Exact critical exponents of the Motzkin and Fredkin Chains

The Motzkin and Fredkin chains are frustration-free spin models with exactly solvable ground states whose $q$-deformations realize an exotic quantum phase transition from a disordered phase to an ordered one under domain-wall boundary conditions. Previous work has mainly focused on their entanglement scaling and spectral gaps, particularly in color-enriched variants. Here we systematically characterize the critical behavior of this transition with three main advances. First, we interpret bulk magnetic order as arising from correlations with boundary spins subject to an effective edge field, directly relating the ordered phase to the domain-wall boundary condition. Second, using the transfer matrix (TM) constructed from the exact matrix product state (MPS) representation, together with a continuum and RG analysis, we derive the algebraic decay of spin correlations and obtain the critical exponent $\eta = \frac{3}{2}$. We further generalize this TM/dual-Hamiltonian method to translationally invariant gapless states with generic power-law correlations, via a zero-dimensional Hamiltonian dual to the one-dimensional TM. Third, the TM spectrum reveals a duality between the ordered and disordered phases which, combined with scale invariance at criticality and the scaling dimension of the spin operator, yields $\nu_\pm = \frac{2}{3}$ from an RG analysis of the $q$-deformed ground states. Both exponents are confirmed numerically by MPS simulations and by direct diagonalization of the TM.