\alpha |  \label{28}$$

where $|i>_{j'}$ for $j=1,\ldots ,k'$ are the $k'$-lowest energy states of $h_{j'}^{(B)}$. Moreover, we suppose that $h_{j'}^{(B)}$ is the same Hamiltonian for each block so that $\epsilon _i$ does not depend on the block.

The truncated Hamiltonian $h_{j'}^{(s)}$ and the intertwiner operator $T_{0,j'}$ are then given by:

$$h_{j'}^{(s')} =  \sum _{i=1}^{k'} |i\rangle'_{j'} \
\epsilon _i \ _{j'} \hspace*{-1pt} \langle i|'   \label{29}$$

$$T_{0,j'} =  \sum _{i=1}^{k'} |i\rangle'_{j'} \
_{j'} \hspace*{-1pt} \langle i|'       \label{30}$$

Later on we shall show examples of these relations.

The final outcome of this analysis is that the effective Hamiltonian $H'$ has a similar structure to the one we started with, namely $H$. The operators involved in $H'_{s'}$ and $H'_{s's'}$ may by all means differ from those of $H_{s}$ and $H_{ss}$, but in some cases the only difference shows up as a change in the coupling constants. This is known as [*the renormalization of the bare coupling constants*]{}. When this is the case, one may easily iterate the RG-transformation and study the RG-flows.

Block RG-Approach to the Heisenberg-Ising Model 
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To exemplify the standard BRG-method we shall study a 1d-lattice Hamiltonian, the Heisenberg-Ising model. Prior to considering in detail the Antiferromagnetic (AF) Heisenberg model we 