Majorana fermion#Majorana bound states in quantum error correction

The concept goes back to Majorana's suggestion in 1937{{cite book |last1=Majorana |first1=Ettore |last2=Maiani |first2=Luciano |year=2006 |chapter=A symmetric theory of electrons and positrons |editor-first=Giuseppe Franco |editor-last=Bassani |title=Ettore Majorana Scientific Papers |isbn=978-3-540-48091-4 |doi=10.1007/978-3-540-48095-2_10 |pages=[https://archive.org/details/ettoremajoranasc00bass/page/n230 201]–233 |s2cid=17529013 |url=https://archive.org/details/ettoremajoranasc00bass |url-access=limited}} Translated from: {{cite journal |last=Majorana |first=Ettore |year=1937 |title=Teoria simmetrica dell'elettrone e del positrone |journal=Il Nuovo Cimento |volume=14 |issue=4 |pages=171–184 |language=it |bibcode=1937NCim...14..171M |s2cid=18973190 |doi=10.1007/bf02961314}} that electrically neutral spin-{{sfrac|1|2}} particles can be described by a real-valued wave 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 $\backslash gamma^\{\backslash dagger\}\_j$ creates a fermion in quantum state $j$ (described by a real wave function), whereas the annihilation operator $\backslash gamma\_j$ annihilates it (or, equivalently, creates the corresponding antiparticle). For a Dirac fermion the operators $\backslash gamma^\{\backslash dagger\}\_j$ and $\backslash gamma\_j$ are distinct, whereas for a Majorana fermion they are identical. The ordinary fermionic annihilation and creation operators $f$ and $f^\{\backslash dagger\}$ can be written in terms of two Majorana operators $\backslash gamma\_1$ and $\backslash gamma\_2$ by

:$f\; =\; \backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\; (\backslash gamma\_1+i\backslash gamma\_2),$

:$f^\{\backslash dagger\}\; =\; \backslash tfrac\{1\}\{\backslash sqrt\{2\}\}\; (\backslash gamma\_1-i\backslash gamma\_2)\; ~.$

In supersymmetry models, neutralinos – superpartners of gauge bosons and Higgs bosons – are Majorana fermions.

Another common convention for the normalization of the Majorana fermion operator $\backslash gamma$ is

:$f\; =\; \backslash tfrac\{1\}\{2\}\; (\backslash gamma\_1\; +\; i\backslash gamma\_2),$

:$f^\backslash dagger\; =\; \backslash tfrac\{1\}\{2\}\; (\backslash gamma\_1\; -\; i\backslash gamma\_2),$

which can be rearranged to obtain the Majorana fermion operators as

:$\backslash gamma\_1\; =\; f^\backslash dagger\; +\; f,$

:$\backslash gamma\_2\; =\; i\; (f^\backslash dagger\; -\; f).$

It is easy to see that $\backslash gamma\_i=\backslash gamma\_i^\backslash dagger$ is indeed fulfilled. This convention has the advantage that the Majorana operator squares to the identity, i.e. $\backslash gamma\_i^2=(\backslash gamma\_i^\backslash dagger)^2=1$. Using this convention, a collection of $2n$ Majorana fermions ($n$ ordinary fermions), $\backslash gamma\_i$ ($i=1,2,..,2n$) obey the following anticommutation identities

: $\backslash \{\backslash gamma\_i,\backslash gamma\_j\backslash \}\; =\; 2\; \backslash delta\_\{ij\}$

and

: $\backslash sum\_\{ijkl\}\; [\backslash gamma\_i\; A\_\{ij\}\; \backslash gamma\_j,\; \backslash gamma\_k\; B\_\{kl\}\; \backslash gamma\_l]\; =\; 4\; \backslash sum\_\{ij\}\; \backslash gamma\_i\; [A,B]\_\{ij\}\; \backslash gamma\_j$

where $A$ and $B$ are antisymmetric matrices. These are identical to the commutation relations for the real Clifford algebra in $n$ dimensions ($\backslash mathrm\{Cl\}(\; \backslash mathbb\; \{R\}^n\; )$).

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, $T\_3\; =\; \backslash pm\; \backslash tfrac\{1\}\{2\}\; ,$ 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).

File:Ettore Majorana.jpg hypothesised the existence of Majorana fermions in 1937]]

The sterile neutrinos introduced to explain neutrino oscillation and anomalously small Standard Model 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,

{{cite journal

|first=Werner |last=Rodejohann

|year=2011

|title=Neutrino-less double beta decay and particle physics

|journal=International Journal of Modern Physics

|volume=E20 |issue=9 |pages=1833–1930

|doi=10.1142/S0218301311020186

|arxiv=1106.1334 |s2cid=119102859

|bibcode=2011IJMPE..20.1833R

}}

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.

