Real-space renormalization examples

Consider retained spins s and s′ separated by one spin sigma. The two adjacent bond weights multiply to $e^{K\sigma(s+s')}$. Summing sigma over its two signs gives $2\cosh[K(s+s')]$. This takes one value when the retained spins agree and another when they disagree. Represent those two weights as a common factor A times a new bond weight $e^{K'ss'}$, then solve their ratio to determine the effective coupling.

The common factor does not affect normalized probabilities of retained configurations, but it contributes to the partition function and free energy. Dropping it is acceptable only if the desired observable is insensitive to that normalization. For a periodic chain with even length, applying this local sum everywhere leaves half as many spins and twice the lattice spacing. A decimation step therefore changes both the coupling parameter and the unit used to measure separation.

Write $q=\tanh K$. The original infinite-chain correlation across j bonds is $q^j$. Retained nearest neighbours were separated by two original bonds, so their correlation is $q^2$. This directly explains the recursion $q'=q^2$ without relying on the algebraic identity for hyperbolic functions. The renormalized chain reproduces all retained-spin correlations at the corresponding physical separations.

The correlation length expressed in new lattice spacings is half the old numerical value. Multiplying by the new spacing restores the same physical length, so the transformation does not make the material lose correlation in metres. It changes the resolution at which that correlation is described. Once the coarse spacing exceeds the physical correlation length, the effective coupling looks weak and the long-distance system appears almost uncorrelated.

Repeated squaring drives every finite-temperature ferromagnetic value $0<q<1$ toward zero. The only endpoint with perfect correlation is q=1, corresponding to zero temperature, and small deviations away from it grow under blocking. There is no intermediate finite-temperature critical fixed point in this exact one-dimensional model. This agrees with the transfer-matrix result, while the normalization factor accumulates the free energy discarded by looking at coupling flow alone.

In higher dimensions, summing or grouping spins typically generates longer-range and multispin interactions. Retaining only the original nearest-neighbour coupling then becomes an approximation in a larger space of Hamiltonians. A majority rule is not automatically an exact partition sum, and an attractive fixed point of a crude recursion is not sufficient evidence for quantitatively correct exponents. Universality explains why some discarded directions are harmless asymptotically, but their irrelevance must be established rather than assumed from convenience.