\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
\begin{equation}\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})
\end{equation}
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}
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