{{cite journal

|first1=J. |last1=Schechter

|first2=J.W.F. |last2=Valle

|year=1982

|title=Neutrinoless double-β decay in SU(2) x U(1) theories

|journal=Physical Review D

|volume=25 |issue=11 |pages=2951–2954

|doi=10.1103/PhysRevD.25.2951

|bibcode=1982PhRvD..25.2951S

|hdl=10550/47205 |hdl-access=free

|url=http://roderic.uv.es/bitstream/10550/47205/1/073631.pdf

}}

The high-energy analog of the neutrinoless double beta decay process is the production of same-sign charged lepton pairs in hadron colliders;{{cite journal |first1=Wai-Yee |last1=Keung |first2= Goran |last2=Senjanović |year=1983 |title=Majorana neutrinos and the production of the right-handed charged gauge boson |journal=Physical Review Letters |volume=50 |issue=19 |pages=1427–1430 |bibcode=1983PhRvL..50.1427K |doi=10.1103/PhysRevLett.50.1427}} 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.{{cite journal |first1=Vladimir |last1=Tello |first2=Miha |last2=Nemevšek |first3=Fabrizio |last3=Nesti |first4=Goran |last4=Senjanović |first5=Francesco |last5=Vissani |year=2011 |title=Left-right symmetry: From LHC to neutrinoless double beta decay |journal=Physical Review Letters |volume=106 |issue=15 |page=151801 |doi=10.1103/PhysRevLett.106.151801 |arxiv=1011.3522 |bibcode=2011PhRvL.106o1801T |pmid=21568545|s2cid=42414212 }} 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.{{cite journal |last1=Kayser |first1=Boris |last2=Goldhaber |first2=Alfred S. |year=1983 |title=CPT and CP properties of Majorana particles, and the consequences |journal=Physical Review D |volume=28 |issue=9 |pages=2341–2344 |doi=10.1103/PhysRevD.28.2341 |bibcode=1983PhRvD..28.2341K|s2cid=1935565 }}{{cite journal |last=Radescu |first=E.E. |year=1985 |title=On the electromagnetic properties of Majorana fermions |journal=Physical Review D |volume=32 |issue=5 |pages=1266–1268 |doi=10.1103/PhysRevD.32.1266 |pmid=9956279 |bibcode=1985PhRvD..32.1266R}}{{cite journal |last1=Boudjema |first1=F. |last2=Hamzaoui |first2=C. |last3=Rahal |first3=V. |last4=Ren |first4=H.C. |year=1989 |title=Electromagnetic Properties of Generalized Majorana Particles |journal=Physical Review Letters |volume=62 |issue=8 |pages=852–854 |doi=10.1103/PhysRevLett.62.852 |bibcode=1989PhRvL..62..852B |pmid=10040354}} Such minimal interaction with electromagnetic fields makes them potential candidates for cold dark matter.{{cite journal |last1=Pospelov |first1=Maxim |last2=ter Veldhuis |first2=Tonnis |year=2000 |title=Direct and indirect limits on the electro-magnetic form factors of WIMPs |journal=Physics Letters B |volume=480 |issue=1–2 |pages=181–186 |arxiv=hep-ph/0003010 |doi=10.1016/S0370-2693(00)00358-0 |bibcode=2000PhLB..480..181P|s2cid=14251522 }}{{cite journal |last1=Ho |first1=Chiu Man |last2=Scherrer |first2=Robert J. |year=2013 |title=Anapole Dark Matter |journal=Physics Letters B |volume=722 |issue=8 |pages=341–346 |arxiv=1211.0503 |doi=10.1016/j.physletb.2013.04.039 |bibcode= 2013PhLB..722..341H|s2cid=15472526 }}

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 $\backslash gamma(E)$ at energy $E$ to the annihilation operator $\{\backslash gamma^\{\backslash dagger\}(-E)\}$ 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.{{cite journal |first = Frank |last = Wilczek |year=2009 |title = Majorana returns |journal = Nature Physics |volume=5 |issue=9 |pages=614–618 |doi = 10.1038/nphys1380 |bibcode=2009NatPh...5..614W |url = http://ctp.lns.mit.edu/Wilczek_Nature/Majorana_return434.pdf}} 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.{{cite journal |first1 = Chetan |last1 = Nayak |first2= Steven H. |last2 = Simon |first3 = Ady |last3 = Stern |first4 = Michael |last4 = Freedman |first5 = Sankar |last5 = Das Sarma |year = 2008 |title = Non-Abelian anyons and topological quantum computation | journal = Reviews of Modern Physics |volume = 80 |issue = 3 |pages = 1083–1159 | bibcode = 2008RvMP...80.1083N |doi = 10.1103/RevModPhys.80.1083 |arxiv = 0707.1889 |s2cid = 119628297 }}

