Unpaired Majorana fermions in quantum wires

2000/10/30 by Alexei Kitaev, A Yu Kitaev · 1 voice · 283 citations
Mathematics · Physics and Astronomy · #Rare-earth and actinide compounds #Spectral Theory in Mathematical Physics #Topological Materials and Phenomena #cond-mat.mes-hall #quant-ph

paper · pdf · doi:10.1070/1063-7869/44/10s/s29

16 pages, 5 figures, Latex and epsf, to be included in the proceedings of the Mesoscopic And Strongly Correlated Electron Systems conference (9-16 July 2000, Chernogolovka, Moscow Region, Russia), one reference added

arxiv created 2000/10/30 · openalex publication_date 2001/10/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/04

Abstract

Certain one-dimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length L possesses two ground states with an energy difference proportional to exp(- L / l 0 ) and different fermionic parities. Such systems can be used as qubits since they are intrinsically immune to decoherence. The property of a system to have boundary Majorana fermions is expressed as a condition on the bulk electron spectrum. The condition is satisfied in the presence of an arbitrary small energy gap induced by proximity of a three-dimensional p-wave superconductor, provided that the normal spectrum has an odd number of Fermi points in each half of the Brillouin zone (each spin component counts separately).

Cited by

• Kibble-Zurek mechanism in a topological phase transition
• Fermionic Villain model with exact lattice chiral symmetries
• Tensor network solvers for ultra-large tight-binding Hamiltonians: algorithms and applications
• Hidden Zeeman Field in Odd-Parity Magnets: An Ideal Platform for Topological Superconductivity
• Local Information Flow in Quantum Quench Dynamics
• Unveiling hidden features of the Kitaev model through a complex-network analysis
• Anyonic interferometry without anyons: how a flux qubit can read out a topological qubit
• Helical Hinge Majorana Modes in Iron-Based Superconductors
• Topology and Edge Modes in Quantum Critical Chains
• Stability of quantum concatenated-code Hamiltonians
• Wilson polygons and the topology of zero-dimensional systems
• Lattice Supersymmetry and Order-Disorder Coexistence in the Tricritical Ising Model
• Selective Equal-Spin Andreev Reflections Induced by Majorana Fermions
• Topological insulators and superconductors
• Non-Abelian statistics versus the Witten anomaly
• Stability of Majorana fermions in proximity-coupled topological insulator nanowires
• Transport through Majorana nanowires attached to normal leads
• トップインタビュー グラスバレー50周年を機に新製品投入--EDIUSと他のGV製品との連携を強化--トムソン・カノープス(株)代表取締役上級副社長・DESビジネスライン統括 平野景由氏 (特集 NAB2009&KOBA2009速報)
• Roadmap on Atomtronics: State of the art and perspective
• Andreev spin qubits in multichannel Rashba nanowires
• Proximity-induced supercurrent through topological insulator based nanowires for quantum computation studies
• Search for Majorana Fermions in Multiband Semiconducting Nanowires
• Classification of symmetry enriched topological phases with exactly solvable models
• Effect of Interaction on the Majorana Zero Modes in the Kitaev Chain at Half Filling
• The eigenstate thermalization hypothesis in constrained Hilbert spaces: A case study in non-Abelian anyon chains
• Search for Majorana Fermions in Superconductors
• Reentrant topological phase transition in a bridging model between Kitaev and Haldane chains
• Anyonization of bosons in one dimension: an effective swap model
• Tunable Topological Superconductivity by Fully Compensated Ferrimagnets
• Magnetononlinear Hall effect from multigap topology in metal-organic frameworks
• Web of dualities on non-orientable surfaces
• From coupled ℤ₃ Rabi models to the ℤ₃ Potts model
• Decoherence of Majorana qubits by noisy gates
• Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model
• Majorana fermions and odd-frequency Cooper pairs in a normal-metal nanowire proximity-coupled to a topological superconductor
• Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover
• Topological superconducting phase in helical Shiba chains
• Majorana edge states in interacting two-chain ladders of fermions
• Tri-Dirac surface modes in topological superconductors
• Majorana fermion exchange in strictly one-dimensional structures
• Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata
• Interaction-enabled topological crystalline phases
• Ballistic Majorana nanowire devices
• Interplay between Coulomb blockade and Josephson effect in a topological superconductor–quantum dot device
• Edge modes in a frustrated quantum Ising chain
• Manipulating unpaired Majorana fermions in a quantum spin chain
• Algorithm 923
• TopologicalπJosephson junction in superconducting Rashba wires
• Edge instabilities of topological superconductors
