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Nonlocal Edge Mode Hybridization in the Long-Range Interacting Kitaev Chain

David Haink, Andreas Alexander Buchheit, Christof Weitenberg, Benedikt Fauseweh

Published Oct 2, 2026• 0 citations
Publisher

Abstract

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

DOI
10.3929/ethz-c-000807668

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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
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.
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 study the competition of disorder and superconductivity for a generalized Kitaev model in incommensurate potentials. The generalized Kitaev model describes one dimensional spinless fermions with long-range p-wave superconducting pairing, which decays with distance l as a power law ∼[Formula: see text]. We focus on the transition from the topological superconducting phase to the topologically trivial Anderson localized phase, and effects of the exponent α on this phase transition. In the topological superconducting phase, for a system under open boundary condition the amplitude of zero-mode Majorana fermion has a hybrid exponential-algebraic decay as the distance increases from the edge. In the Anderson localized phase, some single-particle states remain critical for very strong disorders and the number of critical states increases as α decreases. In addition, except for critical disorders, the correlation function always has an exponential decay at the short range and an algebraic decay at the long range. Phase transition points are also numerically determined and the topological phase transition happens earlier at a smaller disorder strength for a system with smaller α.
Topological phases in superconductors are of great interest due to their potential use for topological quantum computing. They are usually characterized by quasiparticles located at the edges of the superconductor. We investigate how long-range interactions affect the properties of the well-known Kitaev chain, which is solved self-consistently with the mean-field approximation. We find that for weak interactions, the Majorana edge modes have finite energy that decay algebraically to zero with the long-range power law. Strong long-range interactions are a special case due to their slow decay. The solution of the thermodynamic limit without finite-size effects is a local minimum to this problem, but with a larger ground-state energy in contrast to the discussed solution. Our results show the difference between the self-consistent and the Kitaev approach, which has clear constraints on long-range interactions.
We investigate an inter-leg coupled hybrid Kitaev ladder composed of two parallel superconducting chains with distinct pairing interactions. The upper chain of the ladder hosts conventional ppp-wave pairing, while the lower chain exhibits long-range pairing that decays algebraically with distance. We demonstrate that the mutual influence of long-range pairing exponent, chemical potential, and inter-leg coupling strength gives rise to a rich topological phase diagram characterized by multiple Majorana zero modes and massive Dirac modes. In particular, we show that the inter-leg coupling renormalizes the effective energy scales, leading to a systematic shift of the topological phase boundaries and enabling controlled tuning of the Majorana modes. Furthermore, we identify a transition from a two Majorana zero mode phase to a phase encapsulating four Majorana zero modes, as the long-range pairing exponent is varied. This transition is accompanied by a crossover regime in which Majorana zero modes coexist with massive Dirac modes, reflecting hybridization between edge and bulk excitations. This ladder thus provides a minimal and attractive platform for realizing the impact of a long-range pairing on topological phases. Our results highlight the potential of long-range hybrid systems for engineering tunable topological states relevant for quantum information applications.
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.