\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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%