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arXiv:1910.00872 · Physical review. B./Physical review. B

Influence of long-range interaction on Majorana zero modes

Andrzej Więckowski, Andrzej Ptok

Published Oct 17, 2019• 12 citations
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Abstract

Majorana bound states, with their non-Abelian properties, are candidates for the realization of fault-tolerant quantum computation. Here we study the influence of long-range many-body interactions on Majorana zero modes present in Kitaev chains. We show that these interactions can suppress the lifetime of the Majorana zero mode. We also discuss the role of long-range interactions on the Majorana state's spatial structure, and the overlap of the Majorana states localized at opposite ends of the chain. We have determined that increasing the interaction strength leads to decreasing of the stability of the Majorana modes. Moreover, we found out that interaction between particles located at more distant sites plays a more destructive role than the interaction between nearest neighbors.

DOI
10.1103/physrevb.100.144510

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Majorana bound states, with their non-Abelian properties, are candidates for the realization of fault-tolerant quantum computation. Here we study the influence of long-range many-body interactions on Majorana zero modes present in Kitaev chains. We show that these interactions can suppress the lifetime of the Majorana zero mode. We also discuss the role of long-range interactions on the Majorana state's spatial structure, and the overlap of the Majorana states localized at opposite ends of the chain. We have determined that increasing the interaction strength leads to decreasing of the stability of the Majorana modes. Moreover, we found out that interaction between particles located at more distant sites plays a more destructive role than the interaction between nearest neighbors.
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system at zero chemical potential as well as in the symmetric case of Δ = t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion interaction. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.
Abstract In the original Kitaev chain, Majorana zero modes (MZMs) at opposite edges have the same localization length. Here it is shown that a Hermitian finite-range extended Kitaev chain can instead support asymmetrically localized MZMs with different left and right localization lengths. Such asymmetry is excluded when all pairing amplitudes share a common phase, so the effect requires a genuinely extended model beyond the nearest-neighbor Kitaev chain. For the minimal next-nearest-neighbor case, the asymmetry originates from the inequivalence of the left and right boundary equations, which leads to different selections of decaying bulk solutions and yields a necessary-and-sufficient condition for its occurrence. A transfer-matrix reformulation further shows that the asymmetry is robust against sufficiently weak disorder preserving the block-triangular structure. For finite chains, the hybridization-induced energy splitting is exponentially small and controlled by the shorter localization length.
We theoretically investigate Majorana zero modes emerging in an anisotropic XY spin chain with second-neighbor interactions. The spin chain is mathematically equivalent to the Kitaev chain comprised of spinless fermions if only nearest-neighbor interactions are considered. Recent studies on the Kitaev chain with long-ranged interactions, have presented coexistence of several Majorana zero modes. Investigating the topological phase diagram of the anisotropic XY spin chain with second-neighbor interactions, we find coexistence of several Majorana zero modes similar to the Kitaev chain. However, we confirm that one of the zero modes is restricted into a Hilbert subspace. The mode is regarded as a so-called weak zero mode that should exhibit thermodynamical properties different from a (strong) zero mode appearing in the whole Hilbert space. Since the quantum statistics of spin is the same as a hard-core boson, we expect an experimental realization of the weak mode in not only quantum spin materials but also optical lattices for cold atom systems.
The Kitaev chain has been extensively explored in the context of uniform couplings, with studies focusing either on purely nearest-neighbor interactions or on systems dominated by long-range superconducting pairing. Building on these investigations, we introduce a hybrid Kitaev chain in which the lattice is partitioned into two segments: the left segment comprises nearest-neighbor couplings, while the right segment incorporates long-range pairing. To probe the role of the interface, we study two scenarios: a decoupled (suppressed hopping) case, where the segments are isolated, and a coupled case, where they are connected via interface hopping that enables tunneling. Using this setup, we investigate the behavior of Majorana zero modes at the interface between the two segments, finding that in the decoupled case, Majorana zero modes remain sharply localized at the left segment chain edges while massive Dirac modes remain in right segment chain edges, with their energies and localization strongly dependent on the long-range pairing exponent. Introducing a finite interface coupling enables transfer of Majorana zero modes from the edges of the left segment to those of the right segment of the chain. We characterize this dynamics by the fidelity of state transfer, dynamical rotation, and inverse participation ratio. We show the signature of Majorana zero mode transfer across the interface by the spatiotemporal profile of the probability distribution of the time evolved state.
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.
Topological phases in one-dimensional superconducting systems are commonly characterized by symmetry-protected invariants. These invariants determine the number of Majorana zero-energy boundary modes but do not specify their corresponding spatial structure. In this work, we present an analytical study of Majorana zero modes (MZMs) in an extended Kitaev chain with nearest- and next-nearest-neighbor couplings. By expressing the Hamiltonian in the Majorana basis, we derive a recursion relation whose characteristic roots completely determine the spatial structure of the zero modes and yield closed-form expressions for their amplitudes. We show that, even within a single topological phase, the MZMs can exhibit qualitatively distinct decay behaviors-monotonic decay, oscillatory decay, and perfectly localized states. Remarkably, boundary-origin MZMs need not have their maximum probability at the edge of the chain. They can instead exhibit maxima at interior lattice sites with an exponentially decaying envelope from either side of the maxima. Furthermore, the characteristic roots determine the length scale required for finite chains to reproduce the semi-infinite MZM structure, providing a direct link between Hamiltonian parameters, finite-size effects, and experimentally observable spatial profiles.