Abstract
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-Andr\'e-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 initimate 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-Andr\'e-Harper potential also leads to other energy gaps at non-zero 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 for 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.
Highlights
Topological phases of matter have been established as an intriguing research area in condensed matter physics [1,2,3,4,5]
We investigate how energy spectra and winding numbers depend on the superconducting pairing term, and find that the presence of an AAH modulation leads to a critical superconducting pairing C: below this critical value, we observe neither Majorana modes (MMs) nor massive Dirac modes (MDM) in the system
We investigate how the presence of MMs and MDMs depends on the value of the superconducting pairing
Summary
Topological phases of matter have been established as an intriguing research area in condensed matter physics [1,2,3,4,5]. The Kitaev chain exhibits time-reversal, chiral and particle-hole symmetry; it belongs to the BDI class of the topological classification [6] This model can host nonlocal Majorana modes (MMs), i.e., zero-energy modes localized at the two boundaries of the chain. For slow power-law decay, i.e., for decay exponent α given by α 1, the MMs of the short-range model (α 1) coalesce to form massive nonlocal edge states called massive Dirac modes (MDM) These new edge states lie within the bulk gap and are topologically robust against local perturbations that do not violate fermionic parity and particle-hole symmetry. IV, we analyze the effects of the AAH modulation on the central gap behavior of the longrange Kitaev chain This includes characterizing the MMs and MDMs and their corresponding winding numbers, studying the existence of a critical superconducting pairing, and a detailed analysis of the level crossings at zero energy.
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