Majorana fermions Chia-Nan Yeh

Majorana fermions

Chia-Nan Yeh National Taiwan University

(Dated: January 19, 2015)

In this report, we discuss the Majorana fermions, theoretical particles 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

In 1928, British physicist Paul Dirac wrote down an equation that combined quantum theory and special rel- 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- tions (x = 2 or x = −2), so Dirac’s equation could have two solutions, one for an electron with positive energy, and one for an electron with negative energy. But classi- cal physics (and common sense) dictated that the energy of a particle must always be a positive number. Dirac interpreted the equation to mean that for every particle there exists a corresponding antiparticle, exactly match- ing the particle but with opposite charge. For the elec- tron there should be an ”antielectron” identical in every way but with a positive electric charge. Dirac equation connects the four components of a field ψ.

(iγ^µ∂_µ− m)ψ = 0

where γµ are 4 × 4 matrices and obey the rules of Clif- ford algebra. These conditions ensure that the equation properly describes the spin-1/2 particle with mass m. It turns out that γµ is a set of matrices, whose entries con- tain both real and imaginary number. Therefore, it is quite straightforward that ψ must be a complex field, which is now a label of charged particles.

Based on the Dirac equation, for a given ψ, we found that ψc ≡ Cψ^∗ transforms in the same as ψ, where C = γ2γ0. It should be intuitively that if ψ describe a particle, then ψc describes the same particle with the opposite charge. As a result, a particle would be distinct to its antiparticle if it is charged, i.e. complex field.

Majorana inquired whether it might be possible for a spin- particle to be its own antiparticle, by attempting to find the equation that such an object would satisfy. To get an equation of Diracs type (that is, suitable for ¹₂) but capable of governing a real field, requires matrices that satisfy the Clifford algebra and are purely imaginary.

By that kind of gamma matrices, Majorana then get the Majorana equation, which is similar to Dirac equation, to describe the spin-¹₂ Majorana fermions.

(i ˜γ^µ∂_µ− m) ˜ψ = 0

where ˜γ^µ means different set of gamma matrices(purely imaginary) from Dirac’s. Although been proposed over

almost 80 years, the Majorana fermion is still a hypothet- ical particle. Physicists have proposed several conditions that the Majorana fermions might exist, and still work on it.

A. Neutrino oscillation

A neutrino, is an electrically neutral, weakly interact- ing elementary subatomic particle with half-integer spin.

There are three types, or ”flavors”, of neutrinos: electron neutrinos, muon neutrinos and tau neutrinos. In a stan- dard model, it is believed that neutrinos is created with a well-defined flavour, and the transition between fermions in different flavour are highly suppressed. For example,

W⁻→ e⁻+ ντ

W⁻→ e⁻+ νe

the former one is highly suppressed, while the later one is what we measure in usual case. However, in a phe- nomenon known as neutrino flavor oscillation, neutrinos are able to oscillate among the three available flavors while they propagate through space. This is due to the mismatch between the neutrino mass eigenstates and the neutrino flavour eigenstates. This allows for a neutrino that was produced as an electron neutrino at a given lo- cation to have a calculable probability to be detected as either a muon or tau neutrino after it has traveled to another location. This quantum mechanical effect was first hinted by the discrepancy between the number of electron neutrinos detected from the Sun’s core failing to match the expected numbers, dubbed as the ”solar neutrino problem”. In the Standard Model the existence of flavor oscillations implies nonzero differences between the neutrino masses, because the amount of mixing be- tween neutrino flavors at a given time depends on the differences between their squared masses.

B. superconductivity

Superconductivity provides a nice illustration of the Abelian Higgs model. In the superconductor, the Landau-Ginzburg free energy:

£ = 1

2(∇ × A)²+ |(∇ − ieA)φ|²+ m²|φ|²+ λ|φ|⁴

2 where m² = a(T − Tc), Tc is the critical temperature,

and the field qunta ˆφ are electron pairs, which, of course, are bosons. At low temperatures, these fall into the same quantum state(Bose-Eistein condensation). Moreover, if T > Tc, then m² > 0 and hφi = 0. But when T < Tc, m² > 0 and hφi = −^m_2λ² > 0, which is an example of spontaneous symmetry breaking. The electron pair, cre- ated by ˆφ is called the cooper pair and it is analogous to 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 of both types of quantum symmetry originates from the observation that a process in which two particles are adi- abatically interchanged twice is equivalent to a process in which one of the particles is adiabatically taken around the other. However, it is said that two-dimensional sys- tems are qualitatively different from those in three (and higher dimensions) in this respect. This is because in two dimensions a closed loop executed by a particle around another particle is topologically distinct from a 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 (2+1)D.

