Parafermions are exotic quasiparticles with non-Abelian fractional statistics that can be realized and stabilized in 1-dimensional models that are generalizations of the Kitaev p-wave wire. We study the simplest generalization, i.e. the Z3 parafermionic chain. Using a Jordan-Wigner transform we focus on the equivalent three-state chiral clock model, and study its rich phase diagram using the density matrix renormalization group technique. We perform our analyses using quantum entanglement diagnostics which allow us to determine phase boundaries, and the nature of the phase transitions. In particular, we study the transition between the topological and trivial phases, as well as to an intervening incommensurate phase which appears in a wide region of the phase diagram. The phase diagram is predicted to contain a Lifshitz type transition which we confirm using entanglement measures. We also attempt to locate and characterize a putative tricritical point in the phase diagram where the three above mentioned phases meet at a single point.Introduction-There has been concerted effort to engineer systems with stable Majorana bound states, and other anyonic quasiparticles, for use in the topological quantum computation architecture [1][2][3][4][5][6][7]. For example, there has been recent progress in attempts to isolate Majorana bound states in quantum nanowires [5,[8][9][10] and in superconductor surfaces implanted with a line of magnetic impurities [11]. These quasi-1D systems effectively realize a version of the Kitaev p-wave wire model [12], and are predicted to have a gapped topological phase which supports characteristic Majorana bound states at the ends of the wire.While the boundary modes in these heterostructure systems are non-Abelian anyons, they are unfortunately known to be insufficient for universal quantum computation. A possible remedy for this problem has been to look for more exotic nonAbelian excitations. For example, Fendley has recently suggested exploring one-dimensional Z N para-fermionic models which support topological phases with more computationally efficient non-Abelian anyon bound states [13]. Still, the Z N non-Abelian anyons are not able to perform universal quantum computation, however they can be leveraged to create a 2D phase with Fibonaccci anyons, which are universal [14]. These promising features have spurred wide spread interest in these models, and has led to many analytical and numerical studies, including several experimental proposals for realizing these topological phases .In this work, we continue along these lines of research by exploring the rich phase diagram of the Z 3 para-fermionic chain; though for ease of calculation we actually study the Jordan-Wigner transformed para-fermionic chain [40], including chiral interactions. The resulting model is the three state chiral clock model. This model re-surfaced in this context in Ref. 13 as a candidate for exhibiting non-Abelian bound states beyond Majorana fermions. It was shown analytically that para-fermionic boundary zero mod...
We report the experimental results on the magnetism of curvature-induced helical carbon nanotubes (HCNTs). It is demonstrated that without any magnetic impurities in the sample, the as-prepared HCNTs show clear ferromagnetism with a Curie point as high as 970 K
In this article we apply the random forest machine learning model to classify 1D topological phases when strong disorder is present. We show that using the entanglement spectrum as training features the model gives high classification accuracy. The trained model can be extended to other regions in phase space, and even to other symmetry classes on which it was not trained and still provides accurate results. After performing a detailed analysis of the trained model we find that its dominant classification criteria captures degeneracy in the entanglement spectrum.
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