Anomalous Dynamics and Scaling Behaviors in Motzkin Chain

We investigate the critical properties of a frustration-free quantum Motzkin chain, whose unique ground state is a uniform superposition of colored Motzkin walks. This model is gapless, but its low-energy physics is not described by a Conformal Field Theory (CFT). The system exhibits anomalous entanglement scaling, departing from the logarithmic law typical of 1D CFTs, and its spin correlation functions suggest it belongs to a new, previously unidentified 1D universality class. By mapping the quantum Hamiltonian to a classical stochastic process, we numerically determine the dynamical exponent. We systematically characterize the system's anomalous criticality by computing the scaling dimensions of different operators. These operators are classified according to their transformation under the system's discrete symmetries. This symmetry-resolved analysis reveals a rich hierarchy of operator scaling, providing a framework to understand this novel critical phase.