perspective Majorana returns Frank Wilczek In his short career, Ettore Majorana made several profound contributions. One of them, his concept of ‘Majorana fermions’ — particles that are their own antiparticle — is finding ever wider relevance in modern physics.
nrico Fermi had to cajole his friend Indeed, when, in 1928, Paul Dirac number of electrons minus the number of Ettore Majorana into publishing discovered1 the theoretical framework antielectrons, plus the number of electron Ehis big idea: a modification of the for describing spin-½ particles, it seemed neutrinos minus the number of antielectron Dirac equation that would have profound that complex numbers were unavoidable neutrinos is a constant (call it Le). These ramifications for particle physics. Shortly (Box 2). Dirac’s original equation contained laws lead to many successful selection afterwards, in 1938, Majorana mysteriously both real and imaginary numbers, and rules. For example, the particles (muon
disappeared, and for 70 years his modified therefore it can only pertain to complex neutrinos, νμ) emitted in positive pion (π) + + equation remained a rather obscure fields. For Dirac, who was concerned decay, π → μ + νμ, will induce neutron- − footnote in theoretical physics (Box 1). with describing electrons, this feature to-proton conversion νμ + n → μ + p, Now suddenly, it seems, Majorana’s posed no problem, and even came to but not proton-to-neutron conversion + concept is ubiquitous, and his equation seem an advantage because it ‘explained’ νμ + p → μ + n; the particles (muon is central to recent work not only in why positrons, the antiparticles of antineutrinos, ν¯ μ) emitted in the negative − − neutrino physics, supersymmetry and dark electrons, exist. pion decay π → μ + ν¯ μ obey the opposite matter, but also on some exotic states of Enter Ettore Majorana. In his 1937 pattern. Indeed, it was through studies of ordinary matter. paper2, Majorana posed, and answered, the this kind that the existence of different question of whether equations for spin-½ ‘flavours’ of neutrino, corresponding Majorana fermions fields must necessarily, like Dirac’s original to the different types of charged lepton An electrically charged particle is different equation, involve complex numbers. was discovered4. from its antiparticle as it has the opposite Considerations of mathematical elegance Of course, if neutrinos really differ from electric charge, and electric charge is a and symmetry both motivated and guided antineutrinos, then they are not Majorana measurable, stable property. It is possible, his investigation. Majorana discovered fermions. In recent years, however, the however, for an electrically neutral particle that, to the contrary, there is a simple, situation has come to seem less clear-cut, to be its own antiparticle. Photons, which clever modification of Dirac’s equation for it has been discovered that neutrinos have spin 1 in units of the rationalized that involves only real numbers. With oscillate in flavour5. For example, an Planck’s constant ħ, are a familiar case; this discovery, Majorana made the idea electron antineutrino emitted from the Sun neutral pions (spin 0) are a further example, that spin-½ particles could be their own can arrive at Earth as a muon antineutrino and gravitons (spin 2) another. Particles antiparticles theoretically respectable, that or a tau antineutrino. In some sense this that are their own antiparticles must be is, consistent with the general principles is a small effect, but when neutrinos travel created by fieldsφ that obey φ = φ* — of relativity and quantum theory. In a long way they have time to do rare that is, real fields, because the complex- his honour, we call such hypothetical things. These flavour oscillations show conjugate fieldsφ * create their antiparticles. particles Majorana fermions. But are there that the separate ‘laws’ of lepton-number The equations for particles with spin 0, physical examples? conservation do not hold: at best, only the
spin 1 and spin 2 — the Klein–Gordon, sum Le + Lμ + Lτ can be strictly conserved. Maxwell (electromagnetism) and are neutrinos Majorana fermions? Thus awakened from our dogmatic Einstein (general relativity) equations, Majorana speculated that his equation slumber, we re-open Majorana’s question: respectively — readily accommodate real might apply to neutrinos. In 1937, could the distinction between neutrino fields, as these equations are formulated neutrinos were themselves hypothetical, and antineutrino, which seems so plainly using real numbers. and their properties unknown. The apparent, be superficial? (Consider the vast On the other hand, the neutron (which experimental study of neutrinos perceptual disconnect between the morning has spin ½), despite being electrically commenced with their discovery3 in 1956, star and the evening star — yet they’re neutral, is not its own antiparticle: several but their observed properties seemed to both Venus.) neutrons can peacefully coexist within disfavour Majorana’s idea. Specifically, there But how can ν = ν¯ be reconciled