Logarithmic entanglement scaling for arbitrary blocks in the Motzkin spin chain

Let a Motzkin spin chain have length $2n$, and let a block consist of any $L$ consecutive spins with $L\ll 2n$, without requiring the block to be centered. Denote its reduced density matrix by $\rho_B$ and its von Neumann entanglement entropy by

$$S_L=-\operatorname{Tr}(\rho_B\log\rho_B).$$

Logarithmic entanglement-scaling conjecture. The entanglement entropy of any such block, to leading order, scales as

$$S_L\sim\log L.$$

The paper derives this logarithmic scaling, with leading term $\frac{1}{2}\log L$, for blocks centered in a chain with $1\ll L\ll n$, and conjectures that the same leading-order logarithmic behavior holds for blocks in arbitrary positions. The precise asymptotics away from the center are not established here.