Renormalization group

Remarks

A point $K^*$ with $R_b K^* = K^*$ is a fixed point, and clause 3 forces $\xi(K^*) = 0$ or $\xi(K^*) = \infty$. Critical points flow to fixed points with $\xi = \infty$.

Example: decimation of the Ising chain with coupling $K = \beta J$ at zero field, for $b = 2^k$. Summing over a spin $s$ between $s_1$ and $s_2$ gives $\sum_{s = \pm 1} e^{K s (s_1 + s_2)} = A \, e^{K' s_1 s_2}$ with $\tanh K' = \tanh^2 K$ and $A^2 = 4 \cosh 2K$. So $Z_N(K) = A^{N/2} Z_{N/2}(K')$, which is clause 2. Since $\xi(K) = -1/\log \tanh K$, the new coupling has $\xi(K') = \xi(K)/2$, which is clause 3. The fixed points are $K = 0$ and $K = \infty$, and every $0 < K < \infty$ flows to $0$.

Non-example: the identity map taken as $R_2$ fails clause 3 at every $K$ with $0 < \xi(K) < \infty$, since $\xi(K) \ne \xi(K)/2$.

In practice $R_b$ is constructed by block spins or by integrating out momentum shells, and it is usually only approximate, because new couplings appear at each step.