate the free energy ${F_{0}\left(m;T,h\right)}$ assuming that
it is dominated by a single $m$ which is then a variational
parameter. From the minima of ${F_{0}}$ find ${m\left(h,T\right)}$
and a ciritical temperature ${T_{c}}$. Plot qualitatively
${m\left(h\right)}$ above and below the transition.
\item [(c)]
Plot qualitatively ${F_{0}\left(m\right)}$ for ${T>T_{c}}$ and
${T<T_{c}}$ with both ${h=0}$ and ${h\neq 0}$. Explain the
meanings of the various extrema.
\item [(d)]
Expand ${F_{0}\left(m; T, h=0\right)}$ up to order ${m^{4}}$.
What is the meaning of the $m^2$ coefficient?\\

\end{itemize}

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