
\sect{Baruch's  C02.}

Apply the mean field approximation to the classical spin-vector
model
\[{\cal H}=-J\sum_{\langle i,j\rangle}{\bf s}_{i}\cdot {\bf s}_{j}-{\bf h}\cdot \sum_{i}{\bf s}_{i}\]
where ${{\bf s}_{i}}$ is a unit vector and ${i,j}$ are neighboring
sites on a lattice; each pair is counted once. The lattice has
${N}$ sites and each site has ${\gamma}$ neighbors.

\begin{itemize}
\item [(a)]
Define a mean field ${{\bf h}^{eff}}$ and evaluate the partition
function ${Z}$ in terms of ${{\bf h}^{eff}}$.
\item [(b)]
Find an equation for the magnetization ${M(T)=\langle \cos
\theta_{i}\rangle}$ at ${\bf h}=0$ where ${ \theta_i}$ is the
angle relative to the orientation of ${\bf M}$. Find the
transition temperature ${T_{c}}$.
\item [(c)]
The mean field hamiltonian (at ${\bf h}=0$) is ${\cal
H}^{MF}=-J\gamma M\sum_is_i+\half J\gamma M^2N$. Explain this
form, derive the free energy $F$, and re-derive the equation for
$M(T)$ in (b) by a minimum condition.
\item[(d)] Identify exponents $\gamma,\beta$ as $T\rightarrow T_c$ for
the susceptibility
${\chi \sim (T-T_c)^{-\gamma}}$ above $T_c$ and for
$M\sim (T_c-T)^{\beta}$ below $T_c$.
\item[(e)] Show that there is a jump in $C_V$ at
$T_c$.
\end{itemize}

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