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From Majorana fermions to topological quantum computation in semiconductor/superconductor heterostructures

We present a pedagogical review of topological superconductivity and its consequences in spin-orbit coupled semiconductor/superconductor heterostructures. We start by reviewing the historical origins of the notions of Dirac and Majorana fermions in particle physics and discuss how lower dimensional versions of these emerge in one dimensional superconductors. Ultimately, we focus on Majorana zero-modes, which emerge at defects in the Majorana equation. We then review the definition of the topological invariant, and how it allows the prediction of such Majorana modes from the bulk bandstructure of realistic superconductors, which do not have Lorentz invariance. Finally, we end with a discussion of protocols for how to detect such Majorana modes and use them for topological quantum computation.


I. INTRODUCTION
Dirac is reported to have remarked in one of his talks that his equation was more intelligent than its author. But, as noted by Victor Weisskopf 1 , it must be added that it was Dirac himself who found most of the additional insights. The concept of Majorana fermion, which also followed from the Dirac equation and was developed by Ettore Majorana 2 , was a notable exception. In relativistic quantum mechanics, physics of spin-1/2 fermions is described by the Dirac equation 3,4 , Here, we have used the natural units ( = c = 1), and γ µ , with µ = 0, 1, 2, 3, are a set of matrices satisfying the algebra, where η µν is the Minkowski metric (η µν = 0, µ = ν; η 00 = 1; η ii = −1, i = 1, 2, 3), and Eq. 2 follows from the requirement that the energy eigenvalues satisfy the condition E 2 = p 2 +m 2 , while Eq. 3 ensures hermiticity of Dirac Hamiltonian H D that follows from re-casting of Eq. 1 in terms of Schrodinger equation In (3 + 1) dimension, the simplest representation of the Clifford algebra in Eqs. 2,3 can be found in terms of four 4 × 4 matrices γ µ , known as Dirac matrices. There are infinitely many unitarily equivalent representations of the Dirac matrices, each constituting a basis for solving the Dirac equation in Eq. 1.
In general the matrices have both real and imaginary elements, e.g., the so-called chiral or Weyl representation has, where each element itself is a 2 × 2 matrix, and the σ i are the Pauli matrices, Because of both real and imaginary elements in γ µ , Eq. (1) is a set of coupled differential equations with complex coefficients. Thus, the general solution ψ(x) of Eq. (1) where C is the charge conjugation matrix. The charge conjugation matrix is defined as a matrix that satisfies the constraint By considering the coupling to the vector potential, it is clear that the wave-function ψ (c) (x, t) represents a particle with the same mass and spin but opposite charge and magnetic moment to the particles described by the pair of positive energy solutions describing electrons. He suggested that these positive energy solutions would represent holes in the otherwise filled vacuum. The hole excitations were termed positrons, which are the antiparticle of the electron, whose existence was verified by Anderson 6 . The observation of the positron was one of the first examples of a successful prediction of a new fundamental particle, and led to Pauli's memorable comment, "Success seems to have been on the side of Dirac rather than of logic" 7 . With the advent of semiconductor physics, Dirac's argument became the standard description of hole doped semiconductors. Dirac's picture of a filled sea of negative energy electron states is an intrinsically many-particle picture, where, even in the absence of inter-particle interactions, a single particle can no longer be described by the solution of a single particle wave equation, e.g., the Dirac spinor ψ 5 . For example, Klein considered that the Dirac equation in a region of uniform electric field where the elec-trostatic potential drops by 2m, where m is the mass parameter in Eq. 1, would connect propagating negative kinetic energy states in one side to positive energy solutions on the other side. This is called Klein tunneling of the negative energy states and has been seen in condensed matter systems 8 . In Dirac's picture, this process is viewed as the generation of electron-positron pairs by the electric field. While this process conserves charge, it does not conserve number of particles because a pair of particles are created from vacuum. Thus, a more natural and indeed necessary 5 framework is to view the solution ψ(x, t) of Eq. 1 as a second quantized field operator that unifies the description of particles and antiparticles through the introduction of creation and annihilation operators, Here, s = ±1 are the projection of the spin on the z axis, E p = p 0 = p 2 + m 2 , u s (p), v s (p) are positive and negative energy spinor solutions of Eq. 1 evaluated in the momentum space, and b s (p)(c † s (p)) is the annihilation (creation) operator for the particle (antiparticle). The Dirac field ψ(x) annihilates a particle and creates its antiparticle and its hermitian conjugate ψ † (x) creates a particle and annihilates its antiparticle. Imposing anti-commutation relations on the Dirac fields ψ(x), ψ † (y) leads to similar anti-commutation relations among the particle and antiparticle creation and annihilation operators (b † s (p), b r (p), c † s (p), c r (p)). One can also now write the total Hamiltonian as H D = d 3 p (2π) 3 E p (b † s (p)b s (p) + c † s (p)c s (p)) plus an unimportant constant that can be thrown away. The Dirac equation ( 1) thus naturally predicts the existence of anti-fermion for each spin-1/2 fermion, e.g., positively charged positron for negatively charged electron as discovered by Anderson 6 .
In 1937, Ettore Majorana discovered a representation of the γ-matrices that satisfied the Clifford algebra (Eqs. (2,3)) but were purely imaginary. In this so-called Majorana basis C = 1, and the charge conjugation (CC) operation, which takes a particle to its antiparticle 9 , reduces to the complex conjugation operation. Majorana then noticed that since the Dirac equation Eq. 1 becomes completely real, we can require that the solutions ψ(x, t) are also completely real, i.e. ψ = ψ c = Cψ * = ψ * .
