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In particle physics a Majorana fermion (/maɪəˈrɑːnə/[1]) or 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 Dirac fermion, which describes fermions that are not their own antiparticles.

With the exception of neutrinos, all of the Standard Model elementary 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 be either Dirac or Majorana fermions.

In condensed matter physics, quasiparticle excitations can appear like bound Majorana states. However, instead of a single fundamental particle, they are the collective movement of several individual particles (themselves composite) which are governed by non-Abelian statistics.

Theory

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The concept goes back to Majorana's suggestion in 1937[2] that electrically neutral spin-⁠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 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

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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

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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).

Ettore Majorana 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,[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

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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

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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] A 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 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,[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 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 the U.S. 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 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 the bulk of a 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

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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

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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 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.[17]

Topological qubits

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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.[63] The work created a new class of materials called topoconductors, which use topological superconductivity to control hardware-protected topological qubits.[64] 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.[65]

References

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Further reading

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Majorana fermion

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A Majorana fermion is a that is its own , meaning it is a neutral particle indistinguishable from its antiparticle counterpart. This concept was first proposed by Italian physicist in 1937, in his seminal paper introducing a symmetric formulation of the that allows for such self-conjugate solutions. Unlike Dirac fermions, which come in distinct particle-antiparticle pairs (e.g., electrons and positrons), Majorana fermions possess no net or other distinguishing quantum numbers beyond spin, making them potentially massive yet electrically neutral. In , Majorana fermions are of profound interest due to their implications for the nature of neutrinos, which could be Majorana particles if they violate conservation. This hypothesis is central to experiments probing , a rare nuclear process that, if observed, would confirm the Majorana nature of neutrinos and provide insights into the matter-antimatter asymmetry of the universe. In nuclear and , Majorana's theory also influences models of weak interactions and grand unified theories, where such fermions could mediate processes beyond the . Beyond fundamental , Majorana fermions manifest as quasiparticles in condensed matter systems, particularly in topological superconductors and semiconductor-superconductor hybrids. These Majorana zero modes—localized, zero-energy excitations at material edges or defects—exhibit non-Abelian statistics, enabling robust topological where information is stored non-locally and protected from decoherence. Experimental pursuits, including setups and vortex states in superconductors, have reported signatures of these modes since the early , fueling advances in fault-tolerant quantum technologies despite ongoing debates over definitive confirmation.

Fundamentals

Definition and properties

A Majorana fermion is defined as a fermion that is its own , meaning the particle and its antiparticle are identical and indistinguishable. This self-conjugate property arises because the Majorana field satisfies the condition that it equals its own charge conjugate, implying no separate antiparticle exists. first hypothesized such particles in 1937, proposing a symmetric solution to the specifically for neutral fermions, such as those without . Key properties of Majorana fermions include their real-valued wavefunctions in appropriate representations, where the spinor components are real rather than complex, distinguishing them from standard Dirac fermions. They obey Fermi-Dirac statistics, adhering to the and exhibiting fermionic anticommutation relations that ensure half-integer spin and antisymmetric wavefunctions under particle exchange. Under charge conjugation, Majorana fermions remain neutral and unchanged, as the operation maps the field onto itself, which is a direct consequence of their self-conjugate nature. These fermions can be either massless or massive, though massive variants require both left- and right-handed chiral components for consistency with Lorentz invariance. Unlike bosons, Majorana fermions follow fermionic statistics, with their satisfying anticommutation relations that prevent multiple occupancy of the same . Regarding stability, Majorana fermions in can annihilate upon interacting with another identical particle, similar to particle-antiparticle , but in certain condensed matter systems, they manifest as stable excitations.

Relation to Dirac fermions

Dirac fermions represent the standard class of particles in , characterized by distinct particle and states described by complex scalar fields. For instance, the and its , the , form a Dirac fermion pair, where the particle and have opposite electric charges and are created or annihilated independently. In contrast, Majorana fermions are a special case where the particle is identical to its , described by real (self-conjugate) fields that impose a reality condition on the components. This self-conjugation halves the number of independent compared to Dirac fermions: a requires four complex components (eight real ) to describe both particle and polarizations, while a Majorana uses only four real components (two for each polarization, shared between particle and ). Consequently, there is no separate field; the same field annihilates both the particle and what would be its in the Dirac case. The fundamental distinction arises in the application of charge conjugation, a discrete symmetry that interchanges particles with . For Dirac fields, the charge conjugate field ψc\psi^c satisfies CψC−1=ψc≠ψC \psi C^{-1} = \psi^c \neq \psi, where ψc=iγ2ψ∗\psi^c = i \gamma^2 \psi^* in the Dirac representation, confirming the particle and antiparticle as distinct entities. For Majorana fields, however, the field is an eigenstate under charge conjugation such that ψ=ψc\psi = \psi^c, rendering the particle self-conjugate and invariant (up to a phase) under this transformation. Majorana fermions can only exist under specific conditions, such as zero or the presence of symmetries that allow the reality condition without violating conservation laws. Charged particles like electrons cannot be Majorana because their distinct particle-antiparticle charges would be incompatible with self-conjugation; neutral particles, such as neutrinos, are viable candidates if their interactions permit it. This self-conjugate nature has profound implications for conservation laws, particularly , which is strictly conserved for Dirac fermions but can be violated by two units (ΔL=2\Delta L = 2) in theories with Majorana particles. Such violation enables processes like , where a nucleus decays by emitting two electrons without accompanying antineutrinos, serving as a potential of Majorana neutrinos.

