\sect{Mean field for antiferromagnetism}

Consider Ising model on a 2D lattice with antiferromagnetic interaction
${(\epsilon = -\epsilon_0)}$. You can regard the lattice as composed of
two sublattices A and B, such that ${M=\frac{1}{2}(M_A+M_B)}$ is the 
averaged magnetization per spin, and ${M_s=\frac{1}{2}(M_A-M_B)}$ is the
staggered magnetization
  
\Dn

\begin{itemize}

\item[(a)]
Explain the claim: for zero field $(h=0)$, Ising antiferromagnet is the {\em same} 
as Ising ferromagnet, where $M_s$ is the order parameter. Write the expression 
for $M_s(T)$ for ${T \sim T_c}$, based on the familiar solution of the ferromagnetic case.

\item[(b)]
Given $h$ and $\epsilon_0$, find the coupled mean-field equations for $M_A$ and $M_B$.

\item[(c)]
Find the critical temperature ${T_c}$ for $h=0$, and also for small $h$. 
Hints: for~${h=0}$ use the same procedure of expanding $\mbox{arctanh}(x)$ as in the ferromagnetic case; 
for small~$h$ you may use the most extreme simplification that does not give a trivial solution.

\item[(d)]
Find the critical magnetic field $h_c$ above which the system no longer 
acts as an antiferromagnet at zero temperature.

\item[(e)]
Find an expression for the susceptibility $\chi(T)$,  
expressed as a function of the staggered magnetization $M_s(T)$.

\item[(f)]
In the region of ${T \sim T_c}$ give a linear approximation for~$1/\chi$ 
as a function of the temperature~$T$

\end{itemize}

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