With this constraint the Dirac equation would represent particles that are identical to their own antiparticle. Majorana obsered that this constraint of particle being related to its own antiparticle can be ensured in any representation of the Dirac matrices if we replace the Dirac mass term by the so-called Majorana mass m in the Dirac equation Eq. 1 to construct the Majorana equation Because ψ (c) involves complex conjugation, the above equation is no longer invariant under the U (1) gauge transformation ψ → e iΛ ψ needed to couple the particle to electromagnetic field in a gauge invariant way. Therefore, ψ(x, t) must represent a neutral particle, which makes sense for a particle which is its own anti-particle. The physical equivalence of the Majorana particle and its antiparticle becomes clearer in the framework of quantum field theory. In analogy with Eq. 9, a solution to the Majorana equation Eq. 11 can be expanded in terms of plane wave solutions 10 , The field in Eq. 12 satisfies the Majorana self-conjugacy relation ψ = ψ (c) (charge neutral particle that is identical to its own antiparticle). More importantly, only one kind of creation operator b † s (p) operator appears in the field equation as opposed to separate and electron and positron operators as in Eq. 9. An alternative way to see that the Majorana equation Eq. 11 involves half the degrees of freedom (i.e. does not have a separate hole degree of freedom) is to use the Weyl representation of the γ matrices in Eq. 5. In this representation the γ matrices are block off-diagonal (see Eq. 5) and the charge conjugation matrix C is purely block diagonal so that the bottom two components of the Majorana equation Eq. 11 can be written in terms of a two-component spinor ω: The twin properties of charge neutrality and the identification of the particle with its own antiparticle are not particularly uncommon among bosons, with photons and π 0 -mesons being two examples. In fact, as is well known 4 , the Fourier expansion of the photon field A µ (x) also contains creation and annihilation operators of only one kind, a λ (p) and a † λ (p), which is similar to the Fourier expansion in Eq. 12, because photon is a boson that is identical to its own antiparticle. This is not counter-intuitive in bosons, which are considered as fields in the classical limits rather than particles. But these properties are quite special among fermions, which are considered as particles in the classical limit. The neutron, for example, is a charge-neutral fermion, but it has an anti-particle (anti-neutron) that is different from the neutron by the sign of its magnetic moment. Also, neutrinos produced in beta-decay are thought to be charge-neutral, which follows from the conservation of electric charge. They also have a small but non-zero rest mass, so cannot be Weyl fermions 11 which are massless.
But whether neutrinos can be Majorana fermions, so that they are the same as their own antiparticle, is still an unsettled question in particle physics.
Although the jury is still out on whether any fermionic particle known in high energy physics can be a Majorana fermion, in the last few years the concept of a fermionic particle being identical to its own antiparticle has entered the realm of condensed matter physics [12][13][14][15][16][17][18] .
In condensed matter systems, however, the "elementary" particles are necessarily electrons and protons, and so the term "particle" and "antiparticle" refer to the so-called quasiparticles and quasiholes corresponding to excitations of an underlying many-body state. As shown in a seminal paper by N. Read and D. Green 19 , the Bogoliubov-de Gennes (BdG) equations satisfied by the superconducting quasiparticles in the so-called weak-pairing phase of a two-dimensional (2D) spinless p x + ip y superconductor are very similar to the Majorana form of the Dirac equation of relativistic quantum mechanics. These superconducting quasiparticles, then, represent the condensed matter analog of the Majorana fermions of high-energy physics. Similar to the Majorana fermions in high energy physics, in general, the Majorana-like BdG quasiparticles of 2D spinless p x + ip y superconductors are characterized by a finite mass and finite energy. However, in the context of more recent developments with relevance to topological quantum computation (TQC), the term "Majorana fermion" has a slightly different meaning. The Bardeen-Cooper-Schrieffer (BCS) Hamiltonian of a superconductor has a so-called particle-hole (p-h) symmetry, according to which for every energy eigenvalue +E, there exists another energy eigenvalue −E. Additionally, if the hypothetical superconductor is made up of spinless electrons, the second-quantized operators corresponding to the excitations at ±E satisfy γ † E = γ −E . Consequently, if the Hamiltonian admits a non-degenerate zero-energy eigenvalue, the corresponding quasiparticle is identical to its own anti-quasiparticle, with the second quantized operators satisfying γ † 0 = γ 0 . The zero-energy BdG quasiparticle states, should they exist, are bound states localized by defects in the superconductor, e.g., vortices and sample edges 19 , where the superconducting order parameter vanishes. In this chapter, by the terms "Majorana zero mode (MZM)", or "Majorana bound state (MBS)" we will refer to such localized, charge-neutral, zero-energy bound states that may occur at defects and boundaries in appropriate superconductors.
The creation operator for such a zero energy state is a hermitian second quantized operator γ † = γ that anti-commutes with other fermion operators and satisfy the relation γ 2 = 1. A collection of such Majorana bound states satisfy the algebra: In two spatial dimensions, they obey a form of particle statistics known as non-Abelian statistics [19][20][21][22][23] . In non-Abelian statistics, pairwise exchanges of particle coordinates represent non-commutative operations, a fundamental property that can be used to implement  19 as an ideal platform to realize MZMs, provided non-degenerate zero-energy localized states can be realized in such a system.
In a superconductor, a non-degenerate localized zero-energy eigenstate, should it exist, enjoys a form of protection -called topological protection -that makes it immune to weak local perturbations, as long as such perturbations do not close the superconducting gap.