with an atomic nucleus, but an antineutron seemed to be a strict distinction between those many observations that seemed to rapidly annihilates. Neither, of course, neutrinos and antineutrinos. indicate a distinction? The point is that are the most famous spin-½ particles — The distinction is connected with the the ν particles produced in, for example, electrons and protons, which are electrically law of lepton-number conservation, which π+ → μ+ + ν are in a very different state of charged — their own antiparticles. So it applies for each of the leptons — electron motion from the ν¯ particles produced in is not obvious that we need an equation (e), muon (μ) and tau (τ). For example, π− → μ− + ν¯ . The former are left handed, to describe spin-½ particles that are their for electrons, lepton-number conservation spinning in the sense that the fingers of your own antiparticles. means that, in any reaction, the total left hand point, if your thumb aligns with the
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velocity, whereas the latter are right handed. equations if we don’t add antineutrinos as have a heavier fermionic (half-integer spin) So, logically, ν and ν¯ and might be the same separate entities to our fundamental theory. partner; and vice versa for each known particle, having different behaviours when it For if neutrinos in the right-handed state fermion. There is suggestive, although is in different states of motion. of motion are not antineutrinos, they must circumstantial, evidence for the existence If you could bring neutrinos and be something else; and that something of these ‘superpartners’. Specifically, if the antineutrinos to rest, and do experiments else must (as it’s escaped detection so far) superpartners exist and are not too heavy, with them, you could test whether they interact with the kinds of matter we know then in their evanescent form, as virtual behave the same way. That is impractical, very feebly indeed. It is hard to fit such particles, they are computed to modify unfortunately: theoretically, the cosmos is oddball entities within the most attractive (partially screen) the basic units of strong, awash with slow neutrinos, but they are too unified theories, which require symmetry weak and electromagnetic charge so as hard to detect. Although such a direct test among their building blocks. to quantitatively account for the different of Majorana’s hypothesis seems out of reach observed charge values — in a unified field for now, several ambitious experiments are Of supersymmetry and dark matter theory where, fundamentally, those values underway to test one of its implications, Neutrinos were Majorana’s own candidates are equal10. In brief, supersymmetry allows namely, that even the last bastion of for Majorana fermions, and although the unification of the fundamental forces.
lepton-number conservation, Le + Lμ + Lτ, they look more promising than ever in If supersymmetry is valid, then the can be toppled. Searches for neutrino-less that regard, no longer are they unique. photon has as its superpartner a spin-½ double β decay, such as Ge76 → Se76 + 2e, Other problems at the frontier of particle, the photino. As the photino mirrors are launching a promising fusillade6. In fundamental physics seem to call for more the properties of the photon, it must be this decay, total lepton number changes by Majorana fermions. its own antiparticle. Thus the photino is a two, so its occurrence would disprove the Supersymmetry is a leading proposal Majorana fermion. So, for similar reasons, conservation law definitively. to improve the symmetry and coherence are various other superpartners (such as Meanwhile, the leading ideas on of the equations of physics9. It involves neutral gauginos, as well as Higgsinos). In neutrino masses, rooted in unified field the expansion of spacetime into a new, a word, supersymmetry comes chock-a- theories, predict that neutrinos are quantum dimension. Particles that move block with Majorana fermions. If, as widely Majorana fermions7,8. The detailed logic is in that direction change their mass and anticipated, superpartners are produced — complex, but the basic idea is simple: we spin. If supersymmetry is valid, then every as real, not just virtual, particles — at the get more economical, and much prettier, known bosonic (integer spin) particle will Large Hadron Collider, we might quickly
Box 1 | The romance of Ettore Majorana
“There are many categories of scientists: masterpieces: the last, as mentioned, and people of second and third rank, who do another on the quantum theory of spins in their best, but do not go very far; there magnetic fields, which anticipates the later are also people of first-class rank, who brilliant development of molecular-beam make great discoveries, fundamental to and magnetic resonance techniques. the development of science. But then there In recent years, a small industry ana.
are the geniuses, like Galileo and Newton. R has developed, bringing Majorana’s
Well Ettore Majorana was one of them.” ajo unpublished notebooks into print (see m .
