Majorana fermion

Another common convention for the normalization of the Majorana fermion operator is

${\displaystyle f={\tfrac {1}{2}}(\gamma _{1}+i\gamma _{2})}$

${\displaystyle f^{\dagger }={\tfrac {1}{2}}(\gamma _{1}-i\gamma _{2})}$

This convention has the advantage that the Majorana operator squares to the identity.

Using this convention, a collection of ${\displaystyle 2n}$ Majorana fermions (${\displaystyle n}$ ordinary fermions), ${\displaystyle \gamma _{i}}$ (${\displaystyle i=1,2,..,2n}$) obey the following anticommutation identities

${\displaystyle \{\gamma _{i},\gamma _{j}\}=2\delta _{ij}}$

and

${\displaystyle \sum _{ijkl}[\gamma _{i}A_{ij}\gamma _{j},\gamma _{k}B_{kl}\gamma _{l}]=4\sum _{ij}\gamma _{i}[A,B]_{ij}\gamma _{j}}$

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

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 ${\displaystyle {\tfrac {1}{2}}{\tfrac {e^{2}}{h}}}$ 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]

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.

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 Kitaev.[63]