Such perturbations cannot move the state away from zero energy because of the particlehole symmetry and the non-degeneracy condition. Under particle-hole symmetry, if the Hamiltonian of a superconductor has an energy eigenvalue +E, it must also have an energy eigenvalue −E. If the zero energy solution is non-degenerate, it is then its own particle-hole pair. Since small perturbations to the BdG differential equation are not expected to change the total number of solutions, it follows that weak local perturbations (i.e., perturbations that do not couple pairs of MFs) leave the non-degenerate zero energy eigenvalue unperturbed, because perturbing it to a non-zero value (say + ) will necessitate the introduction of its particle-hole counterpart (− ). This argument also implies that the zero-energy solutions are characterized by vanishing expectation values for any local physical observable such as charge, mass, and spin. Otherwise, if there were a non-zero average of any of these observables in the zero energy wave function, a weak local field that couples to it (e.g., a weak magnetic field that couples to spin) would be able to shift the energy of the zero energy state. As zero energy MFs in solid state systems are topologically protected, they can be removed from zero energy only by tuning the system through a topological quantum phase transition (TQPT) 25 at which the bulk energy gap vanishes and the MFs become entangled with other gapless states at the topological quantum critical point. In the past decade, Majorana fermions have been discussed in a variety of low temperature systems [19][20][21][22][23] . Perhaps the most important of these -the semiconductor-superconductor (SM-SC) heterostructure -has attracted intense attention as a result of an abundance of exciting experimental and theoretical results that have appeared steadily in the literature. In what follows, we will first discuss the theoretical background necessary to understand the emergence of MFs in defects and boundaries in condensed matter systems, followed by a detailed discussion of the theory and experiments looking for MFs in SM-SC heterostructures.

II. THEORETICAL BACKGROUND
A. Jackiw-Rebbi solution of a zero energy bound state in one dimension Before we discuss zero energy bound states localized in defects of the order parameter in superconductors, it will be instructive to discuss how such bound state solutions emerge in one-dimensional (1D) Dirac Hamiltonian. The zero energy bound state solutions for the various systems discussed in this chapter can be qualitatively understood by appropriate mapping of the corresponding Hamiltonians on the 1D Dirac problem.
One of the earliest examples of zero energy bound state solutions in a condensed matter system was investigated in domain wall states of polyacetylene 56,57 . In that case, starting with an ansatz for the dimerization order-parameter profile of polyacetylene, it was also possible to demonstrate the existence of a zero energy bound state solution localized at a domain wall of the order parameter by explicitly solving the mean-field equations 58 . Remarkably, this domain wall zero energy bound state was shown to be a condensed matter realization of the zero mode associated with the mass solitons of a 1D Dirac problem investigated by Jackiw and Rebbi 59,60 . The Jackiw and Rebbi soliton solution is a simple example of an index theorem where fermionic zero modes can be used to count the topological defects of a background order parameter.
We begin with the Dirac Hamiltonian H D given in Eq. 4, H D = −iγ 0 γ · ∇ + γ 0 m. In one spatial dimension, we need only two Dirac matrices satisfying To satisfy this algebra with only a pair of matrices, the following 2 × 2 matrices will do the trick: We can compute the charge conjugation matrix to satisfy Eq. 8 to be C = σ x . The one dimensional Dirac Hamiltonian becomes, To discuss the Jakiew-Rebbi zero mode, we begin with the 1D second quantized Dirac Hamiltonian, The second term in Eq. (17) has an effective mass and we assume m(−x) = −m(x) is a spatially varying mass term that changes sign at x = 0 (the location of the domain wall).
We now assume that the quasiparticle operator satisfies the operator equation, Computing the commutator in Eq. 19 we find the following Dirac equation for the two- First we note that because σ y anticommutes with σ x and σ z , the first quantized Hamiltonian is also an eigenfunction of H 1 D with eigenvalue − . As a result, the = 0 solutions of Eq. (20) can be made a simultaneous eigenstate of H 1 D and σ x . Let φ 0 (x) denote such a solution and Setting = 0 and left-multiplying Eq. (20) by iσ z we obtain  It is important to mention here that despite the existence of a single topologically robust zero energy eigenstate localized at the mass domain wall in 1D Dirac theory, it does not represent a Majorana zero mode. In fact, the second quantized creation operator q † in Eq. 18 allows us to define the corresponding annihilation operator, It should be clear that the operators q, q † do not satisfy the Majorana condition q † = q, but, when properly normalized, satisfy the fermion anticommutation relation {q † , q} = 1.
The wave function φ 0 (x) in Eq. 22 and operators q † , q, thus, describe a zero-energy localized conventional fermionic mode. Such a conventional fermion mode can in fact be viewed as the bound state of a pair of Majorana zero modes with strongly overlapping wave functions.
To see this, we define the operators, It should be easy to check that γ + and γ − both individually satisfy the Majorana condition, and further, γ 2 ± = q † q + qq † = 1, and they mutually anticommute with each other {γ + , γ − } = 0. If we define a Fock state |0 with energy E = 0 defined by the condition q|0 = 0 (i.e., the zero energy state at mass domain wall is unoccupied), then the occupied state |1 can be defined as |1 = q † |0 , which is degenerate (E = 0) with |0 .
The conventional fermion occupation number operator n = q † q can be expressed in terms of the Majorana operators γ + , γ − as, n = q † q = 1 2 (1 + iγ + γ − ). Conversely, the operator iγ + γ − = 2q † q − 1, which takes the value 1 in the state |1 and −1 in the state |0 , is called the fermion parity operator. It is now clear that the conventional fermionic mode described by q † , q allows us to define a pair of Majorana zero modes, but the two MZMs γ + , γ − do not occur separately in space, describe the same localized wave function φ 0 (x) in Eq. 22, and should really be viewed as a basis transformation from the creation and annihilation operators q † , q to the Majorana operators γ + , γ − . The goal of the ongoing research on Majorana fermions in condensed matter systems is to create experimental conditions so that individual Majorana zero modes (e.g., γ + or γ − ) can occur spatially well-separated from each other. It is only in this limit they individually acquire topological protection.
