
\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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