A quantum vortex in certain superconductors or superfluids can trap midgap states, which is one source of Majorana bound states.{{cite journal |author1=Kopnin, N.B. |author2=Salomaa, M.M. |year=1991 |title=Mutual friction in superfluid {{sup|3}}He: Effects of bound states in the vortex core |journal=Physical Review B |volume=44 |pages=9667–9677 |bibcode=1991PhRvB..44.9667K |doi=10.1103/PhysRevB.44.9667|issue=17|pmid=9998953}}{{cite journal | first = G.E. | last = Volovik | year=1999 | title = Fermion zero modes on vortices in chiral superconductors | journal = JETP Letters | volume=70 | pages = 609–614 | bibcode = 1999JETPL..70..609V | doi = 10.1134/1.568223 | issue = 9 | arxiv = cond-mat/9909426 | s2cid = 118970615 }}{{cite journal | first1 = N. | last1 = Read | first2 = Dmitry | last2 = Green |date=2000 | title = Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect | journal = Physical Review B | volume=61 |issue=15 | pages=10267–10297 |bibcode=2000PhRvB..6110267R|doi=10.1103/PhysRevB.61.10267 |arxiv = cond-mat/9906453 | s2cid = 119427877 }} Shockley states at the end points of superconducting wires or line defects are an alternative, purely electrical, source. An altogether different source uses the fractional quantum Hall effect as a substitute for the superconductor.{{cite journal |first1 = Gregory |last1 = Moore |first2 = Nicholas |last2 = Read |date = August 1991 |title = Nonabelions in the fractional quantum Hall effect |journal = Nuclear Physics B |volume = 360 |issue = 2–3 |pages = 362–396 |bibcode = 1991NuPhB.360..362M |doi = 10.1016/0550-3213(91)90407-O |doi-access = free }}

