We present a unified study of the effect of periodic, quasiperiodic, and disordered potentials on topological phases that are characterized by Majorana end modes in one-dimensional p-wave superconducting systems. We define a topological invariant derived from the equations of motion for Majorana modes and, as our first application, employ it to characterize the phase diagram for simple periodic structures. Our general result is a relation between the topological invariant and the normal state localization length. This link allows us to leverage the considerable literature on localization physics and obtain the topological phase diagrams and their salient features for quasiperiodic and disordered systems for the entire region of parameter space.
We present a comprehensive study of two of the most experimentally relevant extensions of Kitaev's spinless model of a 1D p-wave superconductor: those involving (i) longer range hopping and superconductivity and (ii) inhomogeneous potentials. We commence with a pedagogical review of the spinless model and, as a means of characterizing topological phases exhibited by the systems studied here, we introduce bulk topological invariants as well as those derived from an explicit consideration of boundary modes. In time-reversal invariant systems, we find that the longer range hopping leads to topological phases characterized by multiple Majorana modes. In particular, we investigate a spin model, which respects a duality and maps to a fermionic model with multiple Majorana modes; we highlight the connection between these topological phases and the broken symmetry phases in the original spin model. In the presence of time-reversal symmetry breaking terms, we show that the topological phase diagram is characterized by an extended gapless regime. For the case of inhomogeneous potentials, we explore phase diagrams of periodic, quasiperiodic, and disordered systems. We present a detailed mapping between normal state localization properties of such systems and the topological phases of the corresponding superconducting systems. This powerful tool allows us to leverage the analyses of Hofstadter's butterfly and the vast literature on Anderson localization to the question of Majorana modes in superconducting quasiperiodic and disordered systems, respectively. We briefly touch upon the synergistic effects that can be expected in cases where long-range hopping and disorder are both present.
Entanglement in quantum XY spin chains of arbitrary length is investigated via a recentlydeveloped global measure suitable for generic quantum many-body systems. The entanglement surface is determined over the phase diagram, and found to exhibit structure richer than expected. Near the critical line, the entanglement is peaked (albeit analytically), consistent with the notion that entanglement-the non-factorization of wave functions-reflects quantum correlations. Singularity does, however, accompany the critical line, as revealed by the divergence of the field-derivative of the entanglement along the line. The form of this singularity is dictated by the universality class controlling the quantum phase transition. [5,6,7,8], where it can play the role of a diagnostic of quantum correlations. Quantum phase transitions [9] are transitions between qualitatively distinct phases of quantum many-body systems, driven by quantum fluctuations. In view of the connection between entanglement and quantum correlations, one anticipates that entanglement will furnish a dramatic signature of the quantum critical point. From the viewpoint of quantum information, the more entangled a state, the more resources it is likely to possess. It is thus desirable to study and quantify the degree of entanglement near quantum phase transitions. By employing entanglement to diagnose many-body quantum states one may obtain fresh insight into the quantum many-body problem.To date, progress in quantifying entanglement has taken place primarily in the domain of bipartite systems [10]. Much of the previous work on entanglement in quantum phase transitions has been based on bipartite measures, i.e., focus has been on entanglement either between pairs of parties [5,6] or between a part and the remainder of a system [7]. For multipartite systems, however, the complete characterization of entanglement requires the consideration of multipartite entanglement, for which a consensus measure has not yet emerged.Singular and scaling behavior of entanglement near quantum critical points was discovered in important work by Osterloh and co-workers [6], who invoked Wootters' bipartite concurrence [11] as a measure of entanglement. In the present letter, we apply a recently-developed global measure that provides a holistic, rather than bipartite, characterization of the entanglement of quantum manybody systems. Our focus is on one-dimensional spin systems, specifically ones that are exactly solvable and
We explore the salient features of the 'Kitaev ladder', a two-legged ladder version of the spin-1/2 Kitaev model on a honeycomb lattice, by mapping it to a one-dimensional fermionic p-wave superconducting system. We examine the connections between spin phases and topologically non-trivial phases of non-interacting fermionic systems, demonstrating the equivalence between the spontaneous breaking of global Z2 symmetry in spin systems and the existence of isolated Majorana modes. In the Kitaev ladder, we investigate topological properties of the system in different sectors characterized by the presence or absence of a vortex in each plaquette of the ladder. We show that vortex patterns can yield a rich parameter space for tuning into topologically non-trivial phases. We introduce a new topological invariant which explicitly determines the presence of zero energy Majorana modes at the boundaries of such phases. Finally, we discuss dynamic quenching between topologically non-trivial phases in the Kitaev ladder, and in particular, the post-quench dynamics governed by tuning through a quantum critical point. 31]. Some analogous studies in the Kitaev honeycomb system have identified Majorana modes bound to vortices and schemes for their manipulation [15]. In the ladder system, the isolated modes, as opposed to being present at vortices, completely parallel the 1D p-wave superconducting system in being present at the interface between topologically trivial and non-trivial segments. Moreover, we find that the periodic patterns mentioned above provide a new route for finding phases and configurations that support these modes. In principle, for translationally invariant systems, such phases can be characterized by a Z 2 topological index that considers the Berry phase accumulated by the eigenvectors in traversing the first Brillouin zone. In practice, we find that for complex periodic patterns, such a treatment proves to be rather involved, and can be replaced by the evaluation of a much more direct topological invariant derived from the equations of motion. Our method generalizes the identification of zero energy plane waves by Wen and Zee [32] to the case of evanescent modes. We use this scheme to pinpoint several different configurations of periodic patterns that yield localized Majorana modes in the bulk or at the ends of the Kitaev ladder. Our analysis also provides an alternate route for isolating Majorana modes in 1D p-wave superconductors by applying appropriate periodic potentials.The manipulation of the Majorana modes requires dynamically changing the parameters associated with the Hamiltonian describing the system. We consider dynamics from two angles. Based on the parallel between spin and fermionic systems, we briefly outline the first steps necessary for realizing schemes in the Kitaev ladder that are analogs of those recently proposed in the 1D p-wave superconductor. Our second study of dynamics entails quenching between topologically trivial and non-trivial regions, an operation that is necessary for several ...
With the surge of research in quantum information, the issue of producing entangled states has gained prominence. Here, we show that judiciously bringing together two systems of strongly interacting electrons with vastly differing ground states -the gapped BCS superconductor and the Luttinger liquid, -can result in quantum entanglement. We propose three sets of measurements involving single-walled metallic carbon nanotubes (SWNT) which have been shown to exhibit Luttinger liquid physics, to test our claim and as nanoscience experiments of interest in and of themselves.
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