 Periodic Potts chain

The ferromagnetic Potts model on a labelled ring has $Q$ possible site states, $J>0$ and ${\sigma }_{L+1}={\sigma }_1$. Its spin-chain transfer matrix is $T=(e^{\beta J}-1)I+P$, with $P$ the $Q$-dimensional all-ones matrix. The canonical partition function is $(e^{\beta J}+Q-1{)}^L+(Q-1)(e^{\beta J}-1{)}^L$. The convention counts two bonds when $L=2$.