The 2D spinless (p x + ip y ) superconductor (superfluid) is the canonical system that supports zero energy MZMs localized at the defects of the order parameter, such as vortices and sample edges 19 . In 2D the mean field Hamiltonian describing the quasiparticle excitations for such a system is given by, where ξ p = p − µ with p → p 2 2m * for small p, m * is the effective mass, and µ is the chemical potential. Here, the spin indices of the electron operators are omitted because the system is considered to be spinless (or spin-polarized). Read and Green 19 showed that for the Hamiltonian in Eq. 26, the long distance behavior of Cooper pair wave function g(r) undergoes a dramatic change as a function of µ. For µ < 0, g(r) ∼ e −r/r 0 , indicating that the pairs are tightly bound and the superconductor (superfluid) is in a so-called strong pairing phase. On the other hand, for µ > 0, g(r) ∼ 1 r , and the long tail in the Cooper pair wave function indicates that the system is in a so-called weak pairing phase, which is continuously connected to the BCS weak coupling superconductor. The phase transition at µ = 0, at which the excitation gap vanishes at the momentum space point k = 0, is not associated with any change in symmetry of the superconducting state but is topological in nature.
The weak and strong pairing phases of the Hamiltonian in Eq. 26 are distinguished by distinct integer values of a topological invariant that can be defined as follows: The Hamiltonian in Eq. 26 can be written in the Nambu basis as, where Ψ † (p) = (c † p , c −p ) and Ψ(p) is its hermitian congugate. Here, the 2 × 2 Hamiltonian matrix H p 2D can be cast in the form, where the three-component vector d(p) can be written as, d(p) = (Re(∆ p ), −Im(∆ p ), ξ p ), , and σ is the three-component vector of the Pauli matrices. The unit vector corresponding to d(p), namely,d(p) = d(p) |d(p)| , provides a mapping of the two-dimensional momentum space on the surface of a unit sphere defined by |d(p)| = 1 . As p moves over the 2D momentum space,d(p) sweeps out area over its unit sphere. Starting with |p| → ∞ as one covers the momentum space ending at |p| = 0, the number of times the unit vectord(p) wraps around the unit sphere is a topological invariant called Chern number (C).
Mathematically, the quantity C is given by, For |p| → ∞ in any direction, ξ p ∼ p 2 2m * dominates andd points along the north pole of the unit sphere. In the strong pairing phase of Eq. 26, µ < 0, and for |p| = 0,d To elucidate the topological difference between the weak and the strong pairing phases, Read and Green showed that 19 , in the weak pairing phase (µ > 0), the BdG equations parallel to the y-axis. Using the small-p approximation ξ p = p 2 2m * − µ −µ, that is, ignoring the term quadratic in p in comparison to the linear p term in the order parameter, it can be shown that 19 , for E = 0, the BdG equations for the spinor Ψ(r) = (u(r), v(r)) T near the boundary parallel to the y-axis can be written as, It can be easily checked that this equation, when written in terms of Ψ(r) and using Pauli The simplest vortex excitation in a 2D superconductor can be modelled as a point with vanishing ∆ 0 (vortex core) and the superconducting phase φ having a 2π phase winding around that point. This, in turn, can be modelled as a puncture (hole) in the superconductor, where ∆ 0 is automatically zero, and a 2π phase winding of the order parameter around the hole. Thus, for a vortex excitation in the weak pairing phase (µ > 0) of 2D spinless p x + ip y superconductor with flux quantum hc 2e , we may consider the edge of the vortex core as a circular ring of radius r = r 0 , separating a region (r > r 0 ) with µ > 0 from the region inside the vortex core (r < r 0 ) modelled as vacuum (µ < 0). Assuming azimuthal symmetry in the presence of a single vortex situated at the origin, and writing the superconducting order parameter ∆(r, θ) = |∆(r)|e iθ and µ(r) = µ 0 h(r), the BdG equations near the vortex core can be written in polar coordinates. It can then be shown that 19 , in the weak pairing phase and for a vortex with flux quantum hc 2e , the BdG equations near the vortex core admit a zero energy solution analogous to the Jackiw-Rebbi solution near a mass domain wall as in the case of the boundary or the sample edge. The zero energy solution near the vortex core exists in the orbital angular momentum channel l = 0. More generally, the low-energy BdG solutions describing chiral edge modes propagating along the circular edge at r = r 0 with energy E can be written as, where l is the orbital angular momentum and E = µ 0 l r 0 12 . If the vortex has an odd number of flux quantum hc 2e , the BdG wave function must be anti-periodic upon one full rotation around the vortex core, θ → θ + 2π. However, since the spinor on the right hand side of Eq. 31 is also anti-periodic upon θ → θ + 2π, it follows that l must be an integer (including zero).
Thus, a hc 2e vortex (or vortices with flux nhc 2e with n odd) admit a zero energy solution for l = 0. Conversely, if the flux inside the vortex core is an even multiple of hc 2e , the BdG wave function must be periodic upon θ → θ + 2π. This ensures that l must be a half-odd-integer and there will be no zero energy mode inside the vortex core even in the weak pairing phase of 2D p x + ip y superconductor. Analogous to the zero mode solution near the boundary, the In 1D spinless p-wave superconductor, in the absence of vortex excitations, the zero energy Majorana solutions occur near the boundaries or the ends of the wire. As argued by Kitaev 29 , these zero modes should be observable in a fractional AC Josephson effect-type experiment. Although we could discuss the 1D spinless p-wave superconductor just as the 1D version of Eq. 26, it will be instructive to discuss this system using the real space lattice model introduced by Kitaev 29 . The one-dimensional model of topological superconductivity proposed by Kitaev can be written as a tight binding Hamiltonian as follows, where t > 0, µ > 0, and ∆e iφ with ∆ > 0 are the nearest neighbor hopping amplitude, chemical potential, and superconducting order parameter, respectively, and c † j and c j are second quantized creation and annihilation operators on a 1D lattice with number of sites N .
