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.
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.
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.
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.
The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents α and β, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.
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.
We study the effects of disorder on a Kitaev chain with longer-range hopping and pairing terms which is capable of forming local zero energy excitations and, hence, serves as a minimal model for localization-protected edge qubits. The clean phase diagram hosts regions with 0, 1, and 2 Majorana zero-modes (MZMs) per edge. Using a semianalytic approach corroborated by numerical calculations of the entanglement degeneracy, we show how phase boundaries evolve under the influence of disorder. While in general the 2 MZM region is stable with respect to moderate disorder, stronger values drive transition towards the topologically trivial phase. We uncover regions where the addition of disorder induces local zero-modes absent for the corresponding clean system. Interestingly, we discover that disorder destroys any direct transition between phases with zero and two MZMs by creating a tricritical point at the 2-0 MZM boundary of the clean system. Finally, motivated by recent experiments, we calculate the characteristic signatures of the disorder phase diagram as measured in dynamical local and nonlocal ``qubit'' correlation functions. Our work provides a minimal starting point to investigate the coherence properties of local qubits in the presence of disorder.
We investigate the non-Hermitian Kitaev chain with non-reciprocal hopping amplitudes and asymmetric superconducting pairing. We work out the symmetry structure of the model and show that particle-hole symmetry (PHS) is preserved throughout the entire parameter regime. As a consequence of PHS, the topological phase transition point of a finite open chain coincides with that of the periodic (infinite) system. By explicitly constructing the zero-energy wave functions (Majorana modes), we show that Majorana modes necessarily occur as reciprocal localization pairs accumulating on opposite boundaries, whose combined probability density exhibits an exact cancellation of the non-Hermitian skin effect for the zero energy modes. Excited states, by contrast, generically display skin-effect localization, with particle and hole components accumulating at opposite ends of the system. At the level of bulk topology, we further construct a Z2 topological invariant in restricted parameter regimes that correctly distinguishes the topological and trivial phases. Finally, we present the topological phase diagram of the non-Hermitian Kitaev chain across a broad range of complex parameters and delineate the associated phase boundaries.