tensor product decomposition usually contains different irreps. The criterion to choose a particular irrep, or a collection of irreps is the one of [*minimum energy*]{}. All the states of a given irrep $V'$ will have the same energy.

The summary of the discussion so far is that [*the intertwiner operator $T_0$ can be identified with the Clebsch-Gordan operator*]{},

$$T_0 = C^{V \otimes \stackrel{n_s}{\ldots } \otimes
V}_{V'}  :  V' \longrightarrow   V \otimes \stackrel{n_s}{\ldots }
\otimes V  \label{11xh8}$$

Let us illustrate this ideas with the AF Heisenberg-Ising model whose Hamiltonian is given by:

$$H_N = J \sum _{j=1}^{N-1} (S^x_j S^x_{j+1} + S^y_j
S^y_{j+1} + \Delta S^z_j S^z_{j+1})   \label{11xh9}$$

where $\Delta \geq 0$ is the anisotropic parameter and $J>0$ for the antiferromagnetic case. If $\Delta = 1$ one has the AF-Heisenberg model which was solved by Bethe in 1931. If $\Delta =
0$ one has the XX-model which can be trivially solved using a Jordan-Wigner transformation which maps it onto a free fermion model. For the remaining values of $\Delta $ the model is also solvable by Bethe ansatz and it is the 1D relative of the 2D statistical mechanical model known as the 6-vertex or XXZ-model.

The region $\Delta > 1$ is 