\sect{Baruch's  C28.}

Potts model in 1-dimension (1d)  . A set of N atoms, each with $p$ states is arranged on a 1d chain with periodic boundary conditions. The atom at the $n$-th site is in a state $i_n$ that is chosen from the set $\{1,2,...,p\}$.
Two neighboring atoms at sites $n$ and $n+1$, respectively, have an interaction energy $-J$ ($J>0$) if they are in the same state, i.e. $i_n=i_{n+1}$, and 0 interaction otherwise. The Hamiltonian is therefore
\[ {\cal H}= -J\sum_{i=1}^N \delta_{i_n,i_{n+1}}\]
where $\delta_{i_n,i_{n+1}}$ is the Kronecker symbol, and the boundary conditions are $i_{N+1}=i_1$.
\begin{itemize}
\item[(a)] Derive the free energy for $p=2$.  (Consider here and below the limit $N\rightarrow \infty$.)
\item[(b)] Derive the free energy  for a general $p$. Hint: Show that the eigenvector of the transfer matrix whose all entries are equal has the largest eigenvalue.
\item[(c)] Find the internal energy $E$ at  the low and high temperature limits and interpret the results.\\
\end{itemize}

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