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For a ferromagnetic nearest-neighbour Ising model on a chain let $s_i=\pm 1$, $K=J/(k_{B}T)>0$ and $h=H/(k_{B}T)$. The dimensionless Hamiltonian is $-K{\sum }_i{s}_i{s}_{i+1}-h{\sum }_i{s}_i$. 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:Parallel and antiparallel end spins give $A{e}^{K^{\prime }}=2\cosh 2K$ and $A{e}^{-K^{\prime }}=2$. HenceFor an even periodic chain, $Z_N(K)=A^{N/2}{Z}_{N/2}(K^{\prime })$. Rescaling the lattice spacing $2a$ back to $a$ completes the renormalization step. Decimation of $b$ successive links gives $\tanh K^{\prime }=(\tanh K{)}^b$, as follows by multiplying the transfer matrix $b$ times.

Let $v=\tanh K$. Then $v^{\prime }=v^2$ on $0\le v\le 1$. 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 $K^{\prime }\simeq K^2$ for small $K$, while $K^{\prime }\simeq K-\frac{1}{2}\log 2$ for large finite $K$. Thus 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 areSince $\xi (K^{\prime })/a=[\xi (K)/a]/2$, the recursion has the correct length scaling. Its exponent is one with respect to $w$; 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.

The Ising-chain decimation constant matters for thermodynamics. With dimensionless free energy per spin $f=-\lim N^{-1}\log Z_N$,The second expression follows independently from transfer eigenvalues $2\cosh K$ and $2\sinh K$ and satisfies the recursion. Multiplying it by $k_{B}T$ 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 matrixand read the new couplings fromHere $W_{++}=e^{2K+2h}+e^{-2K}$, $W_{--}=e^{2K-2h}+e^{-2K}$ 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 $h^{\prime }\simeq 2h$, 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 conventionFor $d=1$ 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.^[ref]