= Solution
For a ferromagnetic nearest-neighbour <Ising model> on a chain let $s_i=\pm1$, $K=J/(k_BT)>0$ and $h=H/(k_BT)$. The dimensionless <Hamiltonian> is $-K\sum_i s_is_{i+1}-h\sum_i s_i$. In one dimension, decimating every other spin closes exactly within the nearest-neighbour family, including its additive constant. This is the exact chain building block of the <Migdal-Kadanoff approach>; <bond moving> becomes an additional approximation in higher-dimensional lattices.
For <exact zero-field Ising-chain decimation>, sum over an intermediate spin $\sigma$ between two retained spins:
$$
W(s,s')=\sum_{\sigma=\pm1}e^{K(s\sigma+\sigma s')}
=2\cosh[K(s+s')]=A(K)e^{K'ss'}.
$$
Parallel and antiparallel end spins give $Ae^{K'}=2\cosh2K$ and $Ae^{-K'}=2$. Hence
$$
\boxed{K'=\tfrac12\log\cosh2K,\qquad
A=2\sqrt{\cosh2K},\qquad\tanh K'=(\tanh K)^2.}
$$
For an even periodic chain, $Z_N(K)=A^{N/2}Z_{N/2}(K')$. Rescaling the lattice spacing $2a$ back to $a$ completes the <renormalization> step. Decimation of $b$ successive links gives $\tanh K'=(\tanh K)^b$, as follows by multiplying the <transfer matrix> $b$ times.
Let $v=\tanh K$. Then $v'=v^2$ on $0\le v\le1$. The infinite-temperature <renormalization-group fixed point> $v=0$ attracts every $0\le v<1$; the zero-temperature <renormalization-group fixed point> $v=1$ is unstable to heating, with slope two. There is no intermediate ferromagnetic <renormalization-group fixed point>. Equivalently $K'\simeq K^2$ for small $K$, while $K'\simeq K-\tfrac12\log2$ for large finite $K$. Thus \b[there is no finite-temperature ordered <thermodynamic phase> in the short-range chain].
The <domain-wall scaling coordinate of an Ising chain> near $T=0$ is the domain-wall <Boltzmann factor> $w=e^{-2K}$. Its recursion and the zero-field <correlation length> are
$$
w'=\frac{2w}{1+w^2}\simeq2w,\qquad
\langle s_0s_n\rangle=(\tanh K)^{|n|},\qquad
\boxed{\frac\xi a=-\frac1{\log\tanh K}\sim\frac1{2w}=\frac12e^{2J/(k_BT)}.}
$$
Since $\xi(K')/a=[\xi(K)/a]/2$, the recursion has the correct <length> scaling. Its exponent is one with respect to $w$; \b[it is not a finite power of physical <temperature> about $T_c=0$]. In the <thermodynamic limit>, a bond is a <domain wall> with <probability> $(1-\tanh K)/2\sim e^{-2K}$, giving the same exponentially large <length> scale. Their finite <energy> allows <entropy> to proliferate walls at any positive <temperature>.
\Image[/past-exam-of-the-phase-transitions-and-collective-phenomena-course-of-the-university-of-cambridge/2012/paper-1-ptcp2012-chain-flow.png]
{title=Exact Ising-chain decimation and length}
{description=Original one-dimensional recursion v'=v squared, finite-temperature iterates flowing to zero and the exact correlation length with its low-temperature asymptote. The coordinate v is tanh(J/kBT); no finite-temperature critical point is introduced.}
The <Ising-chain decimation constant> matters for thermodynamics. With dimensionless <free energy> per spin $f=-\lim N^{-1}\log Z_N$,
$$
f(K)=-\tfrac12\log A(K)+\tfrac12 f(K'),\qquad
\boxed{f(K)=-\log(2\cosh K).}
$$
The second expression follows independently from transfer <eigenvalues> $2\cosh K$ and $2\sinh K$ and satisfies the recursion. Multiplying it by $k_BT$ restores the physical <free energy>. Dropping $A$ would preserve normalized retained-spin <probabilities> but lose the eliminated-spin <entropy>.
<Finite-field Ising-chain decimation> is also exact. Use the symmetric <transfer matrix>
$$
T=\begin{pmatrix}e^{K+h}&e^{-K}\\e^{-K}&e^{K-h}\end{pmatrix},\quad W=T^2,
$$
and read the new couplings from
$$
\boxed{K'=\frac14\log\frac{W_{++}W_{--}}{W_{+-}^2},\qquad
h'=\frac12\log\frac{W_{++}}{W_{--}},\qquad
A=(W_{++}W_{--}W_{+-}^2)^{1/4}.}
$$
Here $W_{++}=e^{2K+2h}+e^{-2K}$, $W_{--}=e^{2K-2h}+e^{-2K}$ and $W_{+-}=2\cosh h$. The latter off-diagonal entry, the field convention split between adjacent bonds and the constant must all be tracked consistently. Near the zero-temperature zero-field <renormalization-group fixed point> $h'\simeq2h$, while at $K=0$ the independent-spin field is unchanged.
In the higher-dimensional <Migdal-Kadanoff approximation>, moving bonds creates chains that can be decimated. With <bond moving> first on a $d$-dimensional hypercubic lattice and <length> factor $b$, the strengthened links have $b^{d-1}K$, giving the convention
$$
K'=\operatorname{artanh}\!\left[\tanh^b(b^{d-1}K)\right].
$$
For $d=1$ no transverse bonds move, so this reduces to the exact chain result. In higher dimensions it discards geometric <correlations> and is generally approximate on the original lattice; moving and decimating in another order yields another approximation, not a contradictory exact chain formula.<Position Space Renormalization Group by Mehran Kardar>[lecture 13, printed pp. 79–81, equations V.18–V.27: exact chain; lecture 14, printed pp. 86–87, equations V.51–V.54: bond-moving convention]{ref}
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