A Majorana fermion (/maɪəˈrɑːnə/[1]), 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.
With the exception of
neutrinos, all of the
Standard Model fermions are known to behave as Dirac fermions at low energy (lower than the
electroweak symmetry breaking temperature), and none are Majorana fermions. The nature of neutrinos is not settled – they may turn out to be either Dirac or Majorana fermions.
The concept goes back to Majorana's suggestion in 1937[2] that electrically neutral
spin-1/2 particles can be described by a
real-valuedwave equation (the
Majorana equation), and would therefore be identical to their antiparticle, because the wave functions of particle and antiparticle are related by
complex conjugation, which leaves the Majorana wave equation unchanged.
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 creates a fermion in quantum state (described by a real wave function), whereas the annihilation operator annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators and are distinct, whereas for a Majorana fermion they are identical. The ordinary fermionic annihilation and creation operators and can be written in terms of two Majorana operators and by
In
supersymmetry models,
neutralinos – superpartners of gauge bosons and Higgs bosons – are Majorana fermions.
Identities
Another common convention for the normalization of the Majorana fermion
operator is
which can be rearranged to obtain the Majorana fermion operators as
It is easy to see that is indeed fulfilled. This convention has the advantage that the Majorana operator
squares to the identity, i.e. .
Using this convention, a collection of Majorana fermions ( ordinary fermions), () obey the following
anticommutation identities
and
where and are
antisymmetric matrices. These are identical to the commutation relations for the real
Clifford algebra in dimensions ().
Elementary particles
Because particles and antiparticles have opposite conserved charges, Majorana fermions have zero charge, hence among the fundamental particles, the only fermions that could be Majorana are
sterile neutrinos, if they exist. All the other elementary fermions of the
Standard Model have
gauge charges, so they cannot have fundamental
Majorana masses: Even the Standard Model's left-handed neutrinos and right-handed antineutrinos have non-zero
weak isospin, a
charge-like quantum number. However, if they exist, the so-called "
sterile neutrinos" (left-handed antineutrinos and right-handed neutrinos) would be
truly neutral particles (assuming no other, unknown gauge charges exist).
The
sterile neutrinos introduced to explain
neutrino oscillation and anomalously small S.M.
neutrino masses 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 of them expected to have very high masses (comparable to the
GUT scale) and the other three expected to have very low masses (below 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 of
B − L.
Neutrinoless double beta decay has not (yet) been observed,[3]
but if it does exist, it can be viewed as two ordinary
beta decay events whose resultant antineutrinos immediately annihilate each other, and is only possible if neutrinos are their own antiparticles.[4]
The high-energy analog of the neutrinoless double beta decay process is the production of same-sign charged lepton pairs in
hadron colliders;[5] 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.[6] In the currently most-favored 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.[7][8][9] Such minimal interaction with electromagnetic fields makes them potential candidates for
cold dark matter.[10][11]
Majorana bound states
In
superconducting materials, a
quasiparticle can emerge as a Majorana fermion (non-fundamental), more commonly referred to as a
Bogoliubov quasiparticle in condensed matter physics. Its existence 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 at energy to the annihilation operator at energy . 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.[12] This name is more appropriate than Majorana fermion (although the distinction is not always made in the literature), because 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 that depends only on the order in which the exchange was performed. The non-abelian statistics that Majorana bound states possess allows them to be used as a building block for a
topological quantum computer.[13]
A
quantum vortex in certain superconductors or superfluids can trap midgap states, which is one source of Majorana bound states.[14][15][16]Shockley states at the end points of superconducting wires or line defects are an alternative, purely electrical, source.[17] An altogether different source uses the
fractional quantum Hall effect as a substitute for the superconductor.[18]
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.[19][20] Many proposals of a similar spirit soon followed, where it was shown that Majorana bound states can appear even without any topological insulator. An intense search to provide experimental evidence of Majorana bound states in superconductors[21][22] first produced some positive results in 2012.[23][24] 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.[25] Simultaneously, a group from
Purdue University and
University of Notre Dame reported observation of fractional
Josephson effect (decrease of the
Josephson frequency by a factor of 2) in
indium antimonide nanowires connected to two superconducting contacts and subjected to a moderate magnetic field,[26] another signature of Majorana bound states.[27] Bound state with zero energy was soon detected by several other groups in similar hybrid devices,[28][29][30][31] and fractional Josephson effect was observed in
topological insulator HgTe with superconducting contacts[32]
The aforementioned experiments mark possible verifications of independent 2010 theoretical proposals from two groups[33][34] predicting the solid state manifestation of Majorana bound states in semiconducting wires
proximitized to superconductors. However, it was also pointed out that some other trivial non-topological bounded states[35] could highly mimic the zero voltage conductance peak of Majorana bound state. The subtle relation between those trivial bound states and Majorana bound states was reported by the researchers in Niels Bohr Institute,[36] who can directly "watch" coalescing Andreev bound states evolving into Majorana bound states, thanks to a much cleaner semiconductor-superconductor hybrid system.
