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Majorana

Chia-Nan Yeh National Taiwan University (Dated: January 19, 2015) In this report, we discuss the Majorana fermions, theoretical proposed by Ettore Ma- jorana. In section 1, we would talk about some background knowledge. Later in section 2, there would be some discussion about how to find a Majorana fermions. And at the end, we would give some application of Majorana fermions.

I. BACKGROUND almost 80 years, the Majorana is still a hypothet- ical . Physicists have proposed several conditions In 1928, British physicist Paul Dirac wrote down an that the Majorana fermions might exist, and still work equation that combined quantum theory and special rel- on it. ativity to describe the behaviour of an electron moving at a relativistic speed. But the equation posed a problem: just as the equation x2 = 4 can have two possible solu- A. oscillation tions (x = 2 or x = −2), so Dirac’s equation could have two solutions, one for an electron with positive energy, A neutrino, is an electrically neutral, weakly interact- and one for an electron with negative energy. But classi- ing elementary subatomic particle with half-integer spin. cal (and common sense) dictated that the energy There are three types, or ”flavors”, of : electron of a particle must always be a positive number. Dirac neutrinos, muon neutrinos and tau neutrinos. In a stan- interpreted the equation to mean that for every particle dard model, it is believed that neutrinos is created with a there exists a corresponding , exactly match- well-defined flavour, and the transition between fermions ing the particle but with opposite charge. For the elec- in different flavour are highly suppressed. For example, tron there should be an ”antielectron” identical in every W − → e− + ν way but with a positive . τ connects the four components of a field ψ. − − W → e + νe µ (iγ ∂µ − m)ψ = 0 the former one is highly suppressed, while the later one is what we measure in usual case. However, in a phe- where γ are 4 × 4 matrices and obey the rules of Clif- µ nomenon known as neutrino flavor oscillation, neutrinos ford algebra. These conditions ensure that the equation are able to oscillate among the three available flavors properly describes the spin-1/2 particle with mass m. It while they propagate through space. This is due to the turns out that γ is a set of matrices, whose entries con- µ mismatch between the neutrino mass eigenstates and the tain both real and imaginary number. Therefore, it is neutrino flavour eigenstates. This allows for a neutrino quite straightforward that ψ must be a complex field, that was produced as an electron neutrino at a given lo- which is now a label of charged particles. cation to have a calculable probability to be detected as Based on the Dirac equation, for a given ψ, we found either a muon or tau neutrino after it has traveled to that ψ ≡ Cψ∗ transforms in the same as ψ, where C = c another location. This quantum mechanical effect was γ γ . It should be intuitively that if ψ describe a particle, 2 0 first hinted by the discrepancy between the number of then ψ describes the same particle with the opposite c electron neutrinos detected from the Sun’s core failing charge. As a result, a particle would be distinct to its to match the expected numbers, dubbed as the ”solar antiparticle if it is charged, i.e. complex field. neutrino problem”. In the the existence Majorana inquired whether it might be possible for a of flavor oscillations implies nonzero differences between spin- particle to be its own antiparticle, by attempting to the neutrino masses, because the amount of mixing be- find the equation that such an object would satisfy. To tween neutrino flavors at a given time depends on the get an equation of Diracs type (that is, suitable for 1 ) 2 differences between their squared masses. but capable of governing a real field, requires matrices that satisfy the Clifford algebra and are purely imaginary. By that kind of , Majorana then get the B. superconductivity Majorana equation, which is similar to Dirac equation, to describe the spin- 1 Majorana fermions. 2 Superconductivity provides a nice illustration of the (iγ˜µ∂ − m)ψ˜ = 0 Abelian Higgs model. In the superconductor, the µ Landau-Ginzburg free energy: where γ˜µ means different set of gamma matrices(purely 1 £ = (∇ × A)2 + |(∇ − ieA)φ|2 + m2|φ|2 + λ|φ|4 imaginary) from Dirac’s. Although been proposed over 2 2

2 where m = a(T − Tc), Tc is the critical temperature, lines (or strands) in (2 +1)-dimensional space-time orig- and the field qunta φˆ are electron pairs, which, of course, inating at initial positions and terminating at final posi- are bosons. At low temperatures, these fall into the same tions, as shown in Fig 1. quantum state(Bose-Eistein condensation). Moreover, if T > T , then m2 > 0 and hφi = 0. But when T < T , c c FIG. 1. Top: The two elementary braid operations σ and σ 2 m2 1 2 m > 0 and hφi = − 2λ > 0, which is an example of on three particles. Middle: Shown here σ2σ1 6= σ1σ2; hence spontaneous symmetry breaking. The electron pair, cre- the braid group is non-Abelian. Bottom: The braid relation ated by φˆ is called the cooper pair and it is analogous to σiσi+1σi = σi+1σiσi+1. the Higgs field in the Higgs mechanism. It turns out that below the critical temperature Tc, the cooper pair would condensate in the vacuum, and the electron number in the vacuum is no longer observable. We can therefore treat the vacuum in the superconductor as a sour or a sink of electrons.

