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# Majorana fermion

A Majorana fermion is a fermion that is its own anti-particle. The term is sometimes used in opposition to Dirac fermion, which describes particles that differ from their antiparticles. While it is common that bosons (such as the photon) are their own anti-particle, for fermions this is highly unusual.

Theory

The concept goes back to Ettore Majorana's 1937 suggestion[1] that neutral spin-1/2 particles can be described by a real wave equation (the Majorana equation), and would therefore be identical to their antiparticle (since the wave function of particle and antiparticle are related by complex conjugation).

The difference between Majorana fermions and Dirac fermions can be expressed mathematically in terms of the creation and annihilation operators of second quantization. The creation operator γ†j creates a fermion in quantum state j, while the annihilation operator γj annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators γ†j and γj are distinct, while for a Majorana fermion they are identical.

Elementary particle

No elementary particle is known to be a Majorana fermion. However, the nature of the neutrino is not yet definitely settled; it might be a Majorana fermion or it might be a Dirac fermion. If it is a Majorana fermion, then neutrinoless double beta decay is possible; experiments are underway to search for this type of decay.

The hypothetical neutralino of supersymmetric models is a Majorana fermion.

Quasiparticle

In superconducting materials, Majorana fermions can emerge as (non-fundamental) quasiparticles.[2] The superconductor imposes electron hole symmetry on the quasiparticle excitations, relating the creation operator γ(E) at energy E to the annihilation operator γ†(−E) at energy −E. At the Fermi level E=0, one has γ=γ† so the excitation is a Majorana fermion. Since the Fermi level is in the middle of the superconducting gap, these are midgap states. A quantum vortex in certain superconductors or superfluids can trap midgap states, so this is one source of Majorana fermions.[3][4][5][6] Shockley states at the end points of superconducting wires or line defects are an alternative, purely electrical, source.[7] An altogether different source uses the fractional quantum Hall effect as a substitute for the superconductor.[8]

It was predicted that Majorana fermions in superconductors could be used as a building block for a (non-universal) topological quantum computer, in view of their non-Abelian anyonic statistics.[9]

Experiments in superconductivity

An intense search to provide experimental evidence of Majorana fermions in superconductors[10][11] first produced some positive results in 2012.[12][13] A team from the Kavli Institute of Nanoscience at Delft University of Technology in the Netherlands reported an experiment involving indium antimonide nanowires connected to a circuit with a gold contact at one end and a slice of superconductor at the other. When exposed to a moderately strong magnetic field the apparatus showed a peak electrical conductance at zero voltage that is consistent with the formation of a pair of Majorana quasiparticles, one at either end of the region of the nanowire in contact with the superconductor.[14]

This experimental discovery from Delft is a precise verification of earlier theoretical proposals from the University of Maryland predicting the solid state manifestation of the Majorana fermions in superconductor-semiconductor hybrid sandwich structures in the presence of an applied magnetic field.[15][16][17]

It is important to note that the solid state manifestations of Majorana fermions are emergent low-energy localized modes of the system (quasiparticles) which are not fundamental new elementary particles as originally envisioned by Majorana (or as the neutrino would be if it turns out to be a Majorana fermion), but are effective linear combinations of half-electrons and half-holes which are topological anyonic objects obeying non-Abelian statistics.[9] The terminology "Majorana fermion" is thus not a good nomenclature for these solid state Majorana modes.

References

^ E. Majorana (1937). "Teoria simmetrica dell’elettrone e del positrone" (in Italian). Nuovo Cimento 14: 171. English translation.

^ F. Wilczek (2009). "Majorana returns". Nature Physics 5 (9): 614. Bibcode 2009NatPh...5..614W. doi:10.1038/nphys1380.

^ N.B. Kopnin; Salomaa (1991). "Mutual friction in superfluid 3He: Effects of bound states in the vortex core". Physical Review B 44 (17): 9667. Bibcode 1991PhRvB..44.9667K. doi:10.1103/PhysRevB.44.9667.

^ G.E. Volovik (1999). "Fermion zero modes on vortices in chiral superconductors". JETP Letters 70 (9): 609. Bibcode 1999JETPL..70..609V. doi:10.1134/1.568223.

^ N. Read; Green (2000). "Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect". Physical Review B 61 (15): 10267. Bibcode 2000PhRvB..6110267R. doi:10.1103/PhysRevB.61.10267.

^ L. Fu; Kane (2008). "Superconducting proximity effect and Majorana fermions at the surface of a topological insulator". Physical Review Letters 100 (9): 096407. Bibcode 2008PhRvL.100i6407F. doi:10.1103/PhysRevLett.100.096407.

^ A. Yu. Kitaev (2001). "Unpaired Majorana fermions in quantum wires". Physics-Uspekhi (supplement) 44 (131): 131. Bibcode 2001PhyU...44..131K. doi:10.1070/1063-7869/44/10S/S29.

^ G. Moore; Read (1991). "Nonabelions in the fractional quantum Hall effect". Nuclear Physics B 360 (2–3): 362. Bibcode 1991NuPhB.360..362M. doi:10.1016/0550-3213(91)90407-O.

^ a b C. Nayak, S. Simon, A. Stern, M. Freedman, and S. Das Sarma (2008). "Non-Abelian anyons and topological quantum computation". Reviews of Modern Physics 80: 1083.

^ J. Alicea. New directions in the pursuit of Majorana fermions in solid state systems. arXiv:1202.1293.

^ C. W. J. Beenakker. Search for Majorana fermions in superconductors. arXiv:1112.1950.

^ E. S. Reich (28 February 2012). "Quest for quirky quantum particles may have struck gold". Nature News. doi:10.1038/nature.2012.10124.

^ Jonathan Amos (13 April 2012). "Majorana particle glimpsed in lab". BBC News. Retrieved 15 April 2012.

^ V. Mourik; K. Zuo; S.M. Frolov; S.R. Plissard; E.P.A.M. Bakkers; L.P. Kouwenhoven (12 April 2012). "Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices". Science. arXiv:1204.2792. doi:10.1126/science.1222360.

^ J. Sau; R. Lutchyn; S. Tewari; S. Das Sarma (2010). "Generic New Platform for Topological Quantum Computation Using Semiconductor Heterostructures". Physical Review Letters 104 (4): 040502. Bibcode 2010PhRvL.104d0502S. doi:10.1103/PhysRevLett.104.040502.

^ R. Lutchyn; J. Sau; S. Das Sarma (2010). "Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures". Physical Review Letters 105 (7): 077001. Bibcode 2010PhRvL.105g7001L. doi:10.1103/PhysRevLett.105.077001.

^ J. Sau; S. Tewari; R. Lutchyn; T. Stanescu; S. Das Sarma (2010). "Non-Abelian quantum order in spin-orbit-coupled semiconductors: Search for topological Majorana particles in solid-state systems". Physical Review B 82 (21): 214509. Bibcode 2010PhRvB..82u4509S. doi:10.1103/PhysRevB.82.214509.

External links

Bob Yirka (13 April 2012). "Researchers find possible evidence of Majorana fermions". Phys dot Org. Retrieved 2012-04-20

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