\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}
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