Exact critical exponents of the Motzkin and Fredkin Chains

The Motzkin and Fredkin chains are frustration-free models with exactly solvable ground states. Their $q$-deformations describe an exotic quantum phase transition from a disordered phase to an ordered one subject to domain-wall boundary conditions. Understanding of this phase transition has so far been mainly focused on the entanglement scaling and spectral gap properties, especially for versions of the models enriched with a color degree of freedom. Although the ground states can be exactly represented as a holographic tensor network resembling the multiscale entanglement renormalization ansatz at the critical point, the lack of unitary and isometric tensors in the hierarchical network hinders the characterization of the critical behaviors. Using instead the transfer matrix (TM) obtained from the matrix product state (MPS) representation combined with a renormalization group analysis, we analytically compute the critical exponents $\eta=1/2$ and $\nu_\pm=2/3$, revealing a duality between the ordered and disordered phases. The exact critical exponents are verified numerically by MPS calculations and numerical diagonalization of the TM.