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Iranian Journal of Physics Research

One-dimensional p-wave superconductivity with long-range hopping and pairing

Ansari, S, Rouhollah Jafari

Published Jul 1, 2020• 0 citations
Publisher

Abstract

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.

DOI
10.47176/ijpr.20.2.35211

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•Oct 2, 2026•0 citations
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.
We consider the Kitaev chain model with finite and infinite range in the hopping and pairing parameters, looking in particular at the appearance of Majorana zero-energy modes and massive edge modes. We study the system both in the presence and in the absence of time-reversal symmetry, by means of topological invariants and exact diagonalization, disclosing very rich phase diagrams. In particular, for extended hopping and pairing terms, we can get as many Majorana modes at each end of the chain as the neighbors involved in the couplings. Finally, we generalize the transfer matrix approach useful to calculate the zero-energy Majorana states at the edges for a generic number of coupled neighbors.
We discover novel topological effects in the one-dimensional Kitaev chain modified by long-range Hamiltonian deformations in the hopping and pairing terms. This class of models display symmetry-protected topological order measured by the Berry/Zak phase of the lower-band eigenvector and the winding number of the Hamiltonians. For exponentially decaying hopping amplitudes, the topological sector can be significantly augmented as the penetration length increases, something experimentally achievable. For power-law decaying superconducting pairings, the massless Majorana modes at the edges get paired together into a massive nonlocal Dirac fermion localized at both edges of the chain: a new topological quasiparticle that we call topological massive Dirac fermion. This topological phase has fractional topological numbers as a consequence of the long-range couplings. Possible applications to current experimental setups and topological quantum computation are also discussed.
The power-law profile of hopping and superconductivity introduces rich critical phenomena that are not possible to explore in the nearest-neighbor models. Having this in mind, we construct the Su-Schrieffer-Heeger (SSH)-Kitaev model with mono-, bi-, and tri-partite sub-lattices where superconductivity [hopping] is of power-law Kitaev [SSH] type. The long-range (LR) superconductivity can alone gap out the zero-momentum critical line while LR hopping alters the finite-momentum critical line. There exists open-ended critical line at vanishing superconducting gap on the phase diagram for all versions of the SSH-Kitaev model. Interestingly, LR superconductivity (hopping) promotes massive Dirac (Majorana zero) modes, while in a finite-size system, their emergence in open boundary conditions is caused by the bulk gap of the anti-periodic Hamiltonian in momentum space. This indicates an unconventional bulk-boundary correspondence for power-law models as compared to what is seen in the short-range models. Importantly, the thermodynamic phase boundaries are correctly reproduced by the above unconventional correspondence. Intriguingly, an even (odd) number of sub-lattices yields an integer (half-integer) winding number while the massive Dirac modes continue to survive irrespective of the sub-lattice profile as long as there exists LR pairing only.
We study the static and dynamical properties of a long-range Kitaev chain, i.e. a p -wave superconducting chain in which the superconducting pairing decays algebraically as [Formula: see text], where l is the distance between the two sites and [Formula: see text] is a positive constant. Considering very large system sizes, we show that when [Formula: see text], the system is topologically equivalent to the short-range Kitaev chain with massless Majorana modes at the ends of the system; on the contrary, for [Formula: see text], there exist symmetry protected massive Dirac end modes. We further study the dynamical phase boundary of the model when periodic [Formula: see text]-function kicks are applied to the chemical potential; we specially focus on the case [Formula: see text] and analyze the corresponding Floquet quasienergies. Interestingly, we find that new topologically protected massless end modes are generated at the quasienergy [Formula: see text] (where T is the time period of driving) in addition to the end modes at zero energies which exist in the static case. By varying the frequency of kicking, we can produce topological phase transitions between different dynamical phases. Finally, we propose some bulk topological invariants which correctly predict the number of massless end modes at quasienergies equal to 0 and [Formula: see text] for a periodically kicked system with [Formula: see text].
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.
In one-dimensional p-wave superconductors with short-range interactions, topologically protected Majorana modes emerge, whose mass decays exponentially with system size, as first shown by Kitaev. In this Letter, we extend this prototypical model by including power law long-range interactions within a self-consistent framework, leading to the self-consistent long-range Kitaev chain (seco-LRKC). In this model, the gap matrix acquires a rich structure where short-range superconducting correlations coexist with long-range correlations that are exponentially localized at both chain edges simultaneously. As a direct consequence, the topological edge modes hybridize even if their wave function overlap vanishes, and the edge mode mass inherits the asymptotic scaling of the interaction. In contrast to models with imposed power law pairing, where massive Dirac modes emerge for exponents x^2