C. Non-abelian anyon

In three spatial dimensions and one time dimension (3 + 1)D there are only two possible symmetriesthe wave function of bosons is symmetric under exchange while that of fermions is antisymmetric. The limitation to one Majorana fermions are also intriguing because they are of both types of quantum symmetry originates from the examples of what are called non-Abelian anyons whose observation that a process in which two particles are adi- quantum state can change simply by exchanging par- abatically interchanged twice is equivalent to a process in ticles, unlike standard bosons and fermions, whose ex- which one of the particles is adiabatically taken around change does not have measurable consequences. Once the other. However, it is said that two-dimensional sys- they can be controlled and manipulated, non-Abelian tems are qualitatively different from those in three (and anyons are expected to find application in topological higher dimensions) in this respect. This is because in quantum computing, a radically different computer de- two dimensions a closed loop executed by a particle sign that uses the exchange of non-Abelian anyons to around another particle is topologically distinct from a perform certain computational tasks. loop which encloses no particles. Therefore, it leads to a difference in the possible quantum mechanical proper- ties for quantum systems when particles are confined to II. HAUNTING THE MAJORANA FERMIONS (2+1)D. Suppose that we have two identical particles in two A. Neutrino dimensions. Then, when one particle is exchanged in a counterclockwise manner with the other, the wave func- tion can change by an arbitrary phase, Because the Majorana fermions must be uncharged, all of the Standard Model fermions except the neutrino iθ ψ(~r1, ~r2) → e ψ(~r1, ~r2) behave as Dirac fermions. Although the nature of the neutrino is not settled, many physicists, including Majo- The phase need not be merely a ± sign because a sec- rana, believe that the neutrino might be the Majorana ond counterclockwise exchange need not lead back to the fermions. Fortunately, in the view of the experimental initial state but can result in a nontrivial phase: data, there is an obviously difference between the neu- trino and the antineutrino. The antineutrinos observed i2θ ψ(~r1, ~r2) → e ψ(~r1, ~r2) so far all have right-handed helicity (i.e. only one of the two possible spin states has ever been seen), while the The special cases θ = 0, π correspond to bosons and neutrinos are left-handed. The distinction is connected fermions, respectively. Particles with other values of the with the law of -number conservation. It is defined statistical angle θ are called anyons. We now consider the that e, µ, τ, νe, νµ, ντ each have lepton number 1. And general case of N particles, where a more complex struc- each anti-lepton has lepton number -1. In this sense, ν ture arises. The topological classes of trajectories which is completely different from ν. take these particles from initial positions R1,R2, ..., RN However, in recent years, the situation has come to 0 0 0 at time ti to final positions R1,R2, ..., RN at time tf are seem less tenable, for it has been discovered that neu- in one-to-one correspondence with the elements of the trinos oscillate in flavour. As a result, the former sepa- braid group BN . An element of the braid group can be rate lepton-number conservation law has to become to- visualized by thinking of trajectories of particles as world tal lepton-number conservation law(Le + Lµ + Lτ = 3 constant). More important, the phenomenon of neutri- the so-called Majorana mode (a particle and its antipar- nos oscillation implies that the neutrinos are massive. ticle have opposite energy , so  = 0 is the only possibil- The difference between the neutrinos and antineutrino is ity). The problem is where and how to find a Majorana due to their spin orientation. If we can flip the orienta- mode? They are predicted to occur for s-wave Cooper tion of the neutrinos then they might be the same par- pairing if electrons in normal state obey Dirac-like equa- ticle, which satisfy the condition of Majorana fermions. tion. How do we flip the orientation of a particle? The key Mourik et al. build on a series of theoretical propos- point is just slow it down and this is available only when als, which showed that Majorana fermions can be engi- that particle is massive. neered in nanostructures that combine a superconduc- In addition to this direct test of Majorana’s hypothe- tor and other materials. In this context, an antiparti- sis, some physicists rather choose an indirect but practi- cle is in fact a hole, an excitation that consists of re- cal way, namely, the violation of the total lepton-number moving an electron from the device. In superconductors conservation law, Le + Lµ + Lτ = constant. If the vi- the electrons form bosonic Cooper pairs, which then con- olation is real, it might indicates that the neutrino can dense into a single quantum state. Superconductors are be annihilated by neutrino. The most famous case is the a natural environment for particles that are their own an- neutrino-less . By theoretical analysis, tiparticle because the Cooper pair condensate blurs the demonstrated that phenomenon of the difference between electron-like and hole-like excitations. neutrino-less double beta decay would not contradict to Indeed, the theory of superconductivity treats electron- the beta decay only if the neutrino is its own anti-particle, like and hole-like excitations on an equal footing and has i.e. a Majorana particle. Neutrino-less double beta decay all excitations appear as a pair at opposite energies, ±. is just the traditional double beta decay without the an- Particle- hole symmetric (i.e., Majorana) states can oc- tineutrinos. If neutrino-less double beta decay is real, in cur at  = 0 only. Once there is a single excitation with that case, two antineutrinos would become virtual parti- energy  = 0, its existence is said to be topologically pro- cles and annihilate to each other which implies that the tected, because no continuous perturbation can drive it neutrino is the antiparticle of itself. Hence, neutrino-less away from its position at  = 0. In the experiment of double beta decay becomes one of the most active topic Mourik et al., an InSb wire, a semiconductor with strong in current time. spin-orbit coupling, is coated with the superconductor NbTiN and placed in a large magnetic field parallel to its axis. The number of electrons in the wire is tuned via B. Superconductor capacitive coupling to metal gates. At certain electron densities, the combination of the superconducting coat- In superconductors, the absolute distinction between ing, the strong spin-orbit coupling in the InSb wire, and electrons and holes is blurred. This is due to the fact that the magnetic field drive the InSb wire into an unconven- we can treat the cooper pair condensation as a source or tional superconducting state that theorists predict has a sink of electron without contradiction of the electron Majorana bound states at its ends. number conservation(because electron number in the vac- uum is unobservable). Base on this idea, the charged- particle problem in finding the Majorana fermions no FIG. 2. The red stars indi- FIG. 3. Energy, E, versus longer bothers us. By BCS theory, it realized that certain cate the expected locations momentum, k fermionic mode in the superconducting state are created of a Majorana pair by the mixture of the electron normal state and hole nor- † mal state operator, in the form aϕj + bϕk. As a special case that γ + iγ0 ϕj = √ 2 γ − iγ0 ϕ† = √ j 2 the corresponding create operators Measurement of the density of states at the wires end reveal a peak at zero energy that persists over a range ϕ + ϕj† γ = j √ of electron densities and magnetic fields, consistent with 2 the idea of topological protection. No peak was present if † the experiment was repeated without any one of the two ϕj − ϕj γ0 = √ crucial ingredients in the theoretical proposals (super- 2 conductivity, magnetic field perpendicular to spin-orbit would be associated with the so-called partihole. Note- field). Ruling out other known causes for zero-energy worthy, because the is its own antipar- states, they interpret their observation as a signature of ticle, the Majorana bound state always has zero energy, a Majorana bound state(Majorana fermion). 4