In
2014, evidence of Majorana bound states was also observed using a low-temperature
scanning tunneling microscope, by scientists at
Princeton University.[37][38] These experiments resolved the predicted signatures of localized Majorana bound states – zero energy modes – at the ends of ferromagnetic (iron) chains on the surface of a superconductor (lead) with strong spin-orbit coupling. Follow up experiments at lower temperatures probed these end states with higher energy resolution and showed their robustness when the chains are buried by layers of lead.[39] Experiments with spin-polarized STM tips have also been used, in 2017, to distinguish these end modes from trivial zero energy modes that can form due to magnetic defects in a superconductor, providing important evidence (beyond zero bias peaks) for the interpretation of the zero energy mode at the end of the chains as a Majorana bound state.[40] More experiments finding evidence for Majorana bound states in chains have also been carried out with other types of magnetic chains, particularly chains manipulated atom-by-atom to make a spin helix on the surface of a superconductor.[41][42]
Majorana fermions may also emerge as quasiparticles in
quantum spin liquids, and were observed by researchers at
Oak Ridge National Laboratory, working in collaboration with Max Planck Institute and University of Cambridge on 4 April 2016.[43]
Chiral Majorana fermions were claimed to be detected in 2017 by Q.L. He et al., in a
quantum anomalous Hall effect/superconductor hybrid device.[44][45] In this system, Majorana fermions edge mode will give a rise to a conductance edge current. Subsequent experiments by other groups, however, could not reproduce these findings.[46][47][48] In November 2022, the article by He et al. was retracted by the editors,[49] because "analysis of the raw and published data revealed serious irregularities and discrepancies".
On 16 August 2018, a strong evidence for the existence of Majorana bound states (or Majorana
anyons) in an
iron-based superconductor, which many alternative trivial explanations cannot account for, was reported by Ding's and Gao's teams at Institute of Physics,
Chinese Academy of Sciences and
University of Chinese Academy of Sciences, when they used
scanning tunneling spectroscopy on the superconducting Dirac surface state of the iron-based superconductor. It was the first time that indications of Majorana particles were observed in a bulk of pure substance.[50] However, more recent experimental studies in iron-based superconductors show that topologically trivial Caroli–de Gennes–Matricon states [51] and Yu–Shiba–Rusinov states[52] can exhibit qualitative and quantitative features similar to those Majorana zero modes would make. In 2020 similar results were reported for a platform consisting of europium sulfide and gold films grown on vanadium.[53]
Majorana bound states in quantum error correction
One of the causes of interest in Majorana bound states is that they could be used in
quantum error correcting codes.[54][55] This process is done by creating so called 'twist defects' in codes such as the
toric code[56] which carry unpaired Majorana modes.[57] The Majoranas are then "braided" by being physically moved around each other in 2D sheets or networks of nanowires.[58] This braiding process forms a
projective representation of the
braid group.[59]
Such a realization of Majoranas would allow them to be used to store and process
quantum information within a
quantum computation.[60] Though the codes typically have no Hamiltonian to provide suppression of errors, fault-tolerance would be provided by the underlying quantum error correcting code.
Majorana bound states in Kitaev chains
In February 2023[61][62] a study reported the realization of a "poor man's" Majorana that is a Majorana bound state that is not
topologically protected and therefore only stable for a very small range of parameters. It was obtained in Kitaev chain consisting of two
quantum dots in a superconducting nanowire strongly coupled by normal
tunneling and
Andreev tunneling with the state arising when the rate of both processes match confirming a prediction of
Alexei Kitaev.[63]
^Das, A.; Ronen, Y.; Most, Y.; Oreg, Y.; Heiblum, M.; Shtrikman, H. (11 November 2012). "Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions". Nature Physics. 8 (12): 887–895.
arXiv:1205.7073.
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2012NatPh...8..887D.
doi:
10.1038/nphys2479.
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^Churchill, H.O.H.; Fatemi, V.; Grove-Rasmussen, K.; Deng, M.T.; Caroff, P.; Xu, H.Q.; Marcus, C.M. (6 June 2013). "Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover". Physical Review B. 87 (24): 241401(R).
arXiv:1303.2407.
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S2CID118487534.