Decimation of an alternating mixed-spin chain ($V_{st}=1+2\cosh [\beta (J_{1}s+J_{2}t)]$)

For Ising spins $s,t=\pm 1$ separated by a decorating spin $\tau =0,\pm 1$, summing its Boltzmann weights removes that spin exactly. If its two exchange interactions are $-\tau (J_{1}s+J_{2}t)$, the effective transfer matrix has entries $1+2\cosh [\beta (J_{1}s+J_{2}t)]$. Its eigenvalues are ${\lambda }_{+}=2+4\cosh (\beta J_1)\cosh (\beta J_2)$ and ${\lambda }_{-}=4\sinh (\beta J_1)\sinh (\beta J_2)$. A transfer step contains two physical sites, so the limiting Helmholtz free energy per site is $-(2\beta {)}^{-1}\log {\lambda }_{+}$.

An observable on a spin removed by decimation is inserted before its local sum. For ${\tau }^2$, replace $V_{st}={\sum }_{\tau }{e}^{\beta \tau (J_{1}s+J_{2}t)}$ by $Q_{st}={\sum }_{\tau }{\tau }^2{e}^{\beta \tau (J_{1}s+J_{2}t)}=2\cosh [\beta (J_{1}s+J_{2}t)]$. Contracting the unchanged ring gives the displayed thermal expectation value. In the infinite chain it becomes $v_{+}^{\mathsf{T}}Q{v}_{+}/{\lambda }_{+}$ for normalized dominant eigenvector $v_{+}$.