• Local Convertibility and the Quantum Simulation of Edge States in Many-Body Systems
• Proximity-induced superconductivity and Josephson critical current in quantum spin Hall systems
• Topological quantum phase transitions in the spin–singlet superconductor with Rashba and Dresselhaus (110) spin–orbit couplings
• Effects of disorder on Coulomb-assisted braiding of Majorana zero modes
• Dynamics of a quantum phase transition and relaxation to a steady state
• Manipulating Topological Edge Spins in a One-Dimensional Optical Lattice
• Floquet engineering of long-rangep-wave superconductivity
• Topological Quantum Buses: Coherent Quantum Information Transfer between Topological and Conventional Qubits
• One-dimensional itinerant interacting non-Abelian anyons
• Limits of thermal conductance quantization in chiral topological Josephson junctions
• Regularized universal topological markers for Dirac systems
• Commensurate and incommensurate states of topological quantum matter
• Degenerate quantum gases with spin–orbit coupling: a review
• Topologically trivial zero-bias conductance peak in semiconductor Majorana wires from boundary effects
• Vortex and Meissner phases of strongly interacting bosons on a two-leg ladder
• Dissipative preparation of Chern insulators
• Engineering one-dimensional topological phases onp-wave superconductors
• Interplay of magnetic order, pairing, and phase separation in a one-dimensional spin-fermion model
• Layer Construction of 3D Topological States and String Braiding Statistics
• Flux insertion, entanglement, and quantized responses
• Supercurrent Interference in Few-Mode Nanowire Josephson Junctions
• Intrinsic topological superconductivity with exactly flat surface bands in the quasi-one-dimensional A2Cr3As3 (A=Na, K, Rb, Cs) superconductors
• Fragmentation, Zero Modes, and Collective Bound States in Constrained Models
• Leakage at zero temperature from changes in chemical potential in Majorana qubits
• Nontopological nature of the edge current in a chiralp-wave superconductor
• Disentanglement in the macroscopic limit
• Non-abelian Quantum Statistics on Graphs
• On the Fermion Level Crossing in the Electroweak Instanton Background
• Non-Hermitian Majorana modes protect degenerate steady states
• Doubled Shapiro steps in a topological Josephson junction
• Many-body computations by stochastic sampling in Hartree-Fock-Bogoliubov space
• Quantized zero-bias conductance plateau in semiconductor-superconductor heterostructures without topological Majorana zero modes
• Supercurrent Detection of Topologically Trivial Zero-Energy States in Nanowire Junctions
• Decaying spectral oscillations in a Majorana wire with finite coherence length
• Topologically nontrivial Andreev bound states
• Topological π Junctions from Crossed Andreev Reflection in the Quantum Hall Regime
• Formation and detection of Majorana modes in quantum spin Hall trenches
• Majorana-like localized spin density without bound states in topologically trivial spin-orbit coupled nanowires
• Majorana zero modes in a quantum wire platform without Rashba spin-orbit coupling
• Josephson junctions via anodization of epitaxial Al on an InAs heterostructure
• Supersymmetric Free Fermions and Bosons: Locality, Symmetry and Topology
• Minimal Zeeman field requirement for a topological transition in superconductors
• Two-dimensional Fermi gases near a p-wave resonance: Effect of quantum fluctuations
• Spatiotemporal evolution of topological order upon quantum quench across the critical point
• Resonant Andreev reflections probed by photon-assisted tunnelling at the atomic scale
• Quantum Dynamical Characterization and Simulation of Topological Phases With High-Order Band Inversion Surfaces
• Jordan–Wigner transformations for tree structures
• Multi-qudit states generated by unitary braid quantum gates based on Temperley-Lieb algebra
• Double-frequency Aharonov-Bohm effect and non-Abelian braiding properties of Jackiw-Rebbi zero-mode
• Origin of families andSO(18)grand unification
• Universal current correlations induced by the Majorana and fermionic Andreev bound states
• Free parafermions
• Tunneling spectroscopy and Majorana modes emergent from topological gapless phases in high-Tccuprate superconductors
• Breakdown of local convertibility through Majorana modes in a quantum quench
• Topological Invariants for Spin-Orbit Coupled Superconductor Nanowires
• Topological current divider
• Real-space topological invariant for time-quasiperiodic Majorana modes
• Corner states, hinge states and Majorana modes in SnTe nanowires
• Scheme for topological single electron pumping assisted by Majorana fermions
• Realizing universal quantum gates with topological bases in quantum-simulated superconducting chains
