\sect{Baruch's C23.} \begin{itemize} \item[(a)] Consider the Ising spin model on a bipartite lattice, i.e. it has two sublattices A,B such that each site on lattice A has its nearest neighbors on sublattice B, and vice versa. The Hamiltonian is $$\label{H} {\cal H}=-J\sum_{{\bf n}, \bm{\delta}}\sigma({\bf n})\sigma({\bf n}+\bm{\delta}) -h\sum_{{\bf n}}\sigma({\bf n})$$ where ${\bf n}$ are the lattice sites, $\bm{\delta}$ labels the nearest neighbors, $h$ is proportional to a magnetic field and $\sigma({\bf n})=\pm$. For $h=0$ show that the free energy satisfies $F(J,T)=F(-J,T)$, hence the critical temperatures satisfy $T_c^F=T_c^{AF}$ for the ferromagnetic ($J>0$) and anti-ferromagnetic ($J<0$) transitions. Define the order parameters at $TT_c$ and find their relationship, if any. \item[(b)] Consider the one dimensional Ising model with the Hamiltonian ${\cal H}=-\sum_{n,n'}J(n-n')\sigma(n)\sigma(n')$ with $\sigma(n)=\pm 1$ at each site n and $J(n)=b/n^{\gamma}$ is a long range interaction and $b>0$. Find the energy of a domain wall (i.e. $n<0$ spins are $-$ and $n\geq 0$ are $+$) and show that the argument for the absence of spontaneous magnetization at finite temperatures fails when $\gamma <2$. \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%