Suppose that we have two identical particles in two dimensions. Then, when one particle is exchanged in a counterclockwise manner with the other, the wave func- tion can change by an arbitrary phase,

ψ( ~r1, ~r2) → e^iθψ( ~r1, ~r2)

The phase need not be merely a ± sign because a sec- ond counterclockwise exchange need not lead back to the initial state but can result in a nontrivial phase:

ψ( ~r1, ~r2) → e^i2θψ( ~r1, ~r2)

The special cases θ = 0, π correspond to bosons and fermions, respectively. Particles with other values of the statistical angle θ are called anyons. We now consider the general case of N particles, where a more complex struc- ture arises. The topological classes of trajectories which take these particles from initial positions R₁, R₂, ..., R_N at time t_ito final positions R⁰₁, R₂⁰, ..., R⁰_N at time t_f are in one-to-one correspondence with the elements of the braid group BN. An element of the braid group can be visualized by thinking of trajectories of particles as world

lines (or strands) in (2 +1)-dimensional space-time orig- inating at initial positions and terminating at final posi- tions, as shown in Fig 1.

FIG. 1. Top: The two elementary braid operations σ1 and σ2

on three particles. Middle: Shown here σ2σ1 6= σ1σ2; hence the braid group is non-Abelian. Bottom: The braid relation σiσi+1σi= σi+1σiσi+1.

Majorana fermions are also intriguing because they are examples of what are called non-Abelian anyons whose quantum state can change simply by exchanging par- ticles, unlike standard bosons and fermions, whose ex- change does not have measurable consequences. Once they can be controlled and manipulated, non-Abelian anyons are expected to find application in topological quantum computing, a radically different computer de- sign that uses the exchange of non-Abelian anyons to perform certain computational tasks.

II. HAUNTING THE MAJORANA FERMIONS

A. Neutrino

Because the Majorana fermions must be uncharged, all of the Standard Model fermions except the neutrino behave as Dirac fermions. Although the nature of the neutrino is not settled, many physicists, including Majo- rana, believe that the neutrino might be the Majorana fermions. Fortunately, in the view of the experimental data, there is an obviously difference between the neu- trino and the antineutrino. The antineutrinos observed so far all have right-handed helicity (i.e. only one of the two possible spin states has ever been seen), while the neutrinos are left-handed. The distinction is connected with the law of lepton-number conservation. It is defined that e, µ, τ, νe, νµ, ντ each have lepton number 1. And each anti-lepton has lepton number -1. In this sense, ν is completely different from ν.

However, in recent years, the situation has come to seem less tenable, for it has been discovered that neu- trinos oscillate in flavour. As a result, the former sepa- rate lepton-number conservation law has to become to- tal lepton-number conservation law(Le + Lµ + Lτ =

3 constant). More important, the phenomenon of neutri-

nos oscillation implies that the neutrinos are massive.

The difference between the neutrinos and antineutrino is due to their spin orientation. If we can flip the orienta- tion of the neutrinos then they might be the same par- ticle, which satisfy the condition of Majorana fermions.

How do we flip the orientation of a particle? The key point is just slow it down and this is available only when that particle is massive.

In addition to this direct test of Majorana’s hypothe- sis, some physicists rather choose an indirect but practi- cal way, namely, the violation of the total lepton-number conservation law, Le+ Lµ + Lτ = constant. If the vi- olation is real, it might indicates that the neutrino can be annihilated by neutrino. The most famous case is the neutrino-less double beta decay. By theoretical analysis, Ettore Majorana demonstrated that phenomenon of the neutrino-less double beta decay would not contradict to the beta decay only if the neutrino is its own anti-particle, i.e. a Majorana particle. Neutrino-less double beta decay is just the traditional double beta decay without the an- tineutrinos. If neutrino-less double beta decay is real, in that case, two antineutrinos would become virtual parti- cles and annihilate to each other which implies that the neutrino is the antiparticle of itself. Hence, neutrino-less double beta decay becomes one of the most active topic in current time.