Theoretical description

Majorana equation

The Majorana equation is a relativistic describing neutral spin-1/2 fermions that are their own antiparticles, originally proposed by in 1937 as part of a symmetric theory for electrons, positrons, and neutral particles like neutrinos. In this framework, Majorana addressed issues in Dirac's theory by introducing real wave functions satisfying anticommutation relations, avoiding the need for distinct antiparticles for neutral fermions. The equation derives from the (iγμ∂μ−m)ψ=0(i \gamma^\mu \partial_\mu - m) \psi = 0 by imposing the Majorana condition ψ=ψc=CψˉT\psi = \psi^c = C \bar{\psi}^T, where CC is the charge conjugation matrix satisfying C−1γμC=−γμTC^{-1} \gamma^\mu C = -\gamma^{\mu T} and ψˉ=ψ†γ0\bar{\psi} = \psi^\dagger \gamma^0. This condition ensures the field is self-conjugate, restricting solutions to those with real coefficients in an appropriate representation (e.g., the Majorana representation where γμ\gamma^\mu matrices are chosen such that the equation admits real solutions). Consequently, the ψ\psi satisfies ψ∗=Bψ\psi^* = B \psi, with B=γ0C∗B = \gamma^0 C^* being the charge conjugation matrix in this basis, reducing the independent components while preserving Lorentz invariance. The explicit form remains (iγμ∂μ−m)ψ=0(i \gamma^\mu \partial_\mu - m) \psi = 0, identical to the but solved under the self-conjugacy constraint. Solutions are expanded in plane waves as ψ(x)=∑s∫d3p(2π)3[as(p)us(p)e−ip⋅x+as†(p)vs(p)eip⋅x]\psi(x) = \sum_s \int \frac{d^3 p}{(2\pi)^3} \left[ a_s(\mathbf{p}) u_s(\mathbf{p}) e^{-i p \cdot x} + a_s^\dagger(\mathbf{p}) v_s(\mathbf{p}) e^{i p \cdot x} \right], where usu_s and vs=γ0Cus∗v_s = \gamma^0 C u_s^* are basis spinors, and the coefficients asa_s are real due to the Majorana condition. For massive particles, this yields two corresponding to the two helicity states, as the self-conjugacy halves the independent components from four to two complex (or four real) . The associated Lagrangian density is L=ψˉ(iγμ∂μ−m)ψ\mathcal{L} = \bar{\psi} (i \gamma^\mu \partial_\mu - m) \psi, which is invariant under charge conjugation for Majorana fields since ψ→ψc=ψ\psi \to \psi^c = \psi implies L→L\mathcal{L} \to \mathcal{L}. In quantized field theory, a factor of 1/21/2 is sometimes included in the mass term to account for the identical particle-antiparticle nature, but the standard form ensures Hermiticity and positive definiteness of the Hamiltonian.