Enrico Fermi, not known for flightiness e for example ref. 31). They are impressive or overstatement, is the source of these documents, full of original calculations much-quoted lines. and expositions covering a wide range The bare facts of Majorana’s life are of physical problems. They leave an . Recami and . Recami
briefly told. Born in Catania, Italy, on e overwhelming impression of gathering 5 August 1906, into an accomplished family, strength; physics might have advanced he rose rapidly through the academic ranks, more rapidly on several fronts had became a friend and scientific collaborator Majorana pulled this material together and of Fermi, Werner Heisenberg and other shared it with the world. luminaries, and produced a stream of How did he vanish? There are two high-quality papers. Then, beginning in leading theories. According to one, he 1933, things started to go terribly wrong. of © kind concession retired to a monastery, to escape a spiritual He complained of gastritis, became which he took up in January 1938. Two crisis and accept the embrace of his deep reclusive, with no official position, and months later, he embarked on a mysterious Catholic faith (not unlike another tortured published nothing for several years. In trip to Palermo, arrived, then boarded a ship scientific genius, Blaise Pascal). According 1937, he allowed Fermi to write-up and straight back to Naples and disappeared to another, he jumped overboard, an act of submit, under his (Majorana’s) name, his without a trace. suicide recalling the alienated supermind last and most profound paper — the point Majorana published only nine papers of fiction, Odd John32. Fermi’s appreciation of departure of this article — containing in his lifetime, none very lengthy. They had a wistful conclusion, which is less well results he had derived some years before. are collected, with commentaries, all in known: “Majorana had greater gifts than At Fermi’s urging, Majorana applied both Italian and English versions, in a slim anyone else in the world. Unfortunately for professorships and was awarded the volume30. Each is a substantial contribution he lacked one quality which other men Chair in Theoretical Physics at Naples, to quantum physics. At least two are generally have: plain common sense.”
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Box 2 | The Majorana equation
0 In 1928, Dirac proposed his relativistic (in which I have adopted units such that γ˜ = σ2 ⊗ σ1 33 0 1 wave equation for electrons . This ħ = c = 1). Furthermore, we require that γ be γ˜ = iσ1 ⊗ 1 2 was a watershed event in theoretical Hermitian, and the remaining marices anti- γ˜ = iσ3 ⊗ 1 3 physics, leading to a new understanding Hermitian. These conditions ensure that the γ˜ = iσ2 ⊗ σ2 of spin, predicting the existence of equation properly describes the wavefunction antimatter, and impelling — for its of a spin-½ particle with mass m. or alternatively, as ordinary matrices: adequate interpretation — the creation of Dirac found a suitable set of 4 × 4 quantum field theory. It also inaugurated γ matrices, whose entries contain both real 00 0 –i a new method in theoretical physics, and imaginary numbers. For the equation to 0 00–i γ 0 = emphasizing mathematical aesthetics as make sense, ψ must then be a complex field. 00i 0 a source of inspiration. Majorana’s most Dirac and most other physicists regarded influential work is especially poetic, in this consequence as a good feature, because i 00 0 that it applies Dirac’s method to Dirac’s electrons are electrically charged, and the 00i 0 equation itself, to distill from it an description of charged particles requires 000 i equation both elegant and new. For many complex fields, even at the level of the γ 1 = years, Majorana’s idea seemed to be an Schrödinger equation. This is also true in the i 000 ingenious but unfulfilled speculation. language of quantum field theory. In quantum 00i 0 Recently, however, it has come into its field theory, if a given fieldφ creates the own, and now occupies a central place particle A (and destroys its antiparticle Ā), the i 0 0 0 in several of the most vibrant frontiers of complex conjugate φ* will create Ā and destroy 0 i 00 γ 2 = modern physics. A. Particles that are their own antiparticles 000 –i Dirac’s equation connects the four must be associated with fields obeyingφ = φ*, components of a fieldψ . In modern that is, real fields. Because electrons and 00 0 –i (covariant) notation it reads positrons are distinct, the associated fieldsψ 000 –i and ψ* and must therefore be different; this μ 3 0 00i (iγ ∂μ − m)ψ = 0 feature appeared naturally in Dirac’s equation. γ = Majorana inquired whether it might 00i 0 The γ matrices are required to obey the be possible for a spin-½ particle to be its –i 000 rules of Clifford algebra, that is own antiparticle, by attempting to find the equation that such an object would satisfy. Majorana’s equation, then, is simply {γμγυ} ≡ γμγυ + γυγμ = 2ημυ To get an equation of Dirac’s type (that is, μ suitable for spin-½) but capable of governing (i γ˜ ∂μ – m)ψ˜ = 0) where ημυ is the metric tensor of flat space. a real field, requiresγ matrices that satisfy Spelling it out, we have the Clifford algebra and are purely imaginary. Because the γ˜ μ matrices are purely Majorana found such matrices. Written as imaginary, the matrices i γ˜ μ are real, and (γ0)2 = −(γ1)2 = −(γ2)2 = −(γ3)2 = 1 tensor products of the usual Pauli matrices σ, consequently this equation can govern a γjγk = −γkγj for i ≠ j they take the form: real fieldψ˜ .