Figure 2 shows the tendency Energy, E, versus momen- bations can cause a quantum particle to decohere and tum, k, for a 1D wire with Rashba spin-orbit interaction, introduce errors in the computation, but such small per- which shifts the spin-down band (blue) to the left and the turbations do not change the braids’ topological proper- spin-up band (red) to the right. Blue and red parabolas ties. This is like the effort required to cut a string and are for B = 0; black curves are for B 6= 0, illustrating the reattach the ends to form a different braid, as opposed to formation of a gap near k = 0 of size Ez (µ is the Fermi a ball (representing an ordinary quantum particle in four- energy with µ = 0 defined at the crossing of parabolas at dimensional spacetime) bumping into a wall. And exper- k = 0). iments in fractional quantum Hall systems indicate these elements may be created using semiconductors made of gallium arsenide at a temperature of near absolute zero and subjected to strong magnetic fields. III. APPLICATION In the real world, anyons form from the excitations in an electron gas in a very strong magnetic field, and carry fractional units of magnetic flux in a particle-like man- A. Topological quantum computation ner. This phenomenon is called the fractional quantum Hall effect. The electron ”gas” is sandwiched between A topological quantum computer is a theoretical quan- two flat plates of aluminium gallium arsenide, which cre- tum computer that employs two-dimensional quasiparti- ate the two-dimensional space required for anyons, and cles called anyons, whose world lines cross over one an- is cooled and subjected to intense transverse magnetic other to form braids in a three-dimensional spacetime fields. When anyons are braided, the transformation of (i.e., one temporal plus two spatial dimensions). These the quantum state of the system depends only on the braids form the logic gates that make up the computer. topological class of the anyons’ trajectories (which are The advantage of a quantum computer based on quan- classified according to the braid group). Therefore, the tum braids over using trapped quantum particles is that quantum information which is stored in the state of the the former is much more stable. The smallest pertur- system is impervious to small errors in the trajectories.

[1] Frank Wilczek, Nature Physics 5, 614 - 618 (2009) P. A. M. Bakkers, L. P. Kouwenhoven, Science 336, 1003 [2] Chetan Nayak, Steven H. Simon, Ady Stern, Michael (2012) Freedman, Sankar Das Sarma, RevModPhys.80.1083 [3] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E.