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.{{cite journal |first1 = Liang |last1 = Fu |first2 = Charles L. |last2 = Kane |year=2008 |title=Superconducting proximity effect and Majorana fermions at the surface of a topological insulatorn|journal=Physical Review Letters |volume=10 |issue=9 |page=096407 |doi=10.1103/PhysRevLett.100.096407| pmid = 18352737 |bibcode = 2008PhRvL.100i6407F |arxiv = 0707.1692 | s2cid = 7618062 }}{{cite journal |first1 = Liang |last1 = Fu |first2 = Charles L. |last2 = Kane |year=2009 |title=Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction |journal=Physical Review B |volume=79 |issue=16 |page=161408 |doi=10.1103/PhysRevB.79.161408 |bibcode = 2009PhRvB..79p1408F |arxiv = 0804.4469 |s2cid = 15398390 }} 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{{cite journal | first = Jason | last = Alicea | year = 2012 | title = New directions in the pursuit of Majorana fermions in solid state systems | journal = Reports on Progress in Physics | volume=75 | issue=7 | page=076501 | arxiv=1202.1293 | bibcode = 2012RPPh...75g6501A | doi = 10.1088/0034-4885/75/7/076501 | pmid=22790778 | s2cid = 206021454 }}{{cite journal | first = C.W.J. | last = Beenakker |date = April 2013 |title=Search for Majorana fermions in superconductors |journal=Annual Review of Condensed Matter Physics | volume = 4 | issue = 113 | pages = 113–136 | arxiv = 1112.1950 | doi = 10.1146/annurev-conmatphys-030212-184337 | bibcode = 2013ARCMP...4..113B | s2cid = 54577113}} first produced some positive results in 2012.{{cite journal |first = Eugenie Samuel |last = Reich |date = 28 February 2012 |title = Quest for quirky quantum particles may have struck gold |journal = Nature News |doi=10.1038/nature.2012.10124}}{{cite news |first = Jonathan |last = Amos |date=13 April 2012 |title = Majorana particle glimpsed in lab |website=BBC News |url = https://www.bbc.co.uk/news/science-environment-17695944 |access-date=15 April 2012}} 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.{{cite journal |first1 = V. | last1 = Mourik | first2 = K. | last2 = Zuo | first3 = S.M. | last3 = Frolov | first4 = S.R. | last4 = Plissard | first5 = E.P.A.M. | last5 = Bakkers | first6 = L.P. | last6 = Kouwenhoven | date = 12 April 2012 | title = Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices | journal = Science | volume = 336 | issue = 6084 | pages = 1003–1007 | doi = 10.1126/science.1222360 | pmid = 22499805 | arxiv = 1204.2792 | bibcode = 2012Sci...336.1003M | s2cid = 18447180 }} 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,{{cite journal|first1 = L.P. | last1 = Rokhinson | first2 = X. | last2 = Liu | first3 = J.K. | last3 = Furdyna | year = 2012 | title = Observation of the fractional ac Josephson effect: the signature of Majorana particles | journal = Nature Physics |volume = 8 | issue = 11 | pages = 795–799 | doi = 10.1038/nphys2429 | arxiv = 1204.4212 | bibcode = 2012NatPh...8..795R | s2cid = 119104344 }} another signature of Majorana bound states.{{cite journal|first1 = H.-J. | last1 = Kwon | first2 = K. | last2 = Sengupta | first3 = V. M. | last3 = Yakovenko | title = Fractional ac Josephson effect in p- and d-wave superconductors | journal = The European Physical Journal B | date = 2004 |volume = 37| issue = 3 | pages = 349–361 | doi = 10.1140/epjb/e2004-00066-4 | arxiv = cond-mat/0210148| bibcode = 2004EPJB...37..349K| s2cid = 119549172 }} A bound state with zero energy was soon detected by several other groups in similar hybrid devices,{{cite journal |last1=Deng |first1=M.T. |last2=Yu |first2=C.L. |last3=Huang |first3=G.Y. |last4=Larsson |first4=M. |last5=Caroff |first5=P. |last6=Xu |first6=H.Q. |date=28 November 2012 |title=Anomalous zero-bias conductance peak in a Nb-InSb nanowire-Nb hybrid device |journal=Nano Letters |volume=12 |issue=12 |pages=6414–6419 |doi=10.1021/nl303758w |bibcode = 2012NanoL..12.6414D |pmid=23181691 |arxiv=1204.4130|s2cid=119240318 }}{{cite journal |last1=Das |first1=A. |last2=Ronen |first2=Y. |last3=Most |first3=Y. |last4=Oreg |first4=Y. |last5=Heiblum |first5=M. |last6=Shtrikman |first6=H. |date=11 November 2012 |title=Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions |journal=Nature Physics |volume=8 |issue=12 |pages=887–895 |doi=10.1038/nphys2479 |arxiv = 1205.7073 |bibcode = 2012NatPh...8..887D |s2cid=119297473 }}{{cite journal |last1=Churchill |first1=H.O.H. |last2=Fatemi |first2=V. |last3=Grove-Rasmussen |first3=K. |last4=Deng |first4=M.T. |last5=Caroff |first5=P. |last6=Xu |first6=H.Q. |last7=Marcus |first7=C.M. |date=6 June 2013 |title=Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover |journal=Physical Review B |volume=87|issue=24 |page=241401(R) |doi=10.1103/PhysRevB.87.241401 |arxiv = 1303.2407 |bibcode = 2013PhRvB..87x1401C |s2cid=118487534}}{{cite journal |last1=Deng |first1=M.T. |last2=Yu |first2=C.L. |last3=Huang |first3=G.Y. |last4=Larsson |first4=Marcus |last5=Caroff |first5=P. |last6=Xu |first6=H.Q. |date=11 November 2014 |title=Parity independence of the zero-bias conductance peak in a nanowire based topological superconductor-quantum dot hybrid device |journal=Scientific Reports |volume=4 |page=7261|doi=10.1038/srep07261|arxiv = 1406.4435 |bibcode = 2014NatSR...4.7261D |pmid=25434375|pmc=4248274}} and fractional Josephson effect was observed in topological insulator HgTe with superconducting contacts.{{cite journal |first1 = J. | last1 = Wiedenmann | first2 = E. | last2 = Bocquillon | first3 = R.S. | last3 = Deacon | first4 = S. | last4 = Hartinger | first5 = O. | last5 = Herrmann | first6 = T.M. | last6 = Klapwijk | first7 = L. | last7 = Maier | first8 = C. | last8 = Ames | first9 = C. | last9 = Brune | first10 = C. | last10 = Gould | first11 = A. | last11 = Oiwa | first12 = K. | last12 = Ishibashi | first13 = S. | last13 = Tarucha | first14 = H. | last14 = Buhmann | first15 = L.W. | last15 = Molenkamp |display-authors=6 | year = 2016 | title = 4-pi-periodic Josephson supercurrent in HgTe-based topological Josephson junctions | journal = Nature Communications | volume = 7 | pages = 10303 | doi = 10.1038/ncomms10303 | pmid = 26792013 | pmc = 4735757 | arxiv = 1503.05591 | bibcode = 2016NatCo...710303W }}