Self-conjugate representations

In quantum field theory, the self-conjugate nature of Majorana fermions is realized through specific representations of the Dirac gamma matrices that impose a reality condition on the spinor fields. In four-dimensional Minkowski spacetime, the Majorana representation chooses gamma matrices γ~μ\tilde{\gamma}^\mu that are purely imaginary, satisfying γ~μ=−(γ~μ)†\tilde{\gamma}^\mu = -(\tilde{\gamma}^\mu)^\dagger, with explicit forms γ~0=(0σ2σ20)\tilde{\gamma}^0 = \begin{pmatrix} 0 & \sigma_2 \\ \sigma_2 & 0 \end{pmatrix}, γ~1=(iσ100iσ1)\tilde{\gamma}^1 = \begin{pmatrix} i\sigma_1 & 0 \\ 0 & i\sigma_1 \end{pmatrix}, γ~2=(0σ2−σ20)\tilde{\gamma}^2 = \begin{pmatrix} 0 & \sigma_2 \\ -\sigma_2 & 0 \end{pmatrix}, γ~3=(iσ300iσ3)\tilde{\gamma}^3 = \begin{pmatrix} i\sigma_3 & 0 \\ 0 & i\sigma_3 \end{pmatrix}, where σi\sigma_i are the Pauli matrices. This basis ensures that solutions to the Dirac equation for Majorana fields are real: ψ~=ψ~∗\tilde{\psi} = \tilde{\psi}^*. The charge conjugation matrix CC in this representation satisfies C=iγ~2γ0C = i \tilde{\gamma}^2 \gamma^0, enabling the Majorana condition ψc=CψˉT=ψ\psi^c = C \bar{\psi}^T = \psi, where ψc\psi^c is the charge-conjugate spinor and ψˉ=ψ†γ0\bar{\psi} = \psi^\dagger \gamma^0. The Clifford algebra underpinning these representations is the standard anticommutation relation {γμ,γν}=2gμνI\{\gamma^\mu, \gamma^\nu\} = 2 g^{\mu\nu} I, where gμνg^{\mu\nu} is the Minkowski metric. For Majorana spinors, the reality condition further constrains the spinor components, reducing the 4-component complex to an equivalent 4-component real field in the Majorana basis. This is tied to the properties of the charge conjugation operator, which obeys CγμC−1=−(γμ)TC \gamma^\mu C^{-1} = - (\gamma^\mu)^T for μ=0,1,2,3\mu = 0,1,2,3, ensuring the self-conjugacy. As a result, certain bilinear forms vanish identically for Majorana fields. For instance, the vector current ψˉγμψ=0\bar{\psi} \gamma^\mu \psi = 0, as it is odd under charge conjugation, while the axial vector bilinear ψˉγμγ5ψ\bar{\psi} \gamma^\mu \gamma^5 \psi and scalar ψˉψ\bar{\psi} \psi are allowed but satisfy specific symmetries. The pseudoscalar bilinear ψˉiγ5ψ\bar{\psi} i \gamma^5 \psi also vanishes due to the reality condition in the Majorana representation. In lower dimensions, such as 1+1 spacetime, the structure simplifies to 2-component Majorana spinors. Here, the can be represented as γ0=σ1=(0110)\gamma^0 = \sigma^1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} and γ1=iσ2=(01−10)\gamma^1 = i \sigma^2 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, satisfying the {γm,γn}=2ηmnI\{\gamma^m, \gamma^n\} = 2 \eta^{mn} I with Euclidean or Minkowski metric ηmn\eta^{mn}. These real matrices allow the Majorana spinor to be purely real-valued, with the charge conjugation simply ψc=ψ∗\psi^c = \psi^*. The existence of such representations depends on dimensionality: Majorana spinors are possible in dimensions d=2,3,4mod  8d = 2,3,4 \mod 8 for Minkowski , where the charge conjugation matrix CC is antisymmetric. From the perspective of Lorentz group representations, Majorana, Weyl, and Dirac spinors correspond to distinct equivalence classes. Weyl spinors form irreducible representations (1/2, 0) or (0, 1/2) of the SL(2,ℂ), which are complex and chiral. A Dirac spinor combines two independent Weyl spinors into a reducible (1/2, 0) ⊕ (0, 1/2) representation, allowing complex fields with particle-antiparticle distinction. In contrast, a Majorana spinor is a real representation where the left- and right-chiral components are charge conjugates of each other, ψ=χ+χ^\psi = \chi + \hat{\chi}, effectively halving the independent degrees of freedom compared to a Dirac field while remaining self-conjugate. This equivalence holds in dimensions permitting real spinor structures, such as 4D.

Role in particle physics

Neutrinos as Majorana particles

In extensions of the Standard Model, neutrinos are hypothesized to be Majorana fermions to account for their observed tiny masses, which are much smaller than those of other fermions. The type-I seesaw mechanism provides a natural explanation by introducing three right-handed sterile neutrinos that acquire large Majorana masses at a high energy scale, typically around 10^{14} GeV or higher. The light neutrino masses then arise from the mixing between left-handed active neutrinos and these heavy states, resulting in an effective mass m_ν ≈ m_D^2 / M_R, where m_D is the Dirac mass term from Yukawa couplings and M_R is the heavy Majorana mass. This mechanism suppresses the light masses relative to the electroweak scale, addressing the hierarchy problem without fine-tuning. The seesaw was first proposed by Minkowski in 1977 and independently elaborated by Gell-Mann, Ramond, Slansky, and Yanagida in 1979. The Majorana nature of neutrinos implies they are self-conjugate, enabling processes that violate lepton number conservation by ΔL = 2. A key example is neutrinoless double beta decay (0νββ), in which a nucleus undergoes (Z, A) → (Z+2, A) without emitting neutrinos, mediated by the exchange of virtual Majorana neutrinos in the Standard Model effective theory. This process is forbidden for Dirac neutrinos, which distinguish particles from antiparticles, and its observation would directly confirm the Majorana character. The decay rate is proportional to the square of the effective Majorana mass parameter m_ββ = |∑i U{ei}^2 m_i|, where U_{ei} are elements of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix and m_i are the neutrino mass eigenvalues. Schechter and Valle demonstrated in 1982 that any light Majorana neutrino contribution to 0νββ implies a finite Majorana mass term, establishing a foundational theorem for beyond-Standard-Model physics. Neutrino oscillation experiments, such as those from and SNO, confirm non-zero neutrino masses through flavor mixing but cannot distinguish between Dirac and Majorana types. Oscillation probabilities depend only on mass-squared differences Δm^2_{ij} and mixing angles, remaining insensitive to the two additional Majorana CP-violating phases that arise in the Majorana case, as these phases factor out of the evolution Hamiltonian for standard vacuum or constant-matter propagation. This insensitivity holds because oscillations probe coherent superpositions of mass eigenstates without requiring particle-antiparticle identity. Sterile neutrinos, as the heavy right-handed Majorana components in the seesaw mechanism, extend the by three generations and play a crucial role in generating the light spectrum. These singlets under the SU(2)_L × U(1)_Y gauge group couple weakly to active neutrinos via the Dirac mass terms, remaining otherwise inert and allowing their Majorana masses to form without gauge anomalies. In minimal models, two or three such sterile states suffice to fit oscillation data while predicting one massless light . If sufficiently massive, typically in the keV range, sterile Majorana neutrinos emerge as viable warm dark matter candidates in cosmological models. They can be produced non-thermally in the early through oscillations with active neutrinos, generating a relic that matches observations without overclosing the , as proposed by Dodelson and Widrow in 1994. This mechanism yields a velocity dispersion intermediate between hot and , potentially resolving small-scale structure tensions in the and galaxy formation simulations.

