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Majorana Fermions As Emergent Quasiparticles

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Majorana Fermions As Emergent Quasiparticles Majorana fermions as emergent quasiparticles Abhisek Sahu1 1Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., V6T 1Z1, Canada (Dated: December 19, 2020) Majorana fermions are a special type of fermion predicted by the Dirac’s equation that are their own antiparticles. Currently they have rapidly gained interest in condensed matter physics as emergent quasiparticles in certain systems like topological superconductors. In this article, we review the theory of Majorana fermions starting from the Dirac equation. Then we discuss using Bogoliubov-deGennes formalism how Superconductors form ideal hunting grounds for Majorana particles and introduce the notion of Majorana zero modes. Finally we discuss the Kitaev model- a paradigmatic model to look for unpaired Majorana zero modes. I. INTRODUCTION II. WHAT ARE MAJORANA FERMIONS? In this section, we briefly review how Majorana In 1928 developed the wave equation that describes fermions come about from Dirac’s equations. The relativistic spin 1=2 particles. The solutions of this Dirac equation for a free particle 1is equation are complex valued four component spinors µ which can be interpreted as a spin 1=2 particle- (iγ @µ − m)Ψ(x) = 0 (1) antiparticle pair. It was Etorre Majorana’s insight to T look for a completely real set of solutions for the Dirac Where Ψ(x) = ( 1; 2; 3; 4) is a four-component µ equation in order to create a symmetric theory of par- spinor field and γ are 4 × 4 matrices satisfying the ticles and anti-particles. As a result in the year 1937 following algebra: he introduced the notion of fermions which are their µ ν µν y own anti-particles; known today as Majorana fermions fγ ; γ g = 2η ; γ0γµγ0 = γµ: (2) [1]. As Majorana fermions are their anti-particles, they the choice of the γ matrices is not unique. Any set must be chargeless. of matrices satisfying (2) can be chosen to solve the Dirac equation and the solutions Ψ(x) for a particular While many elementary particles are well-described choice of γµ is related to solutions for other choices of as Dirac fermions, but so far there seem to be no ex- γ through a unitary transformation Majorana himself amples of those that could be thought of as Majo- used a basis known today as the Majorana basis (see rana fermions. Although there are strong theoretical [2]), where all the γ matrices are purely imaginary and reasons to believe that neutrinos could be Majorana hence in this basis, the complex conjugate of a solution fermions[2] (they seem to have a small mass and are Ψ(x) is a also a solution. Another convenient set that chargeless), convincing experimental evidence is yet to is often used is the Weyl basis: be found. 0 σµ γµ = (3) On the other hand, there is considerable interest in σ^µ 0 the field of condensed matter physics about emergent quasi-particles that behave as Majorana fermions[3]. where we have denoted σµ = (I; −σi) and σ^µ = (I; σi). It so happens that excitations in superconducting sys- σi are the usual Pauli matrices. In the following dis- tems carry signatures of Majoranas. What is more in- cussions we shall use the Weyl basis everywhere. teresting is that certain zero energy excitations called Let us now consider the stationary solutions for the Majorana zero modes(MZM) often have the additional Dirac equation with energy E(which can be both pos- feature of being topologically protected; which means itive and negative), which are nothing but Ψ(x) = −iEt that any continuous deformation of the Hamiltonian e ΦE(x): Here, ΦE(x) satisfies the Dirac equation does not destroy the state. Because of the topologi- (1) with i@0 replaced with E everywhere. The station- cal properties and interesting exchange statistics that ary states provide a complete basis, and any general so- MZMs follow, they have also been thought of as candi- lution Ψ(x) can be expanded in terms of it. Moreover, dates for storing quantum information[4]. Thus obser- they help us see an important internal symmetry of vation of Majorana quasi-particles in solid-state sys- the Dirac equation called the Charge conjugation sym- tems is of great interest. metry. It happens that if Φ(x) is a state associated with energy E, we can find a corresponding charge- The aim of this article is to review the emergence of conjugated state defined as Majorana particles in superconducting systems. We begin by briefly reviewing solutions of Dirac equa- Φc(x) = CΦ∗(x) (4) tions and how Majorana fermions can be obtained from them. Subsequently we shall find physical motivation for systems that might show emergent