establish the existence of several Majorana solid-state physics. Recent investigations electrons; they act as if they were positively fermions, even as the status of neutrinos suggest that exotic quasiparticle excitations charged electrons. remains uncertain. in a variety of interesting condensed-matter The particle–antiparticle correspondence, A popular hypothesis11 for the systems are Majorana fermions. Many of as well as the manifestation of the electron’s astronomical dark matter is that it is a these ideas were born of high mathematical and hole’s characteristic fermion statistics, is weakly interacting massive particle, or fantasy, but there is a very real chance that transparent in the mathematical formalism WIMP. Indeed, it could be one of the they may soon mature into a surprisingly of second quantization. Here, ‘particle superpartners just mentioned. The overall tangible, and even useful, form. states’ are associated with creation operators † neutrality of Majorana fermions means The concept of excitations that are their cj , antiparticle (hole) states with their that they can decay, or annihilate in pairs. own antiparticles is not unprecedented conjugate operators, cj. In essence, cj can The debris from such events could produce in solid-state physics. An example is the create a hole, or destroy a particle, in state j, † energetic cosmic rays, which are the object exciton — a quasiparticle formed by bound whereas cj can create a particle, or destroy a of ongoing search experiments. It is entirely states of electrons and holes. The latter hole in state j. Three key relations embody possible that WIMPs, dominating the mass are a familiar concept in modern solid- the characteristics of Fermi–Dirac statistics of the Universe and proclaiming their state physics12, and represent the absence and describe the relationship between existence with cosmic fireworks, will be the of an electron in a mode that is normally particle and hole operators associated with first established Majorana fermions. (in the overall ground state) occupied. In different states. First, rough but more vivid language, holes are † 2 2 Majorana modes in the solid state bubbles of emptiness in the Fermi sea of (c j ) = cj = 0 There is a completely different area of electrons (Fig. 1a). Holes ‘look’ and ‘behave’ physics in which Majorana’s idea is starting like the antiparticles or antimatter to which means that the attempt to cram two to receive more attention — theoretical their corresponding particles, the valence electrons, or two holes, into the same state
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comes to naught — a manifestation of Pauli’s Indeed, already in the earliest days special, symmetric solutions of differential exclusion principle. Second, for two distinct following Bardeen, Cooper and Schrieffer’s equations to the topology of the parameters (orthogonal) states j and k, triumphant theory of superconductivity that appear in those equations16. (BCS theory)14, it was realized that certain The zero modes are mixtures of particles �c†, c � c†c + c c† = �c , c � = �c†, c†� = 0 j k = jjk k j k j k fermionic modes in the superconducting and holes in equal measure, and thus state are created by mixtures of what one can call the quasiparticles associated which is a consequence of the antisymmetry were, in the normal state, electron and with these zero modes ‘partiholes’. Such of Fermi–Dirac statistics. Third, hole operators. In physical terms, a partiholes differ crucially from conventional (normal state) electron mode can lower its excitons. They are created by operators of �c†, c � = �c†, c � = 1 energy, in the superconducting state, by the form γ = c† + c . As γ is left invariant by jj kk j j j j mixing with a (normal state) hole mode the charge conjugation, c ↔ c†, partihole which is the completeness relation. attached to a Cooper pair. Mathematically, operators create localized spin-½ particles In this formalism, particle–hole this phenomenon is encoded in the that are their own antiparticles. In this sense, interchange (charge conjugation) is Bogoliubov–Valatin formalism13. Therein partiholes are a new instance of Majorana’s † implemented by cj ↔ cj . Because electrons one finds that the creation operators for idea, which is why the corresponding zero and holes have opposite charge, they are modes in the superconducting state are modes are called Majorana modes. not their own antiparticles and therefore mixtures of electron and hole creation But