Experimental searches for Majorana nature

The primary experimental probe for the Majorana nature of neutrinos in is the search for neutrinoless double beta (0νββ) decay, a lepton-number-violating process in which a nucleus undergoes the transition (A,Z)→(A,Z+2)+2e−(A, Z) \to (A, Z+2) + 2e^- without neutrino emission. This decay is forbidden for Dirac neutrinos but permitted if neutrinos are their own antiparticles, with the decay rate proportional to the square of the effective Majorana mass parameter mββ=∑iUei2mim_{\beta\beta} = \sum_i U_{ei}^2 m_i, where UeiU_{ei} are the PMNS mixing matrix elements and mim_i the neutrino masses. No observation of 0νββ has been reported, but experiments using isotopes such as 76^{76}Ge, 130^{130}Te, and 136^{136}Xe have established stringent lower limits on the half-life T1/20νT_{1/2}^{0\nu}, corresponding to upper bounds on mββm_{\beta\beta}. Key results come from germanium-based detectors like GERDA and its successor LEGEND. The GERDA experiment, using enriched 76^{76}Ge crystals, reported a final limit of T1/20ν>1.8×1026T_{1/2}^{0\nu} > 1.8 \times 10^{26} years (90% C.L.) based on 127 kg·yr exposure. LEGEND-200, combining data from GERDA, MAJORANA Demonstrator, and its own 200 kg enriched germanium array, improved this to T1/20ν>1.9×1026T_{1/2}^{0\nu} > 1.9 \times 10^{26} years (90% C.L.) in early 2025 analyses. These limits translate to mββ<75−200m_{\beta\beta} < 75-200 meV, depending on nuclear matrix element calculations. Xenon-based experiments provide complementary constraints using 136^{136}Xe. The EXO-200 detector set a limit of T1/20ν>3.5×1025T_{1/2}^{0\nu} > 3.5 \times 10^{25} years (90% C.L.) from its full dataset of 564 kg·yr exposure. KamLAND-Zen, employing a xenon-loaded liquid scintillator, achieved T1/20ν>2.3×1026T_{1/2}^{0\nu} > 2.3 \times 10^{26} years (90% C.L.) with its complete phase-II data in 2025, yielding mββ<28−61m_{\beta\beta} < 28-61 meV. For tellurium, the CUORE bolometer array using 130^{130}Te reported T1/20ν>3.8×1025T_{1/2}^{0\nu} > 3.8 \times 10^{25} years (90% C.L.) in its latest 2025 analysis, corresponding to mββ<36−180m_{\beta\beta} < 36-180 meV and probing below 0.1 eV in the quasi-degenerate neutrino mass regime. Collider experiments at the LHC search for heavy Majorana neutrinos in the , focusing on same-sign dilepton signatures from processes like pp→W→ℓNpp \to W \to \ell N, N→WℓN \to W \ell, where ℓ\ell denotes same-flavor charged leptons. ATLAS and CMS analyses up to 140 fb⁻¹ of 13 TeV data exclude heavy neutrinos with masses up to ~1 TeV for mixing angles ∣VℓN∣2≳10−6|V_{\ell N}|^2 \gtrsim 10^{-6}, with the most stringent bounds from CMS reaching ∣VℓN∣2<10−5|V_{\ell N}|^2 < 10^{-5} for MN=100−500M_N = 100-500 GeV in same-sign electron/muon channels. These limits tighten with 2025 Run-3 data, extending exclusions to 1.27 TeV in vector-like lepton models incorporating heavy Majorana states. Neutrino oscillation experiments like T2K and NOvA do not directly test the Majorana nature but inform the mass hierarchy, which affects 0νββ interpretability. Their 2025 joint analysis of beam neutrino data shows no strong preference for normal or inverted ordering, with a Bayes factor of 1.3 favoring inverted but consistent with both within uncertainties. As of 2025, no evidence for the Majorana nature of neutrinos exists, with the tightest mββ<0.028−0.061m_{\beta\beta} < 0.028-0.061 eV bound from KamLAND-Zen and combined analyses providing constraints below 0.1 eV. Future ton-scale experiments promise enhanced sensitivity: LEGEND-1000 targets T1/20ν∼1028T_{1/2}^{0\nu} \sim 10^{28} years (mββ≲10m_{\beta\beta} \lesssim 10 meV), while nEXO aims for 102810^{28} years with a 5-tonne 136^{136}Xe TPC, potentially covering the inverted hierarchy fully if no signal emerges.