Majoranas and then provide theoretical justification for the same. Fi- 1 Through out the article we use Einstein summation conven- nally, we discuss the Kitaev model which shows how tion; the same raised and lowered index in an expression such unpaired MZMs can arise in a one dimensional system. as µ here, are considered to be summed over. 2 with energy −E. In the above equation, C is a 4 × 4 are the same as their anti-particle. We call them Majo- matrix called the charge conjugation matrix, and in rana fermions. For more information about properties the Weyl basis C = iγ2. of Majorana fermions the reader may refer to [2]3. A standard approach to obtain particle interpreta- tion from Dirac equation is to quantise (1) by elevating Ψ(x) to the status of an operator field and imposing III. EMERGENCE OF MAJORANA the following anti-commutation relations: FERMIONS IN SOLIDS ^ ^ y 0 4 0 ^ ^ 0 A. Ideal hunting grounds fΨa(x); Ψb(x )g = δabδ (x − x ); fΨa(x); Ψb(x )g = 0 (5) As we saw in our course, condensed matter physics Then we can expand a general field operator Ψ(^ x) in has plenty of examples where the system as a whole terms of the stationary state solutions as behaves very differently compared to the constituent particles. The idea that ’more is different’ has been ^ X −iEt X y −iEt beautifully expressed in the article [6] Under certain Ψ(x) = aEe ΦE(x)+ b−Ee ΦE(x) (6) E>0 E<0 approximations, the collective excitations of a given condensed matter system is best described in terms Note that we have separated the positive and negative of quasiparticles - particles that do not exist at the energy part of the expansions. Here aE and bE are op- microscopic level but seem to emerge from the micro- erators that play the role of arbitrary coefficients in the scopic description of the system and explain the phys- expansion. Using the anti-commutation relations (5) ical observations obtained in experiments. Examples we can show that aE and bE also follow the canonical of such particles include phonons, polarons, magnons y y anti-commutation rules. Thus aE and bE are inter- and plasmons, which we are familiar with. In solid- preted as the creation operators for a particle and an state physics the most important fermionic particles- antiparticle with energy E, as they correspond to the electrons, are Dirac fermions. However it turns out positive energy and negative energy parts of the expan- Majorana fermions can occur in certain solids as emer- sion respectively. By reversing the sign of the dummy gent quasiparticles. summation variable in the second term and using the To see how that comes about we recall that in the charge conjugation property (4), we may write (6) as second quantisation formalism, electrons are repre- a sum over positive energy states only: sented by a set of creation and annihilation operators y where cj creates an electron with quantum numbers X h y i ^ −iEt iEt ∗ denoted by index j while cj annihilates it. The index Ψ(x) = aEe ΦE(x) + bEe CΦE(x) : (7) E>0 j includes the quantum degrees of freedom appropriate for the set up we are describing, typically spin, position Equation (7) predicts the most general particle- or crystal momentum etc. These operators satisfy the antiparticle pair of Dirac fermions. The particle is dis- canonical commutation relations. Without any loss of tinguishable from the anti-particle as it has an opposite generality, we can perform a canonical transformation charge2. We find an interesting special case when we of any other operator of interest to the Majorana Basis impose the so-called Majorana condition on Ψ(^ x): defined as: 1 y 1 ^ c ^ ∗ ^ c = (γ + iγ ) c = (γ − iγ ) (9) Ψ (x) = CΨ (x) = Ψ(x): (8) j 2 j1 j2 j 2 j1 j2 fγ ; γ g = 2δ δ ; δy = δ (10) The significance of this constraint can be more clearly iα jβ ij αβ iα iα seen in the Majorana basis in which C turns out to be The hermiticity condition of γiα implies that the par- just the identity matrix. Hence in this basis the Majo- ticle created the gamma operator is the same as its ^ rana condition simply says that Ψ(x) is real. Moreover, anti-particle, and hence corresponds to a Majorana it can be shown that a solution satisfying the Majo- fermion. While mathematically equivalent, the above ^ rana constraint always exists. Consider any Ψ(x) that description in terms of Majorana fermion operators solves the Dirac equation. It can be shown that Ψ^ c(x) does not give any benefit in understanding the physics is also a solution and since the Diracp equation is linear of the system. The primary reason for this is that in ^ ^ ^ c the superposition: ΨMaj(x) = 1 2(Ψ(x)+Ψ (x)) also most cases the two Majoranas corresponding to a single is a solution.
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