• Mixed-Order Symmetry-Breaking Quantum Phase Transition Far from Equilibrium
• Numerical study of PbTe-Pb hybrid nanowires for engineering Majorana zero modes
• Flux-induced topological superconductor in planar Josephson junction
• Quaternary Jordan-Wigner mapping and topological extended-kink phase in the interacting Kitaev ring
• Gapfull and gapless 1D Topological Superconductivity in Spin-Orbit Coupled Bilayer Graphene
• Exotic topological types of Majorana zero modes and their universal quantum manipulation
• Lie-algebraic approach to one-dimensional translationally invariant free-fermionic dissipative systems
• Many-terminal Majorana island: From topological to multichannel Kondo model
• Catalog of Andreev spectra and Josephson effects in structures with time-reversal-invariant topological superconductor wires
• Vectorial Control of the Spin–Orbit Interaction in Suspended InAs Nanowires
• Topological Superconductivity in a Planar Josephson Junction
• Majorana-Hubbard model on the square lattice
• Planar Josephson junction devices with narrow superconducting strips: Topological properties and optimization
• Majorana fermions in finite-size strips with in-plane magnetic fields
• Controllable photonic and phononic edge localization via optomechanically induced Kitaev phase
• Suppression of2πphase slip due to hidden zero modes in one-dimensional topological superconductors
• Prethermal Strong Zero Modes and Topological Qubits
• Topological order in the Kitaev/Majorana chain in the presence of disorder and interactions
• Correlated parity measurements as a probe of non-Abelian statistics in one-dimensional superconducting wires
• Electrically tunable chiral Majorana edge modes in quantum anomalous Hall insulator–topological superconductor systems
• Topological superconducting state and Majorana fermions in carbon nanotubes
• Anomalous quantum information scrambling for Z3 parafermion chains
• Topological phases and Anderson localization in off-diagonal mosaic lattices
• Robust Transport Signatures of Topological Superconductivity in Topological Insulator Nanowires
• Evolution of Density of States and a Spin-Resolved Checkerboard-Type Pattern Associated with the Majorana Bound State
• The quest and hope of Majorana zero modes in topological superconductor for fault-tolerant quantum computing: An introductory overview
• Crystalline Weyl semimetal phase in Quantum Spin Hall systems under magnetic fields
• Dynamics of a Majorana Trijunction in a Microwave Cavity
• Flag fault-tolerant error correction with arbitrary distance codes
• Minimal circuit for a flux-controlled Majorana qubit in a quantum spin-Hall insulator
• Phase-tunable second-order topological superconductor
• Absence of supercurrent sign reversal in a topological junction with a quantum dot
• Fractional Josephson Vortices and Braiding of Majorana Zero Modes in Planar Superconductor-Semiconductor Heterostructures
• Persistent current and zero-energy Majorana modes in a p-wave disordered superconducting ring
• Strong coupling between a topological qubit and a nanomechanical resonator
• Central Charge of Periodically Driven Critical Kitaev Chains
• Control and detection of Majorana bound states in quantum dot arrays
• Weyl points in systems of multiple semiconductor-superconductor quantum dots
• Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation
• Conductance smearing and anisotropic suppression of induced superconductivity in a Majorana nanowire
• Survival probability in a quenched Majorana chain with an impurity
• Chern-Simons fermionization approach to two-dimensional quantum magnets: Implications for antiferromagnetic magnons and unconventional quantum phase transitions
• Symmetry resolved entanglement: exact results in 1D and beyond
• Dynamical topological order parameters far from equilibrium
• Properties of some Hamiltonians describing topologically non-trivial fermionic systems
• Pumped heat and charge statistics from Majorana braiding
• Parafermions, induced edge states, and domain walls in fractional quantum Hall effect spin transitions
• Entanglement dualities in supersymmetry
• End states in a one-dimensional topological Kondo insulator in large-Nlimit
• \mathbb Z₂ Green's function topology of Majorana wires
• Floquet Topological Phase of Nondriven p-Wave Nonequilibrium Excitonic Insulators
• Multiple Andreev reflection and critical current in topological superconducting nanowire junctions
• Topological Superconductor to Anderson Localization Transition in One-Dimensional Incommensurate Lattices
• Molecular Machines for Quantum Error Correction
• Absence of Fibonacci anyons in Rydberg chains
• Time-reversal-invariant C2-symmetric higher-order topological superconductors
• Supercurrent blockade in Josephson junctions with a Majorana wire
