\sect{Potts model in one dimension}
A set of $N$ atoms is arranged on a one-dimensional chain.
Each atom has $p$~possible {\em orientations}, labelled by ${\sigma=1,2,...,p}$.
Two neighboring atoms $\sigma_i$ and $\sigma_j$ have a negative interaction
energy $-\varepsilon$ if they are in the same orientation, and zero otherwise.
It is useful to define bond variables ${s_i=\sigma_{i{+}1}-\sigma_{i} \mod(p)}$.
\Dn
(1) The partition function $Z_{\text{chain}}(\beta)$ of an open chain
can be written as ${Z=Aq^{N-1}}$. Write what are $A$ and $q$.
Tip: the partition sum factorizes in the "bond" representation.
\Dn
(2) The partition function $Z_{\text{ring}}(\beta)$ of a closed chain,
with periodic boundary conditions, can be written as ${Z=\trc(T^N)}$.
Write what is the matrix $T$ for ${p=4}$.
\Dn
(3) Find what are the eigenvalues of the transfer matrix~$T$ for general $p$,
and deduce an explicit expression for $Z_{\text{ring}}(\beta)$.
Tip: The $T$ matrix is diagonal in the "momentum" representation.
\Dn
(4) Find the energy per atom ${E(T)/N}$ at the $N\rightarrow\infty$ limit.
Express your results in terms of ${(p, \varepsilon, T)}$.
% Write the result as ${E(T)/N = \epsilon f(\epsilon-\mu)}$.
% Provide expressions for $\mu$ and for $f()$ using $p$ and the temperature $T$.
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