From Majorana fermions to topological quantum computation in semiconductor/superconductor heterostructures

Abstract

We present a pedagogical review of topological superconductivity and its consequences in spin-orbit coupled semiconductor/superconductor heterostructures. We start by reviewing the historical origins of the notions of Dirac and Majorana fermions in particle physics and discuss how lower dimensional versions of these emerge in one dimensional superconductors. Ultimately, we focus on Majorana zero-modes, which emerge at defects in the Majorana equation. We then review the definition of the topological invariant, and how it allows the prediction of such Majorana modes from the bulk bandstructure of realistic superconductors, which do not have Lorentz invariance. Finally, we end with a discussion of protocols for how to detect such Majorana modes and use them for topological quantum computation.

arXiv commentA pedagogical review submitted as a book chapter

References in corpus (111)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Unpaired Majorana fermions in quantum wires
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• New directions in the pursuit of Majorana fermions in solid state systems
• Random-Matrix Theory of Quantum Transport
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Evidence of Majorana fermions in an Al - InAs nanowire topological superconductor
• Anderson Transitions
• Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor
• Non-abelian statistics of half-quantum vortices in p-wave superconductors
• Search for Majorana fermions in superconductors
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• Observation of Majorana Fermions in a Nb-InSb Nanowire-Nb Hybrid Quantum Device
• Topological superconductors: a review
• Kwant: a software package for quantum transport
• Observation of the fractional ac Josephson effect: the signature of Majorana particles
• Exponential Protection of Zero Modes in Majorana Islands
• Majorana bound states in a coupled quantum-dot hybrid-nanowire system
• Fermionic quantum computation
• Josephson Current and Noise at a Superconductor-Quantum Spin Hall Insulator-Superconductor Junction
• Majorana Fermion Induced Resonant Andreev Reflection
• Introduction to topological superconductivity and Majorana fermions
• Superconductor-Nanowire Devices from Tunneling to the Multichannel Regime: Zero-Bias Oscillations and Magnetoconductance Crossover
• Anomalous modulation of a zero bias peak in a hybrid nanowire-superconductor device
• Non-Abelian Topological Order in S-Wave Superfluids of Ultracold Fermionic Atoms
• Topologically-Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State
• Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes
• Electrically detected interferometry of Majorana fermions in a topological insulator
• Probing Neutral Majorana Fermion Edge Modes with Charge Transport
• $2n$ Quasihole States Realize $2^{n-1}$-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States
• Non-Abelian quantum order in spin-orbit-coupled semiconductors: The search for topological Majorana particles in solid state systems
• Zero-bias peaks in spin-orbit coupled superconducting wires with and without Majorana end-states
• Scaling of Majorana Zero-Bias Conductance Peaks
• Andreev bound states versus Majorana bound states in quantum dot-nanowire-superconductor hybrid structures: Trivial versus topological zero-bias conductance peaks
• Hard gap in epitaxial semiconductor-superconductor nanowires
• Electron Teleportation via Majorana Bound States in a Mesoscopic Superconductor
• $p_x+i{p}_y$ superfluid from s-wave interactions of fermionic cold atoms
• $4\pi$-periodic Josephson supercurrent in HgTe-based topological Josephson junctions
• Near-zero-energy end states in topologically trivial spin-orbit coupled superconducting nanowires with a smooth confinement
• Transport spectroscopy of NS nanowire junctions with Majorana fermions
• Fractional ac Josephson effect in p- and d-wave superconductors
• Detecting Non-Abelian Statistics in the nu=5/2 Fractional Quantum Hall State
• Tuning p-wave interactions in an ultracold Fermi gas of atoms
• The spin quantum Hall effect in unconventional superconductors
• Observing Majorana Bound States in p-wave Superconductors Using Noise Measurements in Tunneling Experiments
• Quantized conductance at the Majorana phase transition in a disordered superconducting wire
• Measurement-Only Topological Quantum Computation
• Zero-voltage conductance peak from weak antilocalization in a Majorana nanowire
• Midgap edge states and pairing symmetry of quasi-one-dimensional organic superconductors
• Proposal to stabilize and detect half-quantum vortices in strontium ruthenate thin films: Non-Abelian braiding statistics of vortex matter in a $p_x+i{p}_y$ superconductor