The aforementioned experiments mark possible verifications of independent 2010 theoretical proposals from two groups{{cite journal| first1 = Roman M. | last1 = Lutchyn | first2 = Jay D. | last2 = Sau | first3 = S. | last3 = Das Sarma | date = August 2010 | title = Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures | journal = Physical Review Letters | volume = 105 | issue = 7 | page = 077001 | bibcode = 2010PhRvL.105g7001L | doi = 10.1103/PhysRevLett.105.077001 | arxiv = 1002.4033 | pmid=20868069 | s2cid = 8863469 }}{{cite journal| first1 = Yuval | last1 = Oreg | first2 = Gil | last2 = Refael | first3 = Felix | last3 = von Oppen | title = Helical Liquids and Majorana Bound States in Quantum Wires | journal = Physical Review Letters | volume = 105 | issue = 17 | page = 177002 | date = October 2010 | doi = 10.1103/PhysRevLett.105.177002 | bibcode = 2010PhRvL.105q7002O | arxiv = 1003.1145 | pmid=21231073 | s2cid = 14736864 }} 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{{cite journal |last1=Lee |first1=E.J.H. |last2=Jiang |first2=X. |last3=Houzet |first3=M. |last4=Aguado |first4=R. |last5=Lieber |first5=C.M. |last6=Franceschi |first6=S.D. |date=15 December 2013 |title=Spin-resolved Andreev levels and parity crossings in hybrid superconductor–semiconductor nanostructures |journal=Nature Nanotechnology |volume=9 |issue=1 |pages=79–84 |doi=10.1038/nnano.2013.267 |pmid=24336403 |arxiv = 1302.2611 |bibcode = 2014NatNa...9...79L |s2cid=9579343}} could highly mimic the zero voltage conductance peak of a Majorana bound state. The subtle relation between those trivial bound states and Majorana bound states was reported by researchers at the Niels Bohr Institute,{{cite journal |author1=Deng, M.T. |author2=Vaitiekėnas, S. |author3=Hansen, E.B. |author4=Danon, J. |author5=Leijnse, M. |author6=Flensberg, K. |author7=Nygård, J. |author8=Krogstrup, P. |author9=Marcus, C.M. |display-authors=6 |year=2016 |title=Majorana bound state in a coupled quantum-dot hybrid-nanowire system |journal=Science |volume=354 |issue=6319 |pages=1557–1562 |doi=10.1126/science.aaf3961 |pmid=28008065|arxiv=1612.07989 |bibcode=2016Sci...354.1557D |s2cid=5219260}} 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.{{cite journal |first1=Stevan |last1=Nadj-Perge |last2=Drozdov |first2=Ilya K. |last3=Li |first3=Jian |last4=Chen |first4=Hua |last5=Jeon |first5=Sangjun |last6=Seo |first6=Jungpil |last7=MacDonald |first7=Allan H. |last8=Bernevig |first8=B. Andrei |last9=Yazdani |first9=Ali |display-authors=6 |date=2 October 2014 |title=Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor |journal=Science |volume=346 |issue=6209 |pages=602–607 |arxiv=1410.0682 |bibcode=2014Sci...346..602N |doi=10.1126/science.1259327 |pmid=25278507 |s2cid=206561257}}{{cite press release |title=Majorana fermion: Physicists observe elusive particle that is its own antiparticle |date=2 October 2014 |website=Phys.org |url=http://phys.org/news/2014-10-majorana-fermion-physicists-elusive-particle.html |access-date=3 October 2014}} 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.{{cite journal |last1=Feldman |first1=Benjamin E. |last2=Randeria |first2=Mallika T. |last3=Li |first3=Jian |last4=Jeon |first4=Sangjun |last5=Xie |first5=Yonglong |last6=Wang |first6=Zhijun |last7=Drozdov |first7=Ilya K. |last8=Andrei Bernevig |first8=B. |last9=Yazdani |first9=Ali |display-authors=6 |date=March 2017 |title=High-resolution studies of the Majorana atomic chain platform |journal=Nature Physics |language=en |volume=13 |issue=3 |pages=286–291 |doi=10.1038/nphys3947 |arxiv=1611.02707 |bibcode=2017NatPh..13..286F |s2cid=85510646 |issn=1745-2473 |url=http://www.nature.com/articles/nphys3947}} 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.{{cite journal |last1=Jeon |first1=Sangjun |last2=Xie |first2=Yonglong |last3=Li |first3=Jian |last4=Wang |first4=Zhijun |last5=Bernevig| first5=B. Andrei |last6=Yazdani |first6=Ali |date=2017-11-10 |title=Distinguishing a Majorana zero mode using spin-resolved measurements |journal=Science |language=en |volume=358 |issue=6364 |pages=772–776 |doi=10.1126/science.aan3670 |pmid=29025997 |arxiv=1710.04662 |bibcode=2017Sci...358..772J |s2cid=29851188 |issn=0036-8075 |url=https://www.science.org/doi/10.1126/science.aan3670}} More experiments finding evidence for Majorana bound states in chains have 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.{{Cite journal |last1=Kim |first1=Howon |last2=Palacio-Morales |first2=Alexandra |last3=Posske |first3=Thore |last4=Rózsa |first4=Levente |last5=Palotás |first5=Krisztián |last6=Szunyogh |first6=László |last7=Thorwart |first7=Michael |last8=Wiesendanger |first8=Roland |display-authors=6 |date=2018-05-30 |title=Toward tailoring Majorana bound states in artificially constructed magnetic atom chains on elemental superconductors |journal=Science Advances |language=en |volume=4 |issue=5 |page=eaar5251 |doi=10.1126/sciadv.aar5251 |issn=2375-2548 |pmc=5947976 |pmid=29756034 |bibcode=2018SciA....4.5251K}}{{cite journal |last1=Jäck |first1=Berthold |last2=Xie |first2=Yonglong |last3=Yazdani |first3=Ali |date=August 2021 |title=Detecting and distinguishing Majorana zero modes with the scanning tunnelling microscope |journal=Nature Reviews Physics |language=en |volume=3 |issue=8 |pages=541–554 |doi=10.1038/s42254-021-00328-z |arxiv=2103.13210 |bibcode=2021NatRP...3..541J |s2cid=232335790 |issn=2522-5820 |url=https://www.nature.com/articles/s42254-021-00328-z}}

