massive with a doubly degenerate ground state in the thermodynamic limit $N \rightarrow \infty$ characterized by the non-zero value of the staggered magnetization,

$$m_{\mbox{st}} = \langle \frac{1}{N} \sum_j S^z_j
(-1)^j  \rangle   \label{11xh9b}$$

The region $0 \leq \Delta \leq 1$ is massless and the ground state is non-degenerate with a zero staggered magnetization. The phase transition between the two phases has an essential singularity.

We would like next to show which of these features are captured by a real-space RG-analysis. The rule of thumb for the RG-approach to half-integer spin model or fermion model is to consider blocks with an [*odd number of sites*]{}. This allows in principle, although not necesarilly, to obtain effective Hamiltonians with the same form as the original ones. Choosing for (\[11xh9\]) blocks of 3 sites we obtain the block Hamiltonian:

$$\frac{1}{J} H = \vec{S}_1 \cdot \vec{S}_2 +   \vec{S}_2 \cdot
\vec{S}_3 +  \epsilon (S^z_1 S^z_2 + S^z_2 S^z_3)$$ $$= \frac{1}{2} \left\{ [\vec{S}_1 + \vec{S}_2 +\vec{S}_3]^2 -
 (\vec{S}_1 + \vec{S}_3)^2 - 3/4 \right\}   + \epsilon (S^z_1 S^z_2 +
S^z_2 S^z_3)  \label{11xh10}$$

$\epsilon \equiv \Delta - 1$.

If $\epsilon = 0$ the block Hamiltonian $H_B$ is invariant under the $SU(2)$ group and according to the introduction to this section, 