Landscape geometry of Majorana zero modes in inhomogeneous superconductors
Abstract
Spatial inhomogeneity breaks translational symmetry and prevents conventional Bloch-band topological invariants from directly answering a practical question: do Majorana zero modes survive in a given inhomogeneous superconducting device? In this Letter, we develop a real-space landscape approach to address this question. Rather than solving the zero-energy equation as a boundary-value problem, we treat the spatial coordinate as an evolution parameter and recast the equation as a first-order dynamical system. A Majorana zero mode is then identified with a stable trajectory that satisfies the physical boundary condition and approaches the origin at large distance. We show that the landscape geometry fixes the dimensions of the stable and unstable subspaces, while the intersection of the stable subspace with the physical boundary subspace determines the number of Majorana zero modes. In the homogeneous limit, the topological phase transition is manifested as a geometric transition of the landscape. Applied to one-dimensional spinless p-wave superconductors and nanowire--superconductor systems, the approach yields sufficient bounds on both the amplitude and spatial gradient of the inhomogeneity, providing quantitative criteria for the design of Majorana devices.
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BibTeX
@misc{qiao2026landscape,
title = {{Landscape geometry of Majorana zero modes in inhomogeneous superconductors}},
author = {Guo-Jian Qiao and Zhi-Lei Zhang and Kang Xu and C. P. Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18138},
eprint = {2609.18138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18138},
}
RIS
TY - UNPB
TI - Landscape geometry of Majorana zero modes in inhomogeneous superconductors
AU - Guo-Jian Qiao
AU - Zhi-Lei Zhang
AU - Kang Xu
AU - C. P. Sun
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18138
UR - https://arxiv.org/abs/2609.18138
AB - Spatial inhomogeneity breaks translational symmetry and prevents conventional Bloch-band topological invariants from directly answering a practical question: do Majorana zero modes survive in a given inhomogeneous superconducting device? In this Letter, we develop a real-space landscape approach to address this question. Rather than solving the zero-energy equation as a boundary-value problem, we treat the spatial coordinate as an evolution parameter and recast the equation as a first-order dynamical system. A Majorana zero mode is then identified with a stable trajectory that satisfies the physical boundary condition and approaches the origin at large distance. We show that the landscape geometry fixes the dimensions of the stable and unstable subspaces, while the intersection of the stable subspace with the physical boundary subspace determines the number of Majorana zero modes. In the homogeneous limit, the topological phase transition is manifested as a geometric transition of the landscape. Applied to one-dimensional spinless p-wave superconductors and nanowire--superconductor systems, the approach yields sufficient bounds on both the amplitude and spatial gradient of the inhomogeneity, providing quantitative criteria for the design of Majorana devices.
ER -