Haldane topological orders in Motzkin spin chains

Motzkin spin chains are frustration-free models whose ground state is a combination of Motzkin paths. The weight of such path contributions can be controlled by a deformation parameter t. As a function of the latter, these models, besides the formation of domain wall structures, exhibit gapped Haldane topological orders with a constant decay of string order parameters for t<1. A behavior compatible with a Berezinskii-Kosterlitz-Thouless phase transition at t=1 is also presented. By means of numerical calculations we show that the topological properties of the Haldane phases depend on the spin value. This allows one to classify different kinds of hidden antiferromagnetic Haldane gapped regimes associated with nontrivial features such as symmetry-protected topological order. On one hand, our results allow one to clarify the physical properties of Motzkin frustration-free chains, and on the other hand, suggest them as an interesting and paradigmatic class of local spin Hamiltonians.