Singularity and universality from von Neumann to Rényi entanglement entropy and disorder operator in Motzkin chains

Rényi entanglement entropy is widely used to study quantum entanglement properties in strongly correlated systems, and its analytic continuation as the Rényi index 𝑛→1 is often believed to yield von Neumann entanglement entropy. However, earlier theoretical analysis indicated that this process exhibits a singularity for the colored Motzkin spin chain problem, leading to different system size 𝑙 scaling behaviors of ∼√𝑙 and ∼ln⁡𝑙 for the von Neumann and Rényi 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 we extract numerically. Moreover, disorder operators can be measured easily in numerics and experiments and always have area-law or volume-law scaling similar to entanglement entropies. We further explored disorder operators under various symmetries of such a system. Both analytical and numerical results demonstrate that the scaling behaviors of disorder operators also follow ln⁡𝑙 as the leading term, matching that of Rényi entropy. Moreover, we find that the coefficient of the term ln⁡𝑙 is a universal constant shared by both the Rényi entropy and disorder operators and propose that it can probe the underlying constraint physics of Motzkin walks.