To analyze the Hamiltonian in Eq. 32, let's first consider the special case µ = 0, t = ∆.
In this limit, the Hamiltonian can be written as, In analogy with Eq. 25, let's now introduce a pair of Majorana modes for each conventional fermionic mode described by c j , c † j , Despite the phase factors in Eq. 34, introduced to take into account the superconducting phase in Eq. 32, it should be easy to check that γ +,j and γ −,j both satisfy the Majorana condition γ † ±,j = γ ±,j . In terms of these operators, the Hamiltonian in Eq. 34 can be written as, is also an eigenstate of H K with the same energy E G . Thus, the ground state energy is two-fold degenerate and the pair of degenerate ground states differ in the total fermion number by one, i.e., they have opposite fermion parity. Introducing new conventional fermion operators d j = (γ −,j+1 +iγ +,j ) and corresponding d † j , we can rewrite the Hamiltonian H K as, Therefore, the system has a bulk gap to conventional fermion excitations and a pair of degenerate ground states which differ by the total fermion parity (i.e., they differ by one in the total fermion number). Note that one of the degenerate ground states, q † |G = (γ −,1 − iγ +,N )|G , is associated with fermion occupation of a non-local state composed out of the pair of dangling MZMs at the two ends. Now let's consider the Hamiltonian in Eq. 32 in another simple limit, µ < 0, t = ∆ = 0.
In this case, using Eq. 34, the Hamiltonian reduces to As shown in Fig More generally, applying periodic boundary conditions, and Fourier transforming the Hamiltonian in Eq. 32 into momentum space, the Bogoliubov-de Gennes Hamiltonian can be written as (for simplicity we take φ = 0 and the lattice constant a = 1), where k is the momentum and τ z , τ y are the Pauli matrices operating in the particle-hole space. The bulk band structure for the wire, found by diagonalizing Eq. 38, is given by, which shows a bulk band gap closure at k = {0, π} for µ = ±2t, representing the topological quantum phase transitions as described in the Kitaev model. The topological superconducting phase with a pair of MZMs at the end occurs for |µ| < 2t while the topologically trivial phase emerges for |µ| > 2t.
We will now describe how to formally characterize the topological phase of superconduc- In order to find a suitable quantity that changes sign at each gap closing we define the Pfaffian of the BdG Hamiltonian as explained below.
Any BdG Hamiltonian anti-commutes with a particle-hole symmetry operator, which, for This relation is analogous to the charge-conjugation constraint for Majorana fermions and guarantees that energy eigenvalues of the BdG Hamiltonian come in ( , − ) pairs. Thus, the low energy projected BdG Hamiltonian near such a zero-energy level crossing where the energy eigenvalue is can be written as which is particle-hole symmetric with the operator τ x K. Such a crossing of a pair of levels can be characterized using a Pfaffian of the BdG Hamiltonian 29 , which is written as and can be seen to change sign when crosses zero. For a two-level system, writing the energy eigenvalue as a Pfaffian appears as a coincidence. To see that this way of writing is more than a coincidence, let us recall that the Pfaffian of a 2n × 2n anti-symmetric matrix where permutations π are constrained so that i k < j k and i k < i k+1 .
This suggests the topological invariant for superconductors constructed by Kitaev 29 , This topological invariant also provides information about the two dimensional topological superconductors, which are usually described by a Chern number topological invariant. To compute this topological invariant for the Kitaev Hamiltonian H K (k) at k = 0, π as Substituting this into the topological invariant Eq. 45 becomes, which produces exactly the criterion |µ| > 2t to obtain topological superconductivity in the Kitaev model. conductor. For small k in Eq. 39, using the approximations sin k ∼ k and cos k ∼ 1 − k 2 2 , the Hamiltonian H K in Eq. 38 can be approximated as, where We note that m plays the role of chemical potentialμ. The equation of motion for the fermion operator is written as Let us now compare this to the Majorana equation in (1 + 1) dimension, which is the generalization of Eq. 11 to one spatial dimension and is written as, Here, the γ matrices are chosen to be the ones shown in Eq. 16 where the spinor Ψ T = (ψ, ψ * ) would be called a Nambu spinor. For stationary states, we can expand Ψ ∝ Ψ e −i t , this reduces to the conventional BdG equation where the charge conjugation constraint for the Majorana . Thus, the operator γ associated with the zero-energy state  into the eigenstate with eigenvalue ζ 12 for the fermion parity operator iγ 1 γ 2 . We can check that the state |Ψ is normalized by noting that where comparing the latter two forms The key observation to understand the transformation of Majorana modes is that the total fermion parity operator iγ 1 γ 2 γ 3 for the three Majorana operators involved is left invariant by any fermion parity measurement involving the Majorana bound states γ 1 , γ 2 or γ 3 . To see how this is true let us compute the expectation of the fermion parity iγ 1 γ 2 γ 3 , together with an operator O that is independent of γ 1 , γ 2 or γ 3 , with respect to |Ψ as where we have used the fact that [Π, (iγ 1 γ 2 )γ 3 O] = 0 and that Π 2 = Π as expected for a projection operator. Continuing the above chain of equations Comparing the first term of Eq. 59 and the last term of Eq. 60, we see that the operator iγ 1 γ 2 γ 3 is indeed preserved as promised. The next step is to observe that |Ψ is an eigenstate of the fermion parity iγ 2 γ 3 , while |Ψ is an eigenstate of iγ 1 γ 2 . Applying this to the invariance relation we proved we see that This shows that γ 1 is transferred into to ζ 23 ζ 12 γ 3 by the fermion parity measurement of γ 1 and γ 2 .