• Topological invariants for spin-orbit coupled superconductor nanowires
• Observation of half-height magnetization steps in Sr2RuO4
• Zero-energy vortex bound state in the superconducting topological surface state of Fe(Se,Te)
• Class D spectral peak in Majorana quantum wires
• Quantum Computation Using Vortices and Majorana Zero Modes of a $p_x$ + $i{p}_y$ Superfluid of Fermionic Cold Atoms
• Majorana box qubits
• Edge modes, edge currents, and gauge invariance in $p_x+i{p}_y$ superfluids and superconductors
• Majorana Fermions in Proximity-coupled Topological Insulator Nanowires
• Multiple topological states in iron-based superconductors
• Quantum point contact as a probe of a topological superconductor
• Nearly quantized conductance plateau of vortex zero mode in an iron-based superconductor
• Multiplet Structure of Feshbach Resonances in non-zero partial waves
• Reproducing topological properties with quasi-Majorana states
• Ballistic superconductivity in semiconductor nanowires
• Physical mechanisms for zero-bias conductance peaks in Majorana nanowires
• Feshbach Resonances in Fermionic Lithium-6
• Controlling non-Abelian statistics of Majorana fermions on Majorana dimer lattices
• Quasiparticle localization in superconductors with spin-orbit scattering
• Quantized zero bias conductance plateau in semiconductor-superconductor heterostructures without non-Abelian Majorana zero modes
• Flux-induced topological superconductivity in full-shell nanowires
• Topological superconductivity in full shell proximitized nanowires
• Flux-induced Majorana modes in full-shell nanowires
• Geometric phases and quantum entanglement as building blocks for nonabelian quasiparticle statistics
• Observation of a Majorana zero mode in a topologically protected edge channel
• Non-Majorana states yield nearly quantized conductance in superconductor-semiconductor nanowire devices
• Braiding without Braiding: Teleportation-Based Quantum Information Processing with Majorana Zero Modes
• Dirac, Majorana and Weyl fermions
• Josephson radiation from gapless Andreev bound states in HgTe-based topological junctions
• Majorana fermions in superconducting helical magnets
• Andreev rectifier: a nonlocal conductance signature of topological phase transitions
• Fusion rules and vortices in $p_x+i{p}_y$ superconductors
• Observation of the 4$\pi$-periodic Josephson effect in indium arsenide nanowires
• Time reversal non-invariant non-Abelian topological order in non-centrosymmetric superconductors
• Dynamics of Majorana States in a Topological Josephson Junction
• Stability of half-quantum vorticies in p_x+ip_y superconductors
• An Index Theorem for the Majorana Zero Modes in Chiral P-Wave Superconductors
• Topological Order and Non-Abelian Statistics in Noncentrosymmetric s-Wave Superconductors
• Effective g-factor in Majorana Wires
• Topological Superconductivity and Majorana Fermions in Metallic Surface-States
• Orbital effect of magnetic field on the Majorana phase diagram
• Generic quantized zero-bias conductance peaks in superconductor-semiconductor hybrid structures
• Missing Shapiro steps in topologically trivial Josephson Junction on InAs quantum well
• Robust low-energy Andreev bound states in semiconductor-superconductor heterostructures: Importance of partial separation of component Majorana bound states
• Fractional quantum Hall effect and nonabelian statistics
• Coherent transport through a Majorana island in an Aharonov-Bohm interferometer
• Annihilation of colliding Bogoliubov quasiparticles reveals their Majorana nature
• Closing of the Induced Gap in a Hybrid Superconductor-Semiconductor Nanowire
• Hole-doped semiconductor nanowire on top of an s-wave superconductor: A new and experimentally accessible system for Majorana fermions
• AC Josephson effect and resonant Cooper pair tunneling emission of a Cooper Pair Transistor
• Superconducting phase with a chiral $f$-wave pairing symmetry and Majorana fermions induced in a hole-doped semiconductor
• Majorana fermions in carbon nanotubes
• Detecting topological superconductivity using low-frequency doubled Shapiro steps
• A proposal to probe quantum non-locality of Majorana fermions in tunneling experiments
• Half-quantum vortex and d-soliton in Sr${}_2$RuO${}_4$
• Vortex states and Majorana fermions in spin-orbit coupled semiconductor-superconductor hybrid structures
• X-shaped and Y-shaped Andreev resonance profiles in a superconducting quantum dot
• Proposal for measuring the parity anomaly in a topological superconductor ring

Cited by in corpus (3)

• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Topological Solitons in Su-Schrieffer-Heeger Chain with periodic hopping modulation, domain walls and disorder
• Edge state behavior in a Su-Schrieffer-Heeger like model with periodically modulated hopping