• Complexity of Quantum Impurity Problems
• A road to reality with topological superconductors
• Combating quasiparticle poisoning with multiple Majorana fermions in a periodically-driven quantum wire
• Andreev reflection without Fermi surface alignment in high-Tcvan der Waals heterostructures
• Probing a topological quantum critical point in semiconductor-superconductor heterostructures
• Second-Order Topological Superconductors with Mixed Pairing
• Probing topological states through the exact non-Markovian decoherence dynamics of a spin coupled to a spin bath in the real-time domain
• Pattern-dependent proximity effect and Majorana edge mode in one-dimensional quasicrystals
• Exactly solvable models of spin liquids with spinons, and of three-dimensional topological paramagnets
• Weyl Josephson circuits
• Chiral topological superconductivity in Josephson junctions
• Teleportation-based quantum information processing with Majorana zero modes
• Probing Majorana localization of a phase-controlled three-site Kitaev chain with an additional quantum dot
• Majorana modes and p-wave superfluids for fermionic atoms in optical lattices
• Observability of a sharp Majorana transition in a few-body model
• Dispersive Readout of Majorana Qubits
• Proposal for Measuring the Parity Anomaly in a Topological Superconductor Ring
• Topological Phase Transition Driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics
• Explicit forms of zero modes in symmetric interacting Kitaev chain without and with dimerization
• Topological phases via engineered orbital hybridization in noncentrosymmetric optical lattices
• Topological Phases of Parafermions: A Model with Exactly Solvable Ground States
• Protocol Discovery for the Quantum Control of Majoranas by Differentiable Programming and Natural Evolution Strategies
• Braiding statistics and congruent invariance of twist defects in bosonic bilayer fractional quantum Hall states
• On sampling determinantal and Pfaffian point processes on a quantum computer
• Ising order in a magnetized Heisenberg chain subject to a uniform Dzyaloshinskii-Moriya interaction
• Majorana Edge States in Atomic Wires Coupled by Pair Hopping
• Interacting Majorana chain: Transport properties and signatures of an emergent two-dimensional weak topological phase
• Continuous-Variable Assisted Thermal Quantum Simulation
• Majorana modes in emergent-wire phases of helical and cycloidal magnet-superconductor hybrids
• Entanglement and Disordered-Enhanced Topological Phase in the Kitaev Chain
• Fidelity and visibility loss in Majorana qubits by entanglement with environmental modes
• Quantum gates for Majoranas zero modes in topological superconductors in one-dimensional geometry
• Exact Nonequilibrium Transport in the Topological Kondo Effect
• Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems
• Disorder-induced zero-bias peaks in Majorana nanowires
• Non-equilibrium current cumulants and moments with a point-like defect
• Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates
• Pairing Obstructions in Topological Superconductors
• Parity-to-charge conversion for readout of topological Majorana qubits
• Majorana Single-Charge Transistor
• Quantum anomalous Hall Majorana platform
• Chiral solution to the Ginsparg-Wilson equation
• Equivalence of topological insulators and superconductors
• Entanglement entropy in multi-leg Kitaev ladders with interface defects
• Electrical Manipulation of Majorana Fermions in an Interdigitated Superconductor-Ferromagnet Device
• Non-Abelian braiding of Majorana-like edge states and topological quantum computations in electric circuits
• Nearly ferromagnetic spin-triplet superconductivity
• Modeling noise and error correction for Majorana-based quantum computing
• Dyonic zero-energy modes
• Clock model and parafermions in Rashba nanowires
• Harnessing symmetry-protected topological order for quantum memories
• Uranium in its natural compounds
• Lifetime of Majorana qubits in Rashba nanowires with nonuniform chemical potential
• Topological Contextuality and Anyonic Statistics of Photonic-Encoded Parafermions
• Topological Entanglement-Spectrum Crossing in Quench Dynamics
• Fermionic matrix product states and one-dimensional topological phases
• Twofold twist defect chains at criticality
• Majorana fermions in T-shaped semiconductor nanostructures
• Do the surface Fermi arcs in Weyl semimetals survive disorder?
• Majorana Fermions and Topology in Superconductors
• Topological quantum computation based on chiral Majorana fermions
• Ising anyons in frustration-free Majorana-dimer models
• Effective theory and breakdown of conformal symmetry in a long-range quantum chain
• Correlation length, universality classes, and scaling laws associated with topological phase transitions
