Finite-ring low-temperature expansion ($F=-JL-{\beta }^{-1}\log Q-{\beta }^{-1}\left(\genfrac{}{}{0ex}{}{L}{2}\right)(Q-1)e^{-2\beta J}+O_{L,Q}({\beta }^{-1}{e}^{-3\beta J})$)

At fixed finite $L$, the Helmholtz free energy of a periodic Potts chain receives its first thermal correction from a pair of domain walls. The two-domain-wall degeneracy of a Potts ring divided by the ground state degeneracy factor supplies the coefficient. Expanding the exact canonical partition function proves the formula. The expansion is not uniform in $L$: taking the thermodynamic limit first gives $F/L=-J-{\beta }^{-1}\log [1+(Q-1)e^{-\beta J}]$, whose leading correction is of order $e^{-\beta J}$.