where and how would one † not Majorana fermions. Excitons, on the operators, in the form cosθ c j + sinθ c k. observe such Majorana modes? In most other hand, are bound states of electrons But electrons in such Bogoliubov– superconductors, in which the Cooper pairs and holes, and thus, in the language of Valatin modes are not exactly their own have orbital angular momentum 0 (s-wave) second quantization, they are created by antiparticles (except accidentally, in the and the electrons obey a Schrödinger-like, combinations of electron and hole operators, specific casej = k and θ = ± π/4) and nonrelativistic equation, zero modes are † † of the general form cj ck + ckcj . Under thus, such modes are not a realization of not predicted to occur. However, they are charge conjugation, this exciton ‘creation’ Majorana fermions. predicted17 to occur if the Cooper pairs have
operator goes over into itself, and therefore However, there are certain types of orbital angular momentum 1 (px + ipy-wave), the excitations it creates are their own superconductor in which Majorana-type or for s-wave Cooper pairing if the electrons antiparticles. But conventional excitons are excitations are predicted to emerge. For in the normal state obey a Dirac-like always bosons, with integer spin, and thus instance, some superconductors can equation18. The former case could occur, can make no call on Majorana’s legerdemain. contain magnetic flux tubes, also known in effect, in certain quantum Hall states — In this sense they are analogous to the as Abrikosov vortices15, the presence of specifically, the so-called Pfaffian or Moore– photons of conventional particle physics. which alters the equations for the electrons. Read state at ν = 5/2 filling19 — and possibly In particular, depending on the kind in some exotic superconductors including superconductors to the rescue of superconductor and the electronic strontium ruthenate20. The case in which So can there ever be a solid-state situation spectrum, the vortices may trap so-called the normal-state electrons obey Dirac-like in which half-integer-spin particles are their zero modes, spin-1/2 ‘excitons’ of very low beahviour is predicted to be induced at the own antiparticles? At first sight it seems (formally, zero) energy. The zero modes surface of a new class of material called hopeless to realize Majorana fermions from are discrete; there are a finite number topological insulators21 or in graphene22, the raw material of electrons in solids, associated with each vortex. The existence of by exploiting the proximity effect to induce simply because electrons are charged, and these modes is related to a profound result superconductivity in those materials. Thus, therefore definitely different from their in mathematics, the Atiyah–Singer index the race is on to find such exotic realizations antimatter counterparts, the (oppositely theorem, which connects the existence of of Majorana’s idea in a variety of systems. charged) holes. But superconductivity changes the picture13, because in ab superconductors the absolute distinction between electrons and holes is blurred (Fig. 1b,c). In such materials, electrons 5 + form so-called Cooper pairs, which, owing to their boson-like nature, can form a dense ‘condensate’, unimpeded by the Pauli exclusion principle. Indeed, it is just this c condensate that, theoretically, is responsible for superconductivity13. As a consequence, electron number o T
is in effect no longer conserved: two o electrons (in a Cooper pair) can be added PH ock
or subtracted from the condensate without T is
substantially changing its properties. © Crucially too, the superconductor screens electric and confines magnetic fields so that Figure 1 | Antimatter matters in the solid state. a, A familiar concept in solid-state physics, holes are charge is no longer observable (Fig. 1c). bubbles of missing electrons in the Fermi sea of the electronic spectrum, behaving like positively charged Thus, in a superconductor the most electrons. b, In a superconductor, the properties of electrons (blue) and holes (grey) are drastically daunting barrier to producing Majorana- modified by their interaction with the surrounding sea of Cooper pairs; a hole can attract or bind to a like excitations — the charge-conjugation Cooper pair, and acquire negative charge. c, More importantly, Cooper pairs cluster around holes and thin hurdle — seems vulnerable. out around electrons, in such a way that no rigorous distinction between them remains.