Emergent states in condensed matter

Majorana bound states

Majorana bound states are quasiparticle excitations that emerge as zero-energy modes localized at defects, edges, or vortices in topological superconductors, behaving as their own antiparticles due to self-conjugacy. These states arise from superconducting pairing mechanisms, particularly in systems exhibiting p-wave or effective p-wave superconductivity, where the topological order of the bulk gap hosts these emergent fermions. The zero-energy nature of these bound states is a direct consequence of the intrinsic particle-hole symmetry in superconductors, which enforces that the energy spectrum is symmetric around zero, pinning the modes at E=0E = 0. A foundational theoretical framework for understanding Majorana bound states is provided by the Jackiw-Rebbi model, which describes a one-dimensional Dirac fermion coupled to a scalar field with a domain wall in the mass term. In this soliton-like configuration, the mass inversion across the domain wall induces a topologically protected zero-energy bound state, interpreted as a Majorana fermion mode localized at the interface. This mechanism illustrates how topological defects can host such states, serving as a paradigm for their emergence in more complex condensed matter systems. The locality of Majorana bound states manifests as exponential decay of their wavefunctions away from the hosting defect or boundary, conferring robustness against local perturbations and disorder due to the topological protection of the bulk gap. When multiple such modes are present, their exchange, or braiding, generates non-Abelian statistics, where the operation of swapping two modes results in a unitary transformation on the degenerate ground state manifold, rather than a simple phase factor. This property stems from the modes' fractional fermion parity and enables fault-tolerant encoding of quantum information in the collective state.

Realizations in topological superconductors

One of the simplest theoretical models for realizing Majorana bound states is the Kitaev chain, a one-dimensional p-wave superconductor described by the Hamiltonian H=−μ∑ici†ci−t∑i(ci†ci+1+h.c.)+Δ∑i(cici+1+h.c.)H = -\mu \sum_i c_i^\dagger c_i - t \sum_i (c_i^\dagger c_{i+1} + \mathrm{h.c.}) + \Delta \sum_i (c_i c_{i+1} + \mathrm{h.c.}), where cic_i (ci†c_i^\dagger) annihilates (creates) a fermion at site ii, μ\mu is the chemical potential, tt is the hopping amplitude, and Δ\Delta is the p-wave pairing strength. In the topological phase where ∣μ∣<2∣t∣|\mu| < 2|t| and Δ≠0\Delta \neq 0, this model hosts unpaired Majorana zero modes localized at the chain ends, which are robust against local perturbations due to the topological protection of the bulk gap. In condensed matter systems, proximity-induced superconductivity in semiconductor nanowires provides a practical route to engineer effective Kitaev-like chains. For instance, InAs nanowires, which exhibit strong spin-orbit coupling, can be coated with an s-wave superconductor such as Al to induce p-wave-like pairing; applying a perpendicular Zeeman field from a magnetic field then splits the spin degeneracy and opens a topological gap when the field strength exceeds the induced superconducting gap, leading to Majorana bound states at the wire ends. This hybrid setup effectively maps to a one-dimensional topological superconductor, with the spin-orbit interaction enabling the necessary p-wave pairing symmetry in the low-energy sector. Iron-based superconductors, such as FeTe_{1-x}Se_x with x ≈ 0.5, offer a platform for bulk Majorana modes due to their intrinsic topological band structure featuring helical Dirac surface states. In these materials, the coexistence of s±-wave superconductivity and nontrivial topology can give rise to helical Majorana modes within vortex cores or along edges, protected by the bulk topological invariant. Vortex states in proximity-induced topological superconductors on topological insulator (TI) surfaces, as described by the Fu-Kane model, host Majorana bound states at the vortex cores. In this setup, an s-wave superconductor in proximity to a strong TI like Bi_2Se_3 induces effective p-wave pairing on the TI surface states; a vortex in the superconductor then traps a Majorana zero mode due to the helical nature of the surface Dirac fermions, which preserves time-reversal symmetry and supports helical Majorana edge modes. Topological superconductors realizing Majorana modes can be classified by their pairing symmetries: chiral types, such as p+ip or d+id, break time-reversal symmetry and host chiral Majorana edge modes with net chirality, akin to quantum Hall-like states; in contrast, helical superconductors maintain time-reversal symmetry with pairing like p_x ± i p_y in opposite spin sectors, leading to counter-propagating Majorana edge modes that form Kramers pairs. This distinction influences the braiding statistics and potential applications, with helical realizations often emerging in TRS-preserving systems like TI-superconductor interfaces.

