Area law violations and quantum phase transitions in modified Motzkin walk spin chains
Fumihiko Sugino, Pramod Padmanabhan
Abstract Area law violations for entanglement entropy in the form of a square root have 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 the elements of a 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 that is sensitive to boundary perturbations. We study the frustration-free spin chains based on three symmetric inverse semigroups, S 1 3 , S 2 3 and S 1 2 . The system based on S 1 3 and S 2 3 provides examples of quantum phase transitions in one dimension, 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 S 1 3 system. The system with S 1 2 is much simpler and produces states that continue to obey the area law.
arXiv:1710.10426•Journal of Statistical Mechanics Theory and Experiment•Jan 1, 2018•13 citations