Majorana fermions may also emerge as quasiparticles in quantum spin liquids, and were observed by researchers at the U.S. Oak Ridge National Laboratory, working in collaboration with Max Planck Institute and University of Cambridge on 4 April 2016.{{cite journal |last1=Banerjee |first1=A. |last2=Bridges |first2=C.A. |last3=Yan |first3=J.-Q. |display-authors=etal |date=4 April 2016 |title=Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet |journal=Nature Materials |volume=15 |issue=7 |pages=733–740 |arxiv=1504.08037 |bibcode=2016NatMa..15..733B |doi=10.1038/nmat4604 |pmid=27043779 |s2cid=3406627}}

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.{{cite journal |last1=He |first1=Qing Lin |last2=Pan |first2=Lei |last3=Stern |first3=Alexander L. |last4=Burks |first4=Edward C. |last5=Che |first5=Xiaoyu |last6=Yin |first6=Gen |last7=Wang |first7=Jing |last8=Lian |first8=Biao |last9=Zhou |first9=Quan |display-authors=6 |date=2017-07-21 |title=Chiral Majorana fermion modes in a quantum anomalous Hall insulator–superconductor structure |journal=Science |language=en |volume=357 |issue=6348 |pages=294–299 |arxiv=1606.05712 |bibcode=2017Sci...357..294H |doi=10.1126/science.aag2792 |issn=0036-8075 |pmid=28729508 |s2cid=3904085}}{{Retracted|doi=10.1126/science.adf7575|pmid=36395244|https://retractionwatch.com/2021/12/16/another-setback-for-majorana-particle-as-science-paper-earns-an-expression-of-concern/ Retraction Watch|https://retractionwatch.com/2022/11/17/another-majorana-particle-paper-retracted-this-time-from-science/ Retraction Watch|intentional=yes}}{{cite news|author=Conover, Emily |date=20 July 2017 |title=Majorana fermion detected in a quantum layer cake |publisher=Science News|url=https://www.sciencenews.org/article/majorana-fermion-detected-quantum-layer-cake}} In this system, Majorana fermions edge mode give a rise to a $\backslash tfrac\{1\}\{2\}\; \backslash tfrac\{e^\{2\}\}\{h\}$ conductance edge current. Subsequent experiments by other groups, however, could not reproduce these findings.{{cite journal |last1=Kayyalha |first1=Morteza |last2=Xiao |first2=Di |last3=Zhang |first3=Ruoxi |last4=Shin |first4=Jaeho |last5=Jiang |first5=Jue |last6=Wang |first6=Fei |last7=Zhao |first7=Yi-Fan |last8=Xiao |first8=Run |last9=Zhang |first9=Ling |last10=Fijalkowski |first10=Kajetan M. |last11=Mandal |first11=Pankaj |last12=Winnerlein |first12=Martin |last13=Gould |first13=Charles |last14=Li|first14=Qi |last15=Molenkamp |first15=Laurens W. |last16=Chan |first16=Moses H.W. |last17=Samarth |first17=Nitin |last18=Chang |first18=Cui-Zu |display-authors=6 |date=2020-01-03 |title=Absence of evidence for chiral Majorana modes in quantum anomalous Hall-superconductor devices |journal=Science |language=en |volume=367 |issue=6473|pages=64–67 |doi=10.1126/science.aax6361 |pmid=31896711 |arxiv=1904.06463 |bibcode=2020Sci...367...64K |s2cid=209677626}}{{cite journal |author=Jelena Stajic |date=2020-01-03 |title=Looking for chiral Majoranas |journal=Science |volume=367 |issue=6473 |pages=36–38 |doi=10.1126/science.2020.367.6473.twis |s2cid=240657983 |doi-access=free }}{{cite press release |title=The case of the elusive Majorana: The so-called 'angel particle' is still a mystery |website=ScienceDaily |date=2020-01-03 |publisher=Penn State |url=https://www.sciencedaily.com/releases/2020/01/200103141035.htm |quote=A 2017 report of the discovery of a particular kind of Majorana fermion – the chiral Majorana fermion, referred to as the 'angel particle' – is likely a false alarm, according to new research.}} In November 2022, the article by He et al. was retracted by the editors,{{cite journal |title=Editorial Retraction |journal=Science |language=en |volume=378 |issue=6621|pages=718}} 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 the bulk of a pure substance.{{Cite journal |last1=Wang |first1=Dongfei |last2=Kong |first2=Lingyuan |last3=Fan |first3=Peng |last4=Chen |first4=Hui |last5=Zhu |first5=Shiyu |last6=Liu |first6=Wenyao |last7=Cao |first7=Lu |last8=Sun |first8=Yujie |last9=Du |first9=Shixuan |display-authors=6 |date=2018-08-16 |title=Evidence for Majorana bound states in an iron-based superconductor |journal=Science |volume=362 |issue=6412 |pages=333–335 |doi=10.1126/science.aao1797 |pmid=30115743 |issn=0036-8075 |bibcode=2018Sci...362..333W |arxiv=1706.06074 |s2cid=52021577}} However, more recent experimental studies in iron-based superconductors show that topologically trivial Caroli–de Gennes–Matricon states{{cite journal |last1=Chen |first1=Mingyang |display-authors=etal |date=2018-03-06 |title=Discrete energy levels of Caroli-de Gennes-Matricon states in quantum limit in FeTe{{sub|0.55}}Se{{sub|0.45}} |journal=Nature Communications |volume=9 |issue=970 |page=970 |pmid=29511191 |pmc=5840178 |arxiv=1706.06074 |bibcode=2018NatCo...9..970C |doi=10.1038/s41467-018-03404-8 |s2cid=3706042}} and Yu–Shiba–Rusinov states{{cite journal |last1=Chatzopoulos |first1=Damianos |display-authors=etal |date=2021-01-12 |title=Spatially dispersing Yu-Shiba-Rusinov states in the unconventional superconductor FeTe{{sub|0.55}}Se{{sub|0.45}} |journal=Nature Communications |volume=12 |issue=298 |page=298 |doi=10.1038/s41467-020-20529-x |pmid=33436594 |pmc=7804303 |arxiv=2006.12840}} 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.{{Cite journal |last1=Manna |first1=Sujit |last2=Wei |first2=Peng |last3=Xie |first3=Yingming |last4=Tuen Law |first4=Kam |last5=Lee |first5=Patrick |last6=Moodera |first6=Jagadeesh |date=2020-04-06 |title=Signature of a pair of Majorana zero modes in superconducting gold surface states |journal=PNAS |volume=117 |issue=16 |pages=8775–8782 |doi=10.1073/pnas.1919753117 |pmid=32253317 |pmc=7183215 |arxiv=1911.03802 |bibcode=2020PNAS..117.8775M |s2cid=207852777|doi-access=free }}

