\sect{Ising antiferromagnet} 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%