paper

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

arXiv:1710.10426 · doi:10.1088/1742-5468/aa9dcb

Abstract

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.

40 pages, 14 figures, A condensed version of this paper has been submitted to the Proceedings of the 2017 Granada Seminar on Computational Physics, Contains minor revisions and is closer to the Journal version. v3 includes an addendum that modifies the final Hamiltonian but does not change the main results of the paper

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