One of the causes of interest in Majorana bound states is that they could be used in quantum error correcting codes.{{cite journal |last1=Nayak |first1=Chetan |last2=Simon |first2=Steven H. |last3=Stern |first3=Ady |last4=Freedman |first4=Michael |last5=Sarma |first5=Sankar Das |date=2008-03-27 |title=Non-Abelian anyons and topological quantum computation |journal=Reviews of Modern Physics |volume=80 |issue=3 |pages=1083–1159 |doi=10.1103/RevModPhys.80.1083 |arxiv=0707.1889 |bibcode=2008RvMP...80.1083N |s2cid=119628297}}{{cite journal |last1=Sarma |first1=Sankar Das |last2=Freedman|first2=Michael|last3=Nayak |first3=Chetan |date=2015-10-27 |title=Majorana zero modes and topological quantum computation |journal=npj Quantum Information |volume=1 |issue=1 |page=15001 |doi=10.1038/npjqi.2015.1 |bibcode=2015npjQI...115001S |s2cid=116918566 |issn=2056-6387 |language=en |doi-access=free |arxiv=1501.02813 }} This process is done by creating so called 'twist defects' in codes such as the toric code{{cite journal|last1=Bombin|first1=H.|title=Topological Order with a Twist: Ising Anyons from an Abelian Model|journal=Physical Review Letters|date=14 July 2010|volume=105|issue=3|pages=030403|doi=10.1103/PhysRevLett.105.030403|pmid=20867748|bibcode=2010PhRvL.105c0403B|arxiv=1004.1838|s2cid=5285193}} which carry unpaired Majorana modes.{{cite journal | last1=Zheng | first1=Huaixiu | last2=Dua | first2=Arpit | last3=Jiang | first3=Liang | year=2015 | title=Demonstrating non-Abelian statistics of Majorana fermions using twist defects | journal=Physical Review B | volume=92 | issue=24 | pages=245139 | arxiv=1508.04166 | doi=10.1103/PhysRevB.92.245139 | bibcode=2015PhRvB..92x5139Z | s2cid=118701510 }} The Majoranas are then "braided" by being physically moved around each other in 2D sheets or networks of nanowires.{{Cite web |title=Why Majoranas are cool: Braiding and quantum computation |website=topocondmat.org |url=https://topocondmat.org/w2_majorana/braiding.html |access-date=2021-10-07}} This braiding process forms a projective representation of the braid group.{{cite journal|doi=10.1103/PhysRevB.87.045130 | volume=87 | issue=4 | pages=045130 | title=Twist defects and projective non-Abelian braiding statistics | journal=Physical Review B| arxiv=1208.4834 | bibcode=2013PhRvB..87d5130B | last1=Barkeshli | first1=Maissam | last2=Jian | first2=Chao-Ming | last3=Qi | first3=Xiao-Liang | year=2013 | s2cid=96451256 }}

