= Ground-state counting of the frustrated sawtooth Ising chain
{title2=$\Omega_L=3^L+(-1)^L$}
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 $C=\left(\begin{smallmatrix}1&2\\2&1\end{smallmatrix}\right)$, with <eigenvalues> $3,-1$, giving $\Omega_L=\operatorname{Tr}C^L=3^L+(-1)^L$. Thus the <residual entropy> per <spin> is $(k_B/2)\log3$. 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.
Back to article page
