Majorana fermion

In physics, a Majorana fermion is a fermion which is its own anti-particle. The term is used in opposition to Dirac fermion, which describes particles that differ from their antiparticles. The concept goes back to Ettore Majorana's 1937 discovery^[1] that neutral spin-1/2 particles can be described by a real wave equation (the Majorana equation).

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.

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 γ_j annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators γ^†_j and γ_j are distinct, while for a Majorana fermion they are identical.

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] So far, these are only theoretical predictions, but there is an active search for experimental observation.^[9][10]

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.^[11]

See also

• Dirac fermion
• Majorana equation
• Majorana spinor

References

1. ^ E. Majorana, Teoria simmetrica dell’elettrone e del positrone, Nuovo Cimento 14, 171 (1937). English translation
2. ^ F. Wilczek, Majorana returns, Nature Physics 5, 614 (2009).
3. ^ N.B. Kopnin and M.M. Salomaa, Mutual friction in superfluid 3He: Effects of bound states in the vortex core, Phys. Rev. B 44, 9667 (1991).
4. ^ G.E. Volovik, Fermion zero modes on vortices in chiral superconductors, JETP Lett. 70, 609 (1999).
5. ^ N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61, 10267 (2000).
6. ^ L. Fu and C.L. Kane, Superconducting proximity effect and Majorana fermions at the surface of a topological insulator, Phys. Rev. Lett. 100, 096407 (2008).
7. ^ A. Yu. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Usp. (suppl.) 44, 131 (2001).
8. ^ G. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360, 362 (1991).
9. ^ M. Franz, Race for Majorana fermions, Physics 3, 24 (2010).
10. ^ R.F. Service, Search for Majorana fermions nearing success at last?, Science 332, 193 (2011).
11. ^ C. Nayak, S. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).