non commutativity or [*correlations*]{} between the blocks completely. A straight forward way to improve the standard RG method is to include somehow the correlations between adjacent system blocks. As visualized in figure \[correlationblock\] we can refer to these correlations as blocks, which we denote therefore as [*correlation blocks*]{}.

Using these correlation blocks enables us to represent the non commutativities between the system blocks in a compact way and we denote these correlation blocks, using the overall notation given in the appendix, as $$\begin{aligned}
\label{grad5}
{\cal O}^{\text{correlation}}_{{\cal H}_{\left\{i,i-1,\dots\right\}}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}}
   ({\bf K})\end{aligned}$$ $$\begin{aligned}
  \text{with}\quad {\cal H}_{\left\{i,i-1,\dots\right\}}
   \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}
  \;\subset\;
   {{\cal H}_{I}\otimes {\left({\cal H}_{\text{aux}}\right)}_{I}
     \otimes {\cal H}_{I-1}\otimes {\left({\cal H}_{\text{aux}}\right)}_{I-1}
      \otimes \dots}\;\;.\nonumber\end{aligned}$$ The subspace $$\begin{aligned}
\label{grad5b}
  {\cal H}_{\left\{i,i-1,\dots\right\}}
   \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,i-1,\dots\right\}}
  \;=\;
   {\cal H}_{i}\otimes {\left({\cal H}_{\text{aux}}\right)}_{i}
     \otimes {\cal H}_{i-1}\otimes {\left({\cal H}_{\text{aux}}\right)}_{i-1}
      \otimes \dots\end{aligned}$$ denotes the tensor product composition of all the block Hilbert spaces used for the construction of the correlation block.

Decomposition rules {#decom}
===================

We are now dealing with the problem how to include these correlation blocks into the RG transformation. One can find approaches in the past where this is performed perturbatively [@gmdsv95] and therefore unsuitable in our case. To find some insight into this problem let us start with the 