Singularity and universality from von Neumann to R\'enyi entanglement entropy and disorder operator in Motzkin chains

The R\'enyi entanglement entropy is widely used in studying quantum entanglement properties in strongly correlated systems, whose analytic continuation as the R\'enyi index $n \to 1$ is often believed to yield the von Neumann entanglement entropy. However, earlier findings indicate that this process exhibits a singularity for the colored Motzkin spin chain problem, leading to different scaling behaviors of $\sim \sqrt{l}$ and $\sim \log{l}$ for the von Neumann and R\'enyi entropies, respectively. Our analytical and numerical calculations confirm this transition, which can be explained by the exponentially increasing density of states in the entanglement spectrum that we extract numerically. Disorder operators are further employed under various symmetries to study such a system. Both analytical and numerical results demonstrate that the scaling of the disorder operators also follows $\log{l}$ as the leading behavior, matching that of the R\'enyi entropy. We propose that the coefficient of the term $\log{l}$ is a universal constant shared by both the R\'enyi entropies and disorder operators. This universal constant could potentially help capture the underlying constraint physics of Motzkin walks.