B. Superconductor

In superconductors, the absolute distinction between electrons and holes is blurred. This is due to the fact that we can treat the cooper pair condensation as a source or a sink of electron without contradiction of the electron 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 longer bothers us. By BCS theory, it realized that certain fermionic mode in the superconducting state are created by the mixture of the electron normal state and hole nor- mal state operator, in the form aϕj+ bϕ^†_k. As a special case that

ϕj =γ + iγ⁰

√ 2 ϕ^†_j =γ − iγ⁰

√2 the corresponding create operators

γ = ϕj+ ϕj^†

√2

γ⁰=ϕj− ϕj^†

√ 2

would be associated with the so-called partihole. Note- worthy, because the Majorana fermion is its own antipar- ticle, the Majorana bound state always has zero energy,

the so-called Majorana mode (a particle and its antipar- ticle have opposite energy , so  = 0 is the only possibil- ity). The problem is where and how to find a Majorana mode? They are predicted to occur for s-wave Cooper pairing if electrons in normal state obey Dirac-like equa- tion.

Mourik et al. build on a series of theoretical propos- als, which showed that Majorana fermions can be engi- neered in nanostructures that combine a superconduc- tor and other materials. In this context, an antiparti- cle is in fact a hole, an excitation that consists of re- moving an electron from the device. In superconductors the electrons form bosonic Cooper pairs, which then con- dense into a single quantum state. Superconductors are a natural environment for particles that are their own an- tiparticle because the Cooper pair condensate blurs the difference between electron-like and hole-like excitations.

Indeed, the theory of superconductivity treats electron- like and hole-like excitations on an equal footing and has all excitations appear as a pair at opposite energies, ±.

Particle- hole symmetric (i.e., Majorana) states can oc- cur at  = 0 only. Once there is a single excitation with energy  = 0, its existence is said to be topologically pro- tected, because no continuous perturbation can drive it away from its position at  = 0. In the experiment of Mourik et al., an InSb wire, a semiconductor with strong 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 capacitive coupling to metal gates. At certain electron densities, the combination of the superconducting coat- ing, the strong spin-orbit coupling in the InSb wire, and the magnetic field drive the InSb wire into an unconven- tional superconducting state that theorists predict has Majorana bound states at its ends.

FIG. 2. The red stars indi- cate the expected locations of a Majorana pair

FIG. 3. Energy, E, versus momentum, k

Measurement of the density of states at the wires end reveal a peak at zero energy that persists over a range of electron densities and magnetic fields, consistent with the idea of topological protection. No peak was present if the experiment was repeated without any one of the two crucial ingredients in the theoretical proposals (super- conductivity, magnetic field perpendicular to spin-orbit field). Ruling out other known causes for zero-energy states, they interpret their observation as a signature of a Majorana bound state(Majorana fermion).

4 Figure 2 shows the tendency Energy, E, versus momen-

tum, k, for a 1D wire with Rashba spin-orbit interaction, which shifts the spin-down band (blue) to the left and the spin-up band (red) to the right. Blue and red parabolas are for B = 0; black curves are for B 6= 0, illustrating the formation of a gap near k = 0 of size Ez (µ is the Fermi energy with µ = 0 defined at the crossing of parabolas at k = 0).

III. APPLICATION

A. Topological quantum computation

A topological quantum computer is a theoretical quan- tum computer that employs two-dimensional quasiparti- cles called anyons, whose world lines cross over one an- other to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates that make up the computer.

The advantage of a quantum computer based on quan- tum braids over using trapped quantum particles is that the former is much more stable. The smallest pertur-

bations can cause a quantum particle to decohere and introduce errors in the computation, but such small per- turbations do not change the braids’ topological proper- ties. This is like the effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball (representing an ordinary quantum particle in four- dimensional spacetime) bumping into a wall. And exper- 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.

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- ner. This phenomenon is called the fractional quantum Hall effect. The electron ”gas” is sandwiched between two flat plates of aluminium gallium arsenide, which cre- ate the two-dimensional space required for anyons, and is cooled and subjected to intense transverse magnetic fields. When anyons are braided, the transformation of the quantum state of the system depends only on the topological class of the anyons’ trajectories (which are classified according to the braid group). Therefore, the quantum information which is stored in the state of the system is impervious to small errors in the trajectories.

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

P. A. M. Bakkers, L. P. Kouwenhoven, Science 336, 1003 (2012)