Experimental progress

Early theoretical predictions and initial claims

The concept of Majorana bound states in condensed matter systems was first theoretically proposed by Alexei Kitaev in 2001, who described a one-dimensional p-wave superconductor model where unpaired Majorana fermions emerge as zero-energy modes at the ends of a finite-length quantum wire. This model, solvable via a Bogoliubov-de Gennes approach, demonstrated topological protection against local perturbations, making it a foundational framework for non-Abelian anyons in quantum computing. Kitaev later extended these ideas in his 2006 honeycomb lattice spin model, where Majorana fermions arise as itinerant excitations in a quantum spin liquid phase, supporting non-Abelian anyonic statistics through vortex-like defects. Building on these theoretical foundations, predictions for realizing Majorana bound states in experimentally accessible semiconductor systems emerged around 2010. Roman Lutchyn, Jay Sau, and S. Das Sarma proposed that a semiconductor nanowire with strong spin-orbit coupling, proximity-induced s-wave superconductivity, and an external magnetic field could enter a topological phase hosting Majorana zero modes at the wire ends. Independently, Yuval Oreg, Gil Refael, and Felix von Oppen described a similar setup using Rashba spin-orbit interaction in InAs nanowires, emphasizing the role of Zeeman splitting to overcome the superconducting pairing gap and stabilize the topological state. These proposals shifted focus from idealized p-wave superconductors to hybrid heterostructures, predicting observable signatures like zero-bias conductance peaks in tunneling spectroscopy. Initial experimental claims followed swiftly, with the first report in 2012 from the Kouwenhoven group at . Using InSb nanowires proximitized by NbTiN superconductors, they observed stable zero-bias conductance peaks in the differential conductance as a function of magnetic field, interpreting these as signatures of Majorana bound states localized at the nanowire ends. Subsequent reports in 2013, including collaborations involving and Stanford researchers, reinforced these findings through similar nanowire devices, noting the peaks' robustness against temperature and gate voltage variations, though with varying peak heights below the quantized value of 2e2/h2e^2/h. These early claims sparked controversies due to the ambiguous nature of zero-bias peaks, which can arise from trivial Andreev bound states, disorder-induced Yu-Shiba-Rusinov states, or other non-topological mechanisms rather than true Majorana modes. High-profile setbacks included the 2018 investigations and eventual retractions of papers claiming chiral Majorana edge modes, such as the 2017 Science article by He et al. claiming chiral Majorana fermion modes in a quantum anomalous Hall insulator–superconductor structure exhibiting half-integer quantized conductance, later found to involve data irregularities inconsistent with Majorana interpretations. By the end of the 2010s, while multiple groups reported partial evidence through correlated zero-bias features and fusion rules in nanowire networks, no consensus emerged on definitive confirmation, with theoretical analyses highlighting the need for stricter topological gap protocols to distinguish genuine Majorana states.

