\sect{Mean field approximation for classical spin model}

\Dn

Apply the mean field approximation to the classical spin vector model
%
\[
{\cal H}=
-\epsilon\sum_{\langle i,j\rangle} \bm{s}_{i} \cdot \bm{s}_{j}
-\bm{h}\cdot \sum_{i}{\bm{s}}_{i}
\]
%
where ${\bm{s}_{i}}$ is a unit vector and ${i,j}$ are neighboring
sites on a lattice with coordination number $c$. The lattice has
${N}$ sites and each site has ${c}$ neighbors.

\Dn

\begin{itemize}

\item [(a)]
Assume that $\bm{h}=(0,0,h)$, define a mean field ${\bm{h}_{eff}}$,
and evaluate the partition function ${Z}$ in terms of ${\bm{h}_{eff}}$.

\item [(b)]
Define $\theta_i$ as the inclination angle of $\bm{s}_i$ 
with respect to $\bm{h}$. Assume that at equilibrium ${\bm{s}_i=(0,0,M)}$, 
where ${M = \langle \cos{\theta} \rangle}$. 
Find the equation for $M$, and find the transition temperature ${T_{c}}$.


\item [(c)]
Write an expression for the mean field energy 
of the system assuming that $M(T)$ is known.

\item[(d)] 
Identify exponents $\gamma$ and $\beta$ that describe 
the susceptibility ${\chi \sim (T-T_c)^{-\gamma}}$ above $T_c$, 
and the magnetization $M\sim (T_c-T)^{\beta}$ below $T_c$.

\item[(e)] 
Find the jump in the heat capacity $C_V$ at $T_c$.

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%