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.Comment: 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 pape

Dyck Paths and Topological Quantum Computation

The fusion basis of Fibonacci anyons supports unitary braid representations that can be utilized for universal quantum computation. We show a mapping between the fusion basis of three Fibonacci anyons, $\{|1\rangle, |\tau\rangle\}$, and the two length 4 Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard $(2,2)$ Young diagrams using the Jones construction. This correspondence helps us construct the fusion basis of the Fibonacci anyons using Dyck paths as the number of standard $(N,N)$ Young tableaux is the Catalan number, $C_N$ . We then use the local Fredkin moves to construct a spin chain that contains precisely those Dyck paths that correspond to the Fibonacci fusion basis, as a degenerate set. We show that the system is gapped and examine its stability to random noise thereby establishing its usefulness as a platform for topological quantum computation. Finally, we show braidwords in this rotated space that efficiently enable the execution of any desired single-qubit operation, achieving the desired level of precision($\sim 10^{-3}$).Comment: 30 pages, 20 figure

Quantum phase transitions and localization in semigroup Fredkin spin chain

We construct an extended quantum spin chain model by introducing new degrees of freedom to the Fredkin spin chain. The new degrees of freedom called arrow indices are partly associated with the symmetric inverse semigroup S13. Ground states of the model fall into three different phases, and quantum phase transition takes place at each phase boundary. One of the phases exhibits logarithmic violation of the area law of entanglement entropy and quantum criticality, whereas the other two obey the area law. As an interesting feature arising by the extension, there are excited states due to disconnections with respect to the arrow indices. We show that these states are localized without disorder. © 2019, Springer Science+Business Media, LLC, part of Springer Nature11Nsciescopu