# Using renormalization group theory on the Ising model using decimation transformations

Consider a 1d Ising model with no external magnetic field $$(h=0)$$ and adopt a decimation transformation in which every other spin is traced out.

So the Hamiltonian $$H$$ is given by $$H = -J\sum_{(i,j)} s_i s_j$$ where $$(i,j)$$ corresponds to the nearest neighbors of $$i$$.

I am trying to sketch the RG flow of this system, and I am struggling. How do you start a problem like this? I know you coarse grain a system by some spin-block tranformation $$\tau$$, but I am unsure of how to use it.

The end goal is to derive the RG equation, so the map $$K' = R_l[K]$$ for this transformation and find the fixed points in the flow.

Any advice would be appreciated!

Take the partition function $$Z = \sum_{\{ S_{j} \}} \exp\left( \beta J \sum_{( i,j )} s_i s_j \right) = \sum_{\{ s_{j} \}} \prod_{(i,j)} \exp\left( \beta J s_i s_j \right) \ ,$$ and sum over every other spin (so explicitly evaluate $$\{s_{2n+1}\} \in \pm 1$$ in the sum, but don't sum over $$s_{2n}$$).
This should be equal to a similar partition function over the coarse-grained lattice (with a new coupling $$J'$$ and new spin variables $$s_i'$$) $$Z = \sum_{\{ S_{j} \}} \exp\left( \beta J' \sum_{( i,j )} s'_i s'_j \right) = \sum_{\{ s_{j} \}} \prod_{(i,j)} \exp\left( \beta J' s'_i s'_j \right) \ .$$ By comparing term-by-term, you can relate $$J$$ to $$J'$$.