# Majorana fermion

A Majorana fermion, also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Ettore Majorana in 1937. The term is sometimes used in opposition to a Dirac fermion, which describes fermions that are not their own antiparticles.

All of the Standard Model fermions except the neutrino behave as Dirac fermions at low energy (after electroweak symmetry breaking), but the nature of the neutrino is not settled and it may be either Dirac or Majorana. In condensed matter physics, Majorana fermions exist as quasiparticle excitations in superconductors and can be used to form Majorana bound states possessing non-abelian statistics.

## Theory

The concept goes back to 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 $\gamma^{\dagger}_j$ creates a fermion in quantum state $j$ (described by a real wave function), while the annihilation operator $\gamma_j$ annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators $\gamma^{\dagger}_j$ and $\gamma_j$ are distinct, while for a Majorana fermion they are identical.

## Elementary particle

Since particles and antiparticles have opposite conserved charges, only an uncharged particle can have a Majorana mass[clarification needed]. All of the elementary fermions of the Standard Model have gauge charges, so they cannot have fundamental Majorana masses. However, the right-handed sterile neutrinos introduced to explain neutrino oscillation could have Majorana masses. If they do, then at low energy (after electroweak symmetry breaking), by the seesaw mechanism, the neutrino fields would naturally behave as six Majorana fields, with three expected to have very high masses (comparable to the GUT scale) and the other three expected to have very low masses (comparable to 1 eV). If right-handed neutrinos exist but do not have a Majorana mass, the neutrinos would instead behave as three Dirac fermions and their antiparticles with masses coming directly from the Higgs interaction, like the other Standard Model fermions.

The seesaw mechanism is appealing because it would naturally explain why the observed neutrino masses are so small. However, if the neutrinos are Majorana then they violate the conservation of lepton number and even B − L.

Neutrinoless double beta decay, which can be viewed as two beta decay events with the produced antineutrinos immediately annihilating with one another, is only possible if neutrinos are their own antiparticles.[2] Experiments are underway to search for this type of decay.[3]

The high energy analog of the neutrinoless double beta decay process is the production of same sign charged lepton pairs at hadron colliders;[4] it is being searched for by both the ATLAS and CMS experiments at the Large Hadron Collider. In theories based on left–right symmetry, there is a deep connection between these processes.[5] In the most accepted explanation of the smallness of neutrino mass, the seesaw mechanism, the neutrino is naturally a Majorana fermion.

Majorana fermions cannot possess intrinsic electric or magnetic moments, only toroidal moments.[6][7][8] Such minimal interaction with electromagnetic fields makes them potential candidates for cold dark matter.[9][10] The hypothetical neutralino of supersymmetric models is a Majorana fermion.

## Majorana bound states

In superconducting materials, Majorana fermions can emerge as (non-fundamental) quasiparticles. This becomes possible because a quasiparticle in a superconductor is its own antiparticle. Mathematically, the superconductor imposes electron hole "symmetry" on the quasiparticle excitations, relating the creation operator $\gamma(E)$ at energy $E$ to the annihilation operator ${\gamma^{\dagger}(-E)}$ at energy $-E$. Majorana fermions can be bound to a defect at zero energy, and then the combined objects are called Majorana bound states or Majorana zero modes.[11] This name is more appropriate than Majorana fermion (although the distinction is not always made in the literature), since the statistics of these objects is no longer fermionic. Instead, the Majorana bound states are an example of non-abelian anyons: interchanging them changes the state of the system in a way which depends only on the order in which exchange was performed. The non-abelian statistics that Majorana bound states possess allows to use them as a building block for a topological quantum computer.[12]

A quantum vortex in certain superconductors or superfluids can trap midgap states, so this is one source of Majorana bound states.[13][14][15] Shockley states at the end points of superconducting wires or line defects are an alternative, purely electrical, source.[16] An altogether different source uses the fractional quantum Hall effect as a substitute for the superconductor.[17]

### Experiments in superconductivity

In 2008 Fu and Kane provided a groundbreaking development by theoretically predicting that Majorana bound states can appear at the interface between topological insulators and superconductors.[18][19] Many proposals of a similar spirit soon followed, where it was shown that Majorana bound states can appear even without topological insulator. An intense search to provide experimental evidence of Majorana bound states in superconductors[20][21] first produced some positive results in 2012.[22][23] 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 bound states, one at either end of the region of the nanowire in contact with the superconductor.[24]

This experiment from Delft marks a possible verification of independent theoretical proposals from two groups[25][26] predicting the solid state manifestation of Majorana bound states in semiconducting wires.

## References

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3. ^ W. Rodejohann (2011). "Neutrino-less Double Beta Decay and Particle Physics". International Journal of Modern Physics E20: 1833. arXiv:1106.1334. Bibcode:2011IJMPE..20.1833R. doi:10.1142/S0218301311020186.
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19. ^ L. Fu; C. L. Kane (2009). "Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction". Physical Review B 79 (16): 161408. arXiv:0804.4469. Bibcode:2009PhRvB..79p1408F. doi:10.1103/PhysRevB.79.161408.
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22. ^ E. S. Reich (28 February 2012). "Quest for quirky quantum particles may have struck gold". Nature News. doi:10.1038/nature.2012.10124.
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