Area Law Violations and Quantum Phase Transitions in Modified Motzkin Walk Spin Chains

Area law violations for entanglement entropy in the form of a square root has recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here we consider a Motzkin walk with a different Hilbert space on each step of the walk spanned by elements of a {\it Symmetric Inverse Semigroup} with the direction of each step governed by its algebraic structure. This change alters the number of paths allowed in the Motzkin walk and introduces a ground state degeneracy sensitive to boundary perturbations. We study the frustration-free spin chains based on three symmetric inverse semigroups, $\cS^3_1$, $\cS^3_2$ and $\cS^2_1$. The system based on $\cS^3_1$ and $\cS^3_2$ provide examples of quantum phase transitions in one dimensions with the former exhibiting a transition between the area law and a logarithmic violation of the area law and the latter providing an example of transition from logarithmic scaling to a square root scaling in the system size, mimicking a colored $\cS^3_1$ system. The system with $\cS^2_1$ is much simpler and produces states that continue to obey the area law.