On the simplest scale invariant Tree-Tensor-States preserving the quantum symmetries of the antiferromagnetic XXZ chain

For the line of critical antiferromagnetic XXZ chains with coupling $J>0$ and anisotropy $0<\Delta \leq 1$, we describe how the block-spin renormalization procedure preserving the $SU_q(2)$ symmetry introduced by Martin-Delgado and Sierra [Phys. Rev. Lett. 76, 1146 (1996)] can be reformulated as the translation-invariant scale-invariant Tree-Tensor-State of the smallest dimension that is compatible with the quantum symmetries of the model. The properties of this Tree-Tensor-State are studied in detail via the ground-state energy, the magnetizations and the staggered magnetizations, as well as the Shannon-Renyi entropies characterizing the multifractality of the components of the wave function.