• Sachdev-Ye-Kitaev circuits for braiding and charging Majorana zero modes
• Non-Abelian braiding in spin superconductors utilizing the Aharonov-Casher effect
• Symmetry-protected non-Fermi liquids, kagome spin liquids, and the chiral Kondo lattice model
• Gap oscillations and Majorana bound states in magnetic chains on superconducting honeycomb lattices
• Properties and application of SO(3) Majorana representation of spin: Equivalence with Jordan-Wigner transformation and exact Z2 gauge theories for spin models
• Template-Assisted Scalable Nanowire Networks
• Quantized charge transport in chiral Majorana edge modes
• Hard gap in epitaxial semiconductor–superconductor nanowires
• Long coherence times for edge spins
• Vacancies in Kitaev quantum spin liquids on the three-dimensional hyperhoneycomb lattice
• Subsystem Ginzburg-Landau and symmetry-protected topological order coexisting on a graph
• Topological superconductivity in quantum Hall–superconductor hybrid systems
• Bulk-edge correspondence of classical diffusion phenomena
• Analysis of Transmission of Methicillin-resistant Staphylococcus Aureus in an Intensive Care Unit by Pulsed-field Gel Electrophoresis
• Tunable Splitting of the Ground-State Degeneracy in Quasi-One-Dimensional Parafermion Systems
• Transport signatures of topological superconductivity in a proximity-coupled nanowire
• Anisotropy of transport in bulk Rashba metals
• Quantum phase transitions of a generalized compass chain with staggered Dzyaloshinskii–Moriya interaction
• Generic Weyl phase in the vortex state of quasi-two-dimensional chiral superconductors
• Optical Responses of Chiral Majorana Edge States in Two-Dimensional Topological Superconductors
• Local Raman spectroscopy of chiral Majorana edge modes in Kitaev spin liquids and topological superconductors
• Emergence of a new symmetry class for Bogoliubov–de Gennes (BdG) Hamiltonians: expanding 10-fold symmetry classes
• Response to recent comments on Phys. Rev. B 107, 245423 (2023) and Subsection S4.3 of the Supp. Info. for Nature 638, 651-655 (2025)
• Hopf Exceptional Points
• Introduction to Topological Phases and Electronic Interactions in (2+1) Dimensions
• Open spin chain realization of a topological defect in a one-dimensional Ising model: Boundary and bulk symmetry
• Few-Electron Single and Double Quantum Dots in an InAs Two-Dimensional Electron Gas
• The germanium quantum information route
• Fragility of the Fractional Josephson Effect in Time-Reversal-Invariant Topological Superconductors
• Ordered and tunable Majorana-zero-mode lattice in naturally strained LiFeAs
• Annular Majorana mode in a superconducting topological insulator
• Fingerprints of Majorana Bound States in Aharonov-Bohm Geometry
• Magnetic anisotropy of critical current in nanowire Josephson junction with spin-orbit interaction
• Symmetry-Enhanced Boundary Qubits at Infinite Temperature
• Microscopic models for Kitaev's sixteenfold way of anyon theories
• Fractional Josephson effect in nonuniformly strained graphene
• Classification of Interacting Topological Floquet Phases in One Dimension
• Symmetry-protected topological order identified via Gutzwiller-guided density-matrix-renormalization-group: SO(n) spin chains
• Visualizing Majorana bound states in one and two dimensions using the generalized Majorana polarization
• Shuttling Majorana zero modes in disordered and noisy topological superconductors
• Majorana fermions in semiconductor nanowires
• Fractional topological phase in one-dimensional flat bands with nontrivial topology
• Near-zero-energy end states in topologically trivial spin-orbit coupled superconducting nanowires with a smooth confinement
• Inverse ac Josephson effect in Josephson diode
• Endstates in multichannel spinlessp-wave superconducting wires
• Accessing the degree of Majorana nonlocality in a quantum dot-optical microcavity system
• 4π-periodic Josephson supercurrent in HgTe-based topological Josephson junctions
• Control of atomic reconstruction and quasi-1D excitons in strain-engineered moiré heterostructures
• Coexistence of topologically trivial and non-trivial Yu-Shiba-Rusinov bands in magnetic atomic chains on a superconductor
• Holonomic implementation of CNOT gate on topological Majorana qubits
• Majorana Zero Modes in Ferromagnetic Wires without Spin-Orbit Coupling
• A perspective on semiconductor-based superconducting qubits
• Intertwining Josephson and Vortex Topologies in Conventional Superconductors
• Quasiclassical theory of disordered multi-channel Majorana quantum wires

Discussions

• My question might be silly but, the spectrum of a $p$ -wave superconductor is $$ E(k) = \sqrt{ \left(\mu + 2\lambda \cos\left(\frac{2\pi}{N} k \right)\right)^2 + 4 \sin\left(k\frac{2\pi}{N}\right)^2 | [stackexchange, 1 points]