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arXiv:2506.20182 · Journal of Physics Condensed Matter

Analytical classification of Majorana zero-mode spatial profiles in extended Kitaev chains: probability maxima can shift inward

Vijay Pathak, Vaishnav Mallya, Sujit Sarkar

Published Apr 24, 2026• 0 citations
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Abstract

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.

DOI
10.1088/1361-648x/ae643a

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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.
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 investigate the spectral and topological properties of a zigzag Kitaev chain constructed from two diagonally coupled one dimensional Kitaev chains with a zero and finite superconducting pairing phase difference. Using a Bogoliubov-de Gennes formulation, we analyze the energy spectrum, distribution of Majorana zero modes (MZMs), the quasi-particle dispersion, and the winding number, respectively. For a zero phase difference, the resulting energy spectrum shows topological phases with two, four MZMs, and trivial regions. The phases of gap closure determine the topological phase boundaries. In particular, for the phase difference between φ=πφ=πφ=π, the degeneracy of MZMs is partially lifted, leading to modified topological phases compared to the case φ=0φ=0φ=0. The topological and trivial phase boundaries are further confirmed by evaluating the quasi-particle dispersion and the topological invariant, namely the winding number. We show that the zigzag Kitaev chain contributes independently to the total winding number ν=1ν= 1ν=1 and 222, giving rise to distinct topological phases that support two and four MZMs. The ν=0ν= 0ν=0 gives rise to a trivial region. The energy spectrum of systems corroborates the analytical phase boundaries and reveals characteristics associated with hybridization, enabling us to obtain the complete phase diagram of the zigzag model. Our results establish the zigzag Kitaev chain as a minimal platform for engineering MZM quantum computations, with potential applications in the study of topological phases and Majorana based qubit physics.
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 ααα and βββ, 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.
In this paper, we study two quasi-one-dimensional (1D) Kitaev models with ladder-like and tube-like spatial structures, respectively. Our results provide the phase diagrams and explicit expressions of the Majorana zero modes. The topological phase diagrams are obtained by decomposing the topological invariants and the topological conditions for topologically nontrivial phases are given precisely. For systems which belongs to topological class BDI, we obtain the regions in the phase diagrams where the topological numbers show even-odd effect. For the Kitaev tube model a phase factor induced by the magnetic flux in the axial direction of the tube is introduced to alter the classification of the tube Hamiltonian from class BDI to D. The Kitaev tube of class D is characterized by the Z 2 index when the number of chains is odd while 0, 1, 2 when the number of chains is even. The phase diagrams show periodic behaviors with respect to the magnetic flux. The bulk-boundary correspondence is demonstrated by the observations that the topological conditions for the bulk topological invariant to take nontrivial values are precisely those for the existence of the Majorana zero modes.
Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. For a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, which has an exact analytical solution. We use this solution to determine the asymptotic behaviors of the Majorana wave functions at large distances from the edge of the chain for several infinite-range models described in the literature with focus on a model with slow power-law falloff of pairing and hopping amplitudes. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.
Majorana zero modes, spontaneous Z2fZ_2^{f}Z2f​ symmetry breaking, and long range edge correlation in Kitaev chains: analytic solutions and density-matrix-renormalization-group study
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system in the half filled limit as well as in the symmetric case of Δ=t\Delta=tΔ=t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion. We show explicitly that the emergence of Majorana zero mode is associated with the spontaneous symmetry broken of fermion number parity operator Z2fZ_2^fZ2f​, which is +1+1+1 for even number of fermions and −1-1−1 for odd number of fermions. Our results are in support of recent proposal by Klassen and Wen\cite{Klassen}, who proposed such spontaneous symmetry broken in systems with Majorana zero modes. 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.