 Exact zero-field Ising-chain decimation

Keep every $b$th spin in a ferromagnetic nearest-neighbour Ising model at zero field, with coupling $K=J/(k_{B}T)$. Multiplying its two-state transfer matrix $b$ times yields the displayed recursion. For $b=2$, direct summation of one intermediate spin gives $2\cosh [K(s+s^{\prime })]=A{e}^{K^{\prime }s{s}^{\prime }}$, with $K^{\prime }=\frac{1}{2}\log \cosh 2K$. The new spacing is $ba$ before coordinate rescaling. All finite positive couplings flow toward the disordered fixed point $K=0$; the ordered zero-temperature endpoint is unstable to a nonzero domain-wall number per bond.