As an example of how to perform an exchange using such a transport process consider the exchange of Majorana modes γ 1,2 , which will require two auxilliary pairs γ 3,4 and γ 5,6 in an eiegnstate with eigenvalues ζ 34 and ζ 56 respectively. For the first step we measure iγ 2 γ 3 to obtain an eigenvalue ζ 23 . This transfers γ 2 → ζ 34 ζ 23 γ 4 . For the next step we measure iγ 1 γ 2 to obtain an eigenvalue ζ 12 . This transfers γ 1 → ζ 23 ζ 12 γ 3 . The result of this pair of measurements is to transfer the pair (1, 2) to the pair (3,4). The next steps repeat this by sending (3,4) to (5,6) followed by sending the pair (5, 6) back to (2,1). The final result of this is the braiding relation where ζ = ζ 34 ζ 23 ζ 56 ζ 45 ζ 21 ζ 62 . Noting that the parity operator (iγ 1 γ 2 ) 2 = 1, we can write the above exchange matrix as a unitary operator Using the relations {γ i , γ j } = 2δ i,j , it should be easy to check that and similarly U 12 (ζ)γ 2 U † 12 (ζ) = −ζγ 1 . In the measurement based scheme the value of ζ for the sign of the exchange would be random but computable from the results of the measurements in the various steps. If one is unhappy with the result one can always continue to measure until one has the correct sign of ζ. Ultimately, this is an overhead in the computation, which is not greatly significant, though it can be avoided in a deterministic Hamiltonian based approach 76 as opposed to the measurement-based approach discussed here.
Note that even though we did each operation in forward and reverse, the combined effect of the exchanges is non-trivial (i.e. not the identity operation) if ζ 12 = 1. By multiplying both sides of Eq. 65 by the U 23 (ζ 23 )U 12 (ζ 12 ) on the right, we see the two exchange operations if produced in different orders produces different outcomes Thus, Majorana modes obey non-Abelian exchange statistics.
The fermion parity degree of freedom associated with a pair of Majorana modes cannot by itself be used as a qubit (i.e. two level system) for quantum information processing.
There are several ways of constructing two level systems from Majorana modes, which is a process known as encoding 77 . To understand how qubits can be constructed from Majorana modes, we first note that the fermion parity of a superconductor is conserved. Therefore the fermion parity of a system of Majorana modes cannot be changed by any of the allowed operations. One efficient, though not the simplest way to avoid this constraint is to use six Majorana modes γ j=1,...,6 to construct two two-level systems described by spin−1/2 operators S (j=1,2) a=1,2,3 78 . Even though we start from an eight-dimensional Hilbert space associated with six Majorana modes, the restriction of a total fermion parity constraint on exchange operators limited to these qubits limits us to the four dimensional Hilbert space associated with the two spin−1/2. One choice for the effective spin components we can use is written compactly as More explicitly, for j = 1, S 1 a=1,2,3 = iγ 1 γ 2 , iγ 2 γ 3 and iγ 3 γ 1 respectively. It is easy to check that the two Pauli spin operations defined commute [S To devise realistic schemes for realizing MZMs, we recall that first and foremost we need an appropriate superconductor. To make the scheme experimentally feasible, an ordinary swave superconductor is most preferable. The second requirement is the existence of spinless fermions, which obviously do not occur naturally in solid state systems. One possible way out is the application of a Zeeman field to the electrons, V z σ z , where V z is the strength of the Zeeman field and σ z is a Pauli spin matrix. Since the energy of the electrons in the parallel and anti-parallel direction to the Zeeman field is different, this would lead to a shift of the parallel and anti-parallel energy bands. If now the Fermi energy could be tuned to fall in the lower band (Fig. 2), we would have spin-polarized fermions on the Fermi surface.
where the normal part of the Hamiltonian density is written as andĉ † σ (r) are the creation operators for electrons with spin σ. The proximity-induced superconducting pairing potential is written as where ∆(r) is the proximity-induced superconducting pair potential. The BdG equations describe the Bogoliubov excitation operators of this Hamiltonian which are of the form and are defined by the equation Here, H BdG is defined as H BdG = H N +H p . By encoding the particle and the hole components of the wave-function u σ (r) and v σ (r) into a Nambu spinor Ψ(r) = (u ↑ (r), u ↓ (r), −v ↓ (r), v ↑ (r)), the above operator equation can be written as a BdG equation for the wave-function Ψ(r) as The spins for the hole components in Ψ(r) are inverted, which leads to the σ y in the lower right corner of the Hamiltonian to ensure the manifest spin-rotation symmetry of the singlet superconductivity. The matrix structure in the particle-hole space of H BdG can be captured by Pauli matrices, which allow one to write the BdG Hamiltonian for the system as where τ + = τ † − = τx+ıτy 2 . By applying a Hermitian conjugation to Eq. 72 we can see that any solution γ † at energy E of Eq. 72 is accompanied by another solution γ with energy −E. In the spinor language, this corresponds to a new spinor wave-function that is related by the particle-hole transformation where Ξ = σ y τ y K is defined as the particle-hole symmetry operator with K being complex conjugation. The existence of spinor solutions that come in (E, −E) pairs is guaranteed by the particle-hole symmetry of the BdG Hamiltonian ΞH BdG Ξ = σ y τ y H * BdG σ y τ y = −H BdG .
As we show in the next section, a detailed analytical solution of the BdG equations following from Eq. 73 near defects of the pair potential ∆, e.g., vortices and sample edges, reveal the existence of MZMs above a critical value of the Zeeman coupling V z .