Recent developments in nanoelectronics and interferometry

Between 2021 and 2023, significant refinements in semiconductor-superconductor nanowire systems improved the observation of zero-bias conductance peaks (ZBPs) associated with Majorana zero modes, primarily through enhanced control over disorder in materials like InAs-Al hybrids. Researchers developed three-terminal conductance spectroscopy setups to distinguish local and non-local transport signatures in both clean and disordered regimes, revealing sharper ZBPs with reduced broadening from impurities. These advances addressed earlier ambiguities by quantifying disorder effects, showing that trivial Andreev bound states could mimic Majorana signals but were separable via temperature and bias dependence. Complementary studies demonstrated nearly quantized conductance plateaus near 2e²/h in proximitized nanowires, attributed to partially resolved Majorana modes despite residual disorder, as reported in high-impact analyses. In 2024, quantum interference experiments in nanoelectronic circuits provided evidence for Majorana-like modes, particularly in graphene-based and InAs setups. High-visibility interference patterns in graphene Fabry-Pérot interferometers at fractional quantum Hall fillings revealed phase shifts consistent with anyon braiding, extending to non-Abelian statistics akin to Majorana zero modes. These observations highlighted nonlocal correlations in transport, with random telegraph noise signaling quasiparticle fluctuations that mimic half-electron behavior under interference. Advancements in interferometry during 2025 further tested Majorana properties through Aharonov-Bohm phase shifts in hybrid devices. In InAs-Al nanowires, single-shot interferometric parity measurements demonstrated protected operations via braiding of non-Abelian anyons, with exponential error suppression inferred from phase coherence up to 2π periodicity. Cotunneling interferometers with three Majorana modes enabled direct probing of non-local braiding statistics, confirming topological encoding without full qubit readout. Reports from quantum computing labs, including Microsoft's Majorana 1 platform, integrated these techniques to stabilize modes against decoherence, though primarily validated in academic setups. Progress in iron-pnictide superconductors from 2022 to 2025 yielded spectroscopic evidence for chiral edge states, leveraging angle-resolved photoemission spectroscopy (ARPES). In FeTe_{1-x}Se_x, ARPES revealed a ~8 meV surface gap at the Dirac point, coexisting with superconductivity and indicative of topologically protected edge modes influenced by magnetism. Dislocation cores trapped non-Abelian quasiparticles, with simulations showing localized zero-energy states that support chiral propagation along edges. Earlier vortex-core spectroscopy in 2017 confirmed zero-bias peaks as signatures in iron-based systems, bridging to edge state realizations. As of 2025, the consensus acknowledges stronger empirical evidence for Majorana modes in nanoelectronics, bolstered by interference and conductance data, yet an ongoing debate persists regarding definitive topological protection. Critics highlight unverified claims in industry prototypes, such as Microsoft's, where protocols like the topological gap test may yield false positives from disorder. No experiment has fully demonstrated braiding-induced protection against local noise, though refinements continue to narrow the gap between signatures and robust applications.

Applications in quantum information

Topological qubits

Topological qubits leverage Majorana zero modes (MZMs) as a platform for robust quantum information storage and processing, capitalizing on their topological protection against local perturbations. In this approach, quantum information is encoded non-locally across pairs of spatially separated MZMs, which are self-conjugate fermionic modes localized at the ends of topological superconducting wires. The qubit state is determined by the fermion parity of the nonlocal Dirac fermion formed by coupling two remote MZMs, ensuring that local noise cannot alter the encoded information without creating high-energy excitations. This encoding exploits the degeneracy inherent in systems hosting an even number of MZMs, where the ground state manifold is protected by the system's topological order. A single logical qubit can be realized using four MZMs, grouped into two pairs, where the fusion rules of the MZMs define the computational basis. Specifically, each pair of MZMs can fuse into either a vacuum (even parity) or a fermion (odd parity) channel, with the overall qubit state corresponding to the combined parity of the two pairs while conserving total fermion parity. This configuration, often termed a tetron, allows the qubit to occupy one of two degenerate states in the topological Hilbert space, enabling dense encoding where multiple qubits share MZMs without direct interference. The fusion outcomes are governed by the non-Abelian nature of MZMs, which follow Ising anyon fusion rules: γi×γj=1\gamma_i \times \gamma_j = 1 or ψ\psi (trivial or fermion), providing a basis for quantum superposition. Quantum gate operations on these qubits are performed through braiding of the MZMs, exploiting their non-Abelian exchange statistics. Unlike Abelian anyons, braiding two MZMs induces a unitary transformation in the degenerate space that depends on the order of exchanges, enabling the implementation of single-qubit gates such as phase shifts or Hadamard rotations without ancillary resources. For instance, a counterclockwise exchange of two MZMs applies a π/2\pi/2 phase to the odd-parity state relative to the even-parity state. These operations are inherently fault-tolerant, as the topological nature of the braiding paths—worldlines in spacetime—protects against errors, requiring only global connectivity rather than precise local control. Multi-qubit gates arise from braiding involving multiple tetron pairs, with the overall computation fault-tolerant due to the absence of error correction overhead during the braiding process itself. Readout of the qubit state involves measuring the fermion parity encoded in the MZM pairs, typically via coupling to a quantum dot or superconducting island for charge sensing, or through tunneling spectroscopy that probes the zero-bias conductance peak associated with MZM overlap. In the charge-sensing approach, the parity-dependent charge on an island coupled to the MZMs shifts the electrochemical potential, detectable via single-electron transistor measurements. Alternatively, interferometric setups can resolve the topological charge by observing Aharonov-Bohm-like oscillations in conductance as a function of magnetic flux threading paths between MZMs. These methods project the qubit onto the computational basis without disturbing the topological protection, allowing repeated measurements essential for quantum algorithms. Scalability to large-scale quantum computation is achieved by arranging multiple topological superconducting segments, such as Kitaev chains, into one- or two-dimensional networks connected by tunable junctions. In a 1D array of Kitaev chains, qubits can be coupled via braiding paths mediated by Y-junctions or T-junctions, enabling universal gate sets through a combination of single-qubit braids and controlled-phase operations. Extending to 2D surfaces, like those in p+ip superconductors or fractional quantum Hall states, supports surface code-like architectures where MZMs at vortices or edges form a lattice for braiding, facilitating error-corrected universal quantum computing. As of 2025, early implementations, such as Microsoft's Majorana 1 processor, have begun demonstrating networked tetron configurations supporting initial multi-qubit operations toward thousands of qubits while maintaining exponential separation between ground and excited states. The primary advantage of MZM-based topological qubits lies in their intrinsic resilience to decoherence, where errors are suppressed exponentially with the separation between MZMs due to the topological energy gap that forbids low-energy local processes. This leads to bit-flip error rates scaling as e−Δ/Te^{-\Delta / T}, where Δ\Delta is the bulk gap and TT the temperature, far surpassing physical qubits without topological order. Braiding operations further enhance fault tolerance by encoding logical information in global topology rather than local states, reducing the need for active error correction and enabling scalable quantum computing with relaxed material requirements.

