Sawtooth Ising chain

An Ising model on a chain of corner-sharing triangles, with one base spin and one apex spin per cell. If the apex interacts only with the two base corners, summing each apex independently gives a two-by-two transfer matrix on base spins. For symmetric Ising spins and energy $Js{s}^{\prime }+\bar{J}\bar{s}(s+s^{\prime })$, its entries are $2e^{-\beta Js{s}^{\prime }}\cosh [\beta \bar{J}(s+s^{\prime })]$.

For positive equal exchange constants and $2L$ symmetric Ising spins on a ring, a minimum-energy triangle has precisely one spin different from its other two. Given adjacent base spins, there is one permitted apex choice if they agree and two if they disagree. The counting transfer matrix is $C=\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)$, with eigenvalues $3,-1$, giving ${\Omega }_L=\mathrm{Tr}{C}^L=3^L+(-1{)}^L$. Thus the residual entropy per spin is $(k_B/2)\log 3$. If the two apex bonds are strictly stronger than the base bond, only the two globally reversed configurations minimize all triangles and this residual entropy vanishes.