Such a realization of Majoranas would allow them to be used to store and process quantum information within a quantum computation.{{cite journal |last1=Hastings |first1=M.B. |last2=Geller |first2=A. |year=2015 |title=Reduced space-time and time costs using dislocation codes and arbitrary ancillas |journal=Quantum Information and Computation |volume=15 |issue=11–12 |pages=0962–0986 |doi=10.26421/QIC15.11-12-6 |bibcode=2014arXiv1408.3379H |arxiv=1408.3379 |s2cid=36122810}} Though the codes typically have no Hamiltonian to provide suppression of errors, fault-tolerance would be provided by the underlying quantum error correcting code.

In February 2023{{Cite journal |last1=Dvir |first1=Tom |last2=Wang |first2=Guanzhong |last3=van Loo |first3=Nick |last4=Liu |first4=Chun-Xiao |last5=Mazur |first5=Grzegorz P. |last6=Bordin |first6=Alberto |last7=ten Haaf |first7=Sebastiaan L. D. |last8=Wang |first8=Ji-Yin |last9=van Driel |first9=David |last10=Zatelli |first10=Francesco |last11=Li |first11=Xiang |last12=Malinowski |first12=Filip K. |last13=Gazibegovic |first13=Sasa |last14=Badawy |first14=Ghada |last15=Bakkers |first15=Erik P. A. M. |date=15 February 2023 |title=Realization of a minimal Kitaev chain in coupled quantum dots |url=https://www.nature.com/articles/s41586-022-05585-1 |journal=Nature |language=en |volume=614 |issue=7948 |pages=445–450 |doi=10.1038/s41586-022-05585-1 |pmid=36792741 |arxiv=2206.08045 |bibcode=2023Natur.614..445D |s2cid=249712114 |issn=1476-4687}}{{Cite journal |last=Wright |first=Katherine |date=2023-02-15 |title=Evidence Found for a Majorana "Cousin" |url=https://physics.aps.org/articles/v16/24 |journal=Physics |language=en |volume=16 |pages=24|doi=10.1103/Physics.16.24 |bibcode=2023PhyOJ..16...24W |s2cid=257616165 |doi-access=free }} 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 a 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.{{Cite journal |last=Kitaev |first=A Yu |date=2001-10-01 |title=Unpaired Majorana fermions in quantum wires |url=https://iopscience.iop.org/article/10.1070/1063-7869/44/10S/S29 |journal=Physics-Uspekhi |volume=44 |issue=10S |pages=131–136 |doi=10.1070/1063-7869/44/10S/S29 |arxiv=cond-mat/0010440 |bibcode=2001PhyU...44..131K |s2cid=250872768 |issn=1468-4780}}

{{see also|Topological quantum computer}}

On 19 February 2025 Microsoft announced the “Majorana 1” processor, for use in quantum computers, claiming to feature Majorana zero modes.{{Cite news|last=Nellis |first=Stephen |date=19 February 2025 |title=Microsoft creates chip it says shows quantum computers are 'years, not decades' away|publisher=Reuters |url=https://www.reuters.com/technology/microsoft-creates-chip-it-says-shows-quantum-computers-are-years-not-decades-2025-02-19/}} The work created a new class of materials called topoconductors, which use topological superconductivity to control hardware-protected topological qubits.{{Cite web |title=Powerful quantum computers in years not decades, says Microsoft |url=https://www.bbc.com/news/articles/cj3e3252gj8o|date=19 Feb 2025 |last1=Vallance |first1=Chris |access-date=2025-02-26 |website=BBC}} The research paper utilized a method to determine fermion parity in Majorana zero modes in a single shot – validating a necessary ingredient for utility-scale topological quantum computation architectures based on measurement.{{cite journal |last1=Aghaee |first1=Morteza |title=Interferometric single-shot parity measurement in InAs–Al hybrid devices |url=https://www.nature.com/articles/s41586-024-08445-2 |date= 19 Feb 2025 |journal=Nature |volume=638 |issue=8051 |page=651–655 |doi=10.1038/s41586-024-08445-2 |arxiv=2401.09549 |access-date=21 Feb 2025}}

{{reflist|30em}}

• {{cite journal

| first = Palash B. | last = Pal

| year = 2011 | orig-date = 12 October 2010

| title = Dirac, Majorana, and Weyl fermions

| journal = American Journal of Physics

| volume = 79 | issue = 5 | pages = 485–498

| arxiv = 1006.1718 | s2cid = 118685467

| doi = 10.1119/1.3549729

| bibcode = 2011AmJPh..79..485P

}}

{{Particles}}

Category:Fermions

Category:Quantum field theory