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Physical review. B./Physical review. B

Topological phases induced by the Aubry-André-Harper potential in the longer-range Kitaev superconducting chain

Chunhui Zhang, Li Sheng, Dingyu Xing

Published Sep 13, 2022• 5 citations
Publisher

Abstract

We investigate the one-dimensional longer-range Kitaev chain with an on-site complex Aubry-Andr\'e-Harper (AAH) potential. For the incommensurate (commensurate) case, by calculating the real-space (Bloch) winding number and energy spectrum we discover the AAH potential-induced phases and present the related phase diagrams. We also find that the imaginary phase shift in AAH potentials can suppress these topological phases for both cases. For the incommensurate case, the inverse participate ratios of the eigenstates with the smallest $|E|\mathrm{s}$ indicate the existence of the incommensurate AAH potential-induced phases, and the result of the mean inverse participate ratio shows that these incommensurate AAH potential-induced phases are not always strongly localized.

DOI
10.1103/physrevb.106.094203

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We investigate the one-dimensional longer-range Kitaev chain with an on-site complex Aubry-Andr'e-Harper (AAH) potential. For the incommensurate (commensurate) case, by calculating the real-space (Bloch) winding number and energy spectrum we discover the AAH potential-induced phases and present the related phase diagrams. We also find that the imaginary phase shift in AAH potentials can suppress these topological phases for both cases. For the incommensurate case, the inverse participate ratios of the eigenstates with the smallest ∣E∣s|E|\mathrm{s}∣E∣s indicate the existence of the incommensurate AAH potential-induced phases, and the result of the mean inverse participate ratio shows that these incommensurate AAH potential-induced phases are not always strongly localized.
We present a detailed study of the topological properties of the Kitaev chain with long-range pairing terms and in the presence of an Aubry-André-Harper on-site potential.Specifically, we consider algebraically decaying superconducting pairing amplitudes; the exponent of this decay is found to determine a critical pairing strength, below which the chain remains topologically trivial.Above the critical pairing, topological edge modes are observed in the central gap.For sufficiently fast decay of the pairing, these modes are identified as Majorana zero modes.However, if the pairing term decays slowly, the modes become massive Dirac modes.Interestingly, these massive modes still exhibit a true level crossing at zero energy, which points towards an intimate relation to Majorana physics.We also observe a clear lack of bulk-boundary correspondence in the long-range system, where bulk topological invariants remain constant, while dramatic changes appear in the behavior at the edge of the system.In addition to the central gap around zero energy, the Aubry-André-Harper potential also leads to other energy gaps at nonzero energy.As for the analogous short-range model, the edge modes in these gaps can be characterized through a 2D Chern invariant.However, in contrast to the short-range model, this topological invariant does not correspond to the number of edge mode crossings anymore.This provides another example of the weakening of the bulk-boundary correspondence occurring in this model.Finally, we discuss possible realizations of the model with ultracold atoms and condensed matter systems.
We investigate localization and topological properties of a dimerized Kitaev chain with ppp-wave superconducting correlations and a quasiperiodically modulated chemical potential. With regard to the localization studies, we demonstrate the existence of distinct phases, such as, the extended phase, the critical (intermediate) phase, and the localized phase that arise due to the competition between the dimerization and the on-site quasiperiodic potential. Most interestingly, the critical phase comprises of two different anomalous mobility edges that are found to exist between the extended to the localized phase, and between the critical (multifractal) and localized phases. We perform our analysis employing the inverse and the normalized participation ratios, fractal dimension, and the level spacing. Subsequently, a finite-size analysis is done to provide support of our findings. Furthermore, we study the topological properties of the zero-energy edge modes via computing the real-space winding number and number of the Majorana zero modes present in the system. We specifically illustrate that our model exhibits a phase transition from a topologically trivial to a nontrivial phase (topological Anderson phase) beyond a critical dimerization strength under the influence of the quasiperiodic potential strength. Finally, in presence of a large potential, we demonstrate that the system undergoes yet another transition from the topologically nontrivial to an Anderson localized phase. Thus, we believe that our results will aid exploration of fundamentally different physics pertaining to the critical and the topological Anderson phases.
Using exact numerical methods, we study the interplay between p-wave superconductivity and off-diagonal modulation in the incommensurate off-diagonal Aubry-André model with p-wave pairing. When chemical potential is zero, the modulation not only leads to an Anderson-like localization with all bulk states changing into critical ones, but also induces a topological phase transition accompanying the disappearing of exponentially localized zero-energy edge Majorana fermions. These two transitions happen at different places and true Anderson localization is absent. With a finite chemical potential, the quasi-periodicity of the modulation causes band splitting and drives the system into a band insulator phase. It has a fixed incommensurate particle density which is determined by the modulation frequency. Phase diagrams are identified, and can be tested with existing proposals for experimental realizations of the effective off-diagonal disorder and the p-wave pairing.
In this paper, we have considered the extended version of the Kitaev model in one dimension, i.e., a long-range p-wave superconducting wire. In the long-range Kitaev chain, the superconducting hopping and pairing terms in the Hamiltonian decay, independently, in a power-law fashion , where l is the distance between the two sites and x is some positive constant. We have studied the appearance of Majorana zero-energy edge modes and also, massive Dirac edge modes by exact diagonalization, as well as analytical computations. Exact diagonalization indicates the existence of both kinds of massless and massive edge modes in the energy spectrum. Furthermore, we obtain the phase diagram and the topological phase transitions by calculating the winding number, which is the topological invariant.
The Anderson localization phase transition in the Aubry–André–Harper (AAH) model with p -wave superconducting (SC) pairing is numerically investigated by suddenly changing the on-site potential from zero to various finite values which fall into the extended, critical and localized phase regimes shown in this model. The time evolution of entanglement entropy (EE), mean width of wave packets and Loschmidt echo of the system exhibit distinct but consistent dynamical signatures in those three phases. Specifically, the EE grows as a power function of time with the exponent of which varies in the extended phase but keeps almost unchanged in the critical phase for different quench parameters. However, if the system is in the localized phase after a quench, the EE grows much slower and will soon get saturated. The time-dependent width of wave packets in the system shows similar behaviors as the EE. In addition, from the perspective of the dynamical phase transition, we find that the Loschmidt echo oscillates and always keeps finite when the system is quenched in the extended phase. In contrast, in the critical or localized phase, the echo will reach to zero at some time intervals or will decay almost to zero after a long-time evolution. The universal features of these quantities in the critical phase of the system with various SC pairing amplitudes are also observed.
We propose a one-dimensional generalized Aubry-Andr\'e-Harper (AAH) model with off-diagonal hopping and staggered on-site potential. We find that the localization transitions could be multiple reentrant with the increasing of staggered on-site potential. The multiple localization transitions are verified by the quantum static and dynamic measurements such as the inverse or normalized participation ratios, fractal dimension, and survival probability. Based on the finite-size scaling analysis, we also obtain an interesting intermediate phase where the extended, localized, and critical states are coexistent in certain regimes of model parameters. These results are quite different from those in the generalized AAH model with off-diagonal hopping, and can help us to find novel quantum phases and new localization phenomena in the disordered systems.