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tomorrow’s qubits? Majorana partihole operators (of quantity n) both disappointing and surprising if real j k Majorana modes open a portal into some obey the algebra {γ , γ } = 2δjk. Majorana fermions, now ardently sought, extremely unusual and interesting aspects This is another Clifford algebra, similar do not soon materialize. ❐ of quantum theory. For instance, Majorana to the one in the Dirac or Majorana modes on Abrikosov vortices have the equations (Box 2). However, in contrast Frank Wilczek is at the Center for Theoretical unique feature that the operators that create to the algebra associated with the Dirac Physics, Massachusetts Institute of Technology, them square to the identity, not to zero — and Majorana equations, the algebra 77 Massachusetts Avenue, Cambridge, 2 that is, γj = 1. Thus the object created by describing Majorana modes describes the Massachusetts 02139, USA. γj is not a conventional fermion. Nor is it geometry of n Euclidean dimensions in e-mail: [email protected] a conventional boson. Indeed, adding a an abstract mode space, rather than the second partihole to a state already occupied paltry 3+1 dimensions of spacetime. In References by one partihole neither annihilates the this mode space, the process of exchanging 1. Dirac, P. A. M. Proc. R. Soc. Lond. A 117, 610–624 (1928). 2. Majorana, E. Nuovo Cimento 5, 171 –184 (1937). state nor creates an essentially new, doubly neighbouring modes characterized by the 3. Cowan, C. Jr, Reines, F., Harrison, F., Kruse, H. & McGuire, A. occupied state; rather, it recreates the state indices j and k induces the transformation Science 124, 103–104 (1956). of zero occupancy. 4. Danby, G. et al. Phys. Rev. Lett. 9, 36–44 (1962). To further explain the quantum statistics γ → γ 5. Fukuda, Y. et al. Phys. Rev. Lett. 81, 1562–1567 (1998). j k 6. http://en.wikipedia.org/wiki/Double_beta_decay of Majorana modes, let us consider several γk → −γj 7. Gell-Mann, M., Ramond, P. & Slansky, R. in Supergravity of them, and see what happens when (eds Freedman, D. & van Nieuwenhuizen, P.) 315 they are interchanged. At this point, it is in which the minus sign is all-important. (North Holland, 1979). 8. Yanagida, T. in Proc. Workshop on Unified Theory and Baryon convenient to restrict our consideration This transformation is just what we would Number in the Universe (eds Sawada, O. & Sugamoto, A.) to two spatial dimensions, such that the get from a π/2 rotation in the j–k plane, 95 (KEK, 1979). vortices trapping the zero modes reduce with γi regarded as a vector. The minimal 9. Weinberg, S. The Quantum Theory of Fields Vol. 3: to point defects rather than lines. In such realization of all these transformations is Supersymmetry (Cambridge Univ. Press, 1999). 10. Dimopoulos, S., Raby, S. & Wilczek, F. Phys. Rev. D two-dimensional settings, the possibility the so-called spinor representation, which 24, 1681–1683 (1981). [(n+1)/2] 27 of quantum statistics more general than is 2 -dimensional . In this way, simple 11. Bertone, G., Hooper, D. & Silk, J. Phys. Rep. 405, 279–390 (2005). bosons or fermions — anyons — has been exchange operations in physical space 12. Anderson, P. W. Concepts in Solids (W. A. Benjamin, 1976). discussed vigorously in recent years23,24. induce complex motions in quantum- 13. Schrieffer, J. R. Theory of Superconductivity (W. A. Benjamin, 1964). The simplest (abelian) anyons can obey mechanical Hilbert space. 14. Bardeen, J., Cooper, L. & Schrieffer, J. R. Phys. Rev. statistics ranging continuously between This power to evolve simple operations 108, 1175–1204 (1957). Fermi–Dirac and Bose–Einstein statistics, in physical space into complex motions 15. Abrikosov, A. Sov. Phys. JETP 5, 1174–1182 (1957). 16. Weinberg, E. J. Phys. Rev. D 24, 2669–2673 (1981). because the anticlockwise exchange of in an exponentially large Hilbert space 17. Read, N. & Green, D. Phys. Rev. B 61, 10267–10297 (2000). one anyon around another gives rise to a is thought to provide qualitatively new 18. Jackiw, R. & Rossi, P. Nucl. Phys. B 190, 681–691 (1981). iθ complex phase, |ψ1ψ2 〉 = e |ψ2ψ1 〉, which and powerful methods for quantum 19. Moore, G. & Read, N. Nucl. Phys. B 360, 362–396 (1991). can be different from ±1 θ( = 2π for bosons information processing28. This is the vision 20. Das Sarma, S., Nayak, C. & Tewari, S. Phys. Rev. B 29 73, 220502 (2006). and θ = π for fermions). Anyons are of ‘topological quantum computing’ . 21. Fu, L. & Kane, C. Phys. Rev. Lett. 100, 096407 (2008). included in the conventional theory of the Many of the systems mentioned 22. Ghaemi, P. & Wilczek, F. Preprint at
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