Quantum error correction with Majorana modes

Majorana modes, particularly zero modes in topological superconductors, offer inherent topological protection against local perturbations, making them promising for quantum error correction in fault-tolerant quantum computing. This protection arises from the non-local encoding of quantum information in the parity of pairs of Majorana bound states, where errors require processes that affect distant modes simultaneously, suppressing decoherence exponentially with system size. Quasiparticle poisoning, where unwanted excitations tunnel into the system and flip the fermion parity, represents the dominant error channel, distinct from standard Pauli errors in qubit systems. Error correction protocols for Majorana-based architectures thus focus on detecting and correcting these parity-violating errors while accounting for correlated noise in multi-mode measurements. Seminal work has developed fermion error-correcting codes tailored to Majorana systems, constructing stabilizers from products of even numbers of Majorana operators to preserve overall parity. For instance, the shortest known fermion code, a [[6,1,3]]_f code derived from weakly self-dual classical binary codes, corrects single quasiparticle poisoning events by measuring local parities across six Majorana modes. This approach enables correction of both parity-violating (fermionic) and parity-conserving (bosonic) errors, with the code distance ensuring robustness against multi-fermion processes. Larger families, such as translationally invariant [[15,1,6]]_f codes, extend this to higher-weight errors, achieving fault tolerance for poisoning probabilities between 10% and 90%. Physical realizations involve mesoscopic superconducting islands hosting localized Majorana zero modes, where syndrome measurements project onto even-parity subspaces without disturbing the encoded information. These codes leverage the spatial separation of Majoranas to exponentially suppress error rates, though finite-temperature effects introduce residual quasiparticle excitations that necessitate active correction. Noise modeling in Majorana architectures bridges physical error processes to abstract fault-tolerance thresholds, using stochastic models that generalize Pauli operators to capture Majorana-specific dynamics. Quasiparticle poisoning induces correlated errors across modes, particularly during joint parity readout, but rapid relaxation to the ground state aligns the effective noise with independent Pauli-like channels. Simulations of subsystem codes, such as the d=5 Bacon-Shor code adapted for Majoranas, yield pseudo-thresholds around 1% for physical error rates, demonstrating viable fault tolerance provided measurement errors are below 0.1%. A one-dimensional repetition code further exemplifies practical implementation, encoding a logical qubit across chains of topological nanowires (e.g., 9 segments of 5 μm each). By measuring local syndromes to correct dominant phase errors—arising from slow parity flips—while tolerating rarer bit-flip errors, this code extends qubit lifetimes from milliseconds to over 1 second at quasiparticle densities of 0.01 μm⁻³, achieving up to 100-fold improvement in coherence time. The hierarchy of error rates, with phase errors far outnumbering flips, underpins the code's efficacy, making it suitable for near-term nanoelectronic platforms. Recent advances in subsystem codes enhance error correction for Majorana tetrons—superconducting islands with four zero modes—by incorporating gauge qubits to detect odd-weight fermionic errors. These [[2n, k_b, k, d_f]] codes, built from Pauli stabilizers and classical linear codes, reduce the number of generators compared to standard stabilizer codes, shortening fault-tolerant measurement sequences (e.g., 8 steps for a [[10,1,2,3]] code). The fermionic distance d_f, bounded by the bosonic distance d_b ≤ d_f ≤ 2d_b, enables correction of both even- and odd-parity errors using belief-propagation decoders like BPOSD. For poisoning rates η=0.1, these codes improve pseudothresholds by up to 84%, outperforming prior constructions and paving the way for scalable arrays in topological quantum processors. As of 2025, Microsoft's Majorana 1 processor has demonstrated tetron-based qubits with distinct parity lifetimes (e.g., X-loop > Z-loop), showing up to milliseconds coherence for certain measurements and validating these codes in hardware toward self-correcting quantum memories where dynamically suppresses errors without continuous intervention.

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