A. Vortex bound states in semiconductor-superconductor structures
In this section, we show how one can derive explicitly the existence of the Majorana bound states in a vortex in a spin-orbit coupled semiconductor superconductor heterostructure 92 .
Let us start by introducing a vortex into ∆(r) by assuming it to be of the form ∆(r, θ) = |∆(r)|e inθ , where n = 0, 1, . . . is the multiplicity of the vortex. In the absence of a vortex (i.e. n = 0), the BdG Hamiltonian has a rotation symmetry generated by the total angular momentum J z = L z +σ z /2. The spin-orbit coupling term proportional to α couples spin and orbital angular momentum so that only the total angular momentum J z is conserved. Adding a vortex changes this angular momentum operator, because application of rotation θ → θ+ϕ shifts the superconductor phase ∆(r, θ) → ∆(r, θ + ϕ) = ∆(r, θ)e iϕ . This phase is generated by application of the unitary operator U = e inτzϕ/2 using the relation U † τ + U = e inϕ τ + . We can include this unitary operator in the rotation operator by modifying the total angular momentum operator to With this choice, it is a straightforward calculation to check that J z is conserved i.e.
[J z , H BdG ] = 0. This allows us to assume the vortex solutions to be eigenstates of total angular momentum J z = m J . The angular (i.e. θ) dependence of such eigenstates is constrained entirely by m J , so that the wave function can be written entirely in terms of a radial spinor: where the radial spinor Ψ We are now at a point where we can write the full BdG Hamiltonian for an n-fold vortex in polar coordinates as (79) where η = 2 2m * , σ + = σ † − = σ x + ıσ y and p + = p x + ıp y = e ıθ (−ı∂ r + 1 r ∂ θ ) and p − = p x − ıp y = e −ıθ (−ı∂ r − 1 r ∂ θ ). With this Hamiltonian the BdG equation can be written as H BdG Ψ m J (r, θ) = EΨ m J (r, θ). We can now use the angular dependence in Eq. 78 to write a purely radial (i.e. one dimensional) BdG equation in terms of a radial BdG Hamiltoniañ By substituting the full Hamiltonian H BdG into this equation, we get the radial Hamiltonian The resulting BdG equations are now much more tractable because they are one dimensional i.e. radial though complex. We can make this Hamiltonian real by performing a π/4 spin rotation via the unitary transformation U = e iπσz/4 so that U σ + U † = iσ + . Following this transformation, the solutions of the BdG equation can be assumed to be real without loss of generality. However, the BdG equations are still quite challenging because they are four component coupled second order differential equations.
We can reduce the complexity of this problem by half by taking advantage of the particlehole symmetry of the zero-energy Majorana mode in the vortex. The particle-hole symmetry operator, Ξ, transforms the J z = m J spinor eigenstate with energy E into a −m J eigenstate with energy −E because Ξe ı(m J −σz/2+nτz/2)θ Ψ m J (r) = e ı(−m J −σz/2+nτz/2)θ ΞΨ m J (r).
Since a Majorana mode is particle-hole symmetric, it must be associated with the quantum numbers m J = 0. Focusing on this channel of the BdG HamiltonianH BdG,m J =0 =H BdG and the fact that particle-hole symmetry changes the sign of the energy E, such a particle-hole symmetry for real Hamiltonians is tantamount to a chiral symmetry σ y τ yHBdG σ y τ y = −H BdG .
It is easy to check that zero-energy solutions Ψ m J =0 (r) = Ψ(r) of such a chiral symmetric Hamiltonian would be an eigenstate of the chiral symmetry operator S = σ y τ y so that where λ = ±1. Since each eigenvalue λ of S is two-fold degenerate, by choosing an eigenvalue λ we can write Ψ(r) in terms of two functions u σ (r) as Ψ(r) = σ u σ (r)η σ where Sη σ = λη σ are the two eigenspinors of S. Thus, we can use the particle-hole constraint to reduce the 4 component BdG equations to two components.
We can see the reduction of the BdG Hamiltonian to two components by replacing τ x iñ H BdG by τ x = ıλσ y τ z , which follows from σ y τ y = λ. Making this substitution, which applies only to the E = 0 states, the BdG Hamiltonian for a given value of λ becomes While this still appears to be a 4 × 4 matrix, the transformed Hamiltonian commutes with τ z , so that we can consider each of the τ z = ±1 sectors (electron and hole) separately. This allows one to write the BdG differential equation in the (τ z = +1) in terms of the spinor Ψ 0 (r) = (u ↑ (r), u ↓ (r)) T for a single vortex (n = 1) in the form of a 2 × 2 matrix differential equation: To make progress towards an analytic solution, we approximate the radial dependence of ∆(r) by ∆(r) = 0 for r < R and ∆(r) = ∆ for r ≥ R. Let us start by considering the range (r < R), which is the non-superconducting region (∆(r) = 0). In this range, the wave function is simply that of a spin-orbit coupled semiconductor. In the absence of spin-orbit coupling, each spin-component has solutions given by Bessel functions J 0 (zr) and J 1 (zr).
This suggests that we can include spin-orbit coupling by trying a spinor of the form We can determine (u ↑ , u ↓ ) and z by substituting Eq. (86) into Eq. (85), which then takes This condition simplifies to The value of z can be determined by setting the determinant of the above matrix to zero.
This leads to the equation for z Note that the solutions of the above equation come in pairs ±z. However, the Bessel functions J 0 (zr) and J 1 (zr) are odd and even functions of z respectively. Therefore, there are two linearly independent solutions that one can obtain for r < R.
