A two-tensor model with order-three

We construct a two-tensor model with order-3 and present its $W$-representation. Moreover we derive the compact expressions of correlators from the $W$-representation and analyze the free energy in large $N$ limit. In addition, we establish the correspondence between two colored Dyck walks in the Fredkin spin chain and tree operators in the ring. Based on the classification Dyck walks, we give the number of tree operators with the given level. Furthermore, we show the entanglement scaling of Fredkin spin chain beyond logarithmic scaling in the ordinary critical systems from the viewpoint of tensor model.