The r > R region includes non-zero amplitude of superconductivity. This region is complicated to solve analytically except by a power-series solution in 1/r. However, our focus of whether there is a normalizable solution in this region depends on the conditions for which there are exponentially decaying solutions as r → ∞. Inspired by the asymptotic form of Bessel functions J n (zr) ∝ e −zr /r 1/2 , we consider an ansatz Substituting this ansatz into the BdG equation we get We notice that the r dependence disappears in the limit r → ∞ so that Similar to the case for r < R, this equation has a non-trivial solution if z satisfies the secular equation: We can separate the interesting topological case C from the cases A and B by considering the sign of the product C 0 = n z n = (µ 2 +∆ 2 −V 2 Z ) of the roots z n , which is the polynomial at z = 0. Note that complex conjugate pairs (z, z * ) do not change the sign of this product C 0 since zz * = |z| 2 > 0. The case B has pairs of real roots with the same sign. These also do not contribute to the sign of C 0 . On the other hand, the topological case C is characterized by pairs of real roots with opposite signs, so that the product C 0 would be negative. This leads to the condition to realize topological zero-energy Majorana modes Therefore Majorana modes are realized only for Zeeman field in excess of a critical value While this gives us the condition for a Majorana mode, we still have not written an explicit form of the solution, although there are many constraints that are derived. We have almost exact solutions for r < R, except for certain coefficients that can be written in terms the matrix equations Eq. 88. However, we considered an asymptotic r → ∞ limit for r > R. This has been 92 extended to a power-series solution in 1/r, which is systematic, but does not directly impact the qualitative structure of the solution. Eq. 74 to one dimension by assuming parameters to vary only along the x−direction. In this case we can assume the BdG spinor to be a plane wave with wave vector k y along the y direction so that the BdG Hamiltonian is Similar to the angular momentum quantum number m J in Eq. 78, k y transforms to −k y under particle-hole symmetry Ξ because of the complex conjugation operator K. As a result H BdG is still particle-hole symmetric and the zero-energy Majorana operators appear at k y = 0. The BdG Hamiltonian at k x = 0, is particle-hole symmetric and real so it is chiral symmetric with the chiral symmetry operator S. As before, we can then assume the zero energy Majorana mode is an eigenstate of S, so that we can replace τ x with σ y τ z . This reduces the BdG Hamiltonian to a two component form similar to Eq. 84 as The corresponding BdG equation, similar to the case of vortices (i.e. Eq. 85) is a 2 × 2 coupled differential equation as well: As in the case of vortices we will consider the parameters µ and ∆ to be constants at different values across a domain wall at x = 0. On each side we can expand Ψ 0 (x) in terms of plane-waves Ψ 0 (x) ∝ e zx Ψ 0 so that the above equation becomes This equation is identical to Eq. 92 and therefore has three solutions with Re(z) with the same sign only in the case of a topological bulk with C 0 < 0. The properties of domainwalls with various boundary conditions depends on the details of the boundary conditions discussed in the two sub-sections.

Edge boundary conditions
Based on analogy with FQHE and chiral p-wave superconductors, one expects a chiral gap-less state confined to the edge of the semiconductor heterostructure. We model an edge with a chemical potential µ towards the edge increases from µ(x) = µ for x < 0 to µ(x) = ∞ > |V z | for x > 0. We will assume that ∆(x) = ∆ is independent of x. The wave-function Ψ(x) must then vanish at x = 0. This leads to the boundary condition where Ψ n and z n satisfy Eq. 98. The resulting wave-function Ψ n (x) with be normalizable if Re(z n ) > 0. The boundary condition for the two component spinor Ψ(x = 0) contains two constraints. These can be solved for three values n. As we found in the case of the vortex, Eq. 98 has three solutions with Re(z n ) > 0 only if C 0 < 0, which was exactly the topological condition that leads to Majorana zero modes in the vortex.
Let us now discuss the solution of the BdG equation associated with the Hamiltonian in Eq. 95 away from k y = 0. This Hamiltonian has one zero-energy Majorana state at k y = 0.
Using perturbation theory, we can calculate the correction to the energy at finite k y as which is linear in k y at this order. This shows that the dispersion of the Majorana zero mode at k y = 0 of the topological superconductor for C 0 < 0 is a chiral Majorana with a linear dispersion relation. wall to x > 0. Using Eq. 93, we note that this change of sign of ∆ also changes the sign of z in the same λ sector. Thus, in the topological regime, C 0 < 0, we obtain three solutions with Re(z n ) < 0. These values correspond to three normalizable plane-wave solutions for x > 0. Combining the solutions for x < 0 and x > 0, in the topological regime C 0 < 0, we have six plane-wave solutions to be used to match four constraints. This leads us to a pair of zero energy Majorana modes for the π Josephson junction at k y = 0.
We can move away from k y = 0 by computing the matrix elements of the k y perturbation proportional to σ x τ z in Eq. 95, similar to the case of the chiral edge states. However, in this case, conjugating the BdG Hamiltonian with σ z τ z together with the application of a mirror symmetry x → −x has the effect of flipping k y without any other change of the system. Therefore, the dispersion of the system resulting from the perturbation of adding k y , in contrast to that of the chiral edge states, must be symmetric E(k y ) = E(−k y ). This dispersion is what is called a non-chiral Majorana mode. The two fold degeneracy of the zero energy Majorana modes at k y = 0 is broken by going away from the π phase difference, similar to those in topological insulators 93 . The resulting dispersion is that of a massive Dirac/Majorana mode: where v, b are constants that would be determined from perturbation theory.

C. Relation to bulk phase transition and topological invariant
The calculation so far showed us that vortices and edges in the semiconductor-superconductor This equation contains an imaginary part ikαλ∆, which automatically forces k to vanish.
The bulk spectrum of the Hamiltonian is The spectrum of this Hamiltonian is These energies vanish precisely when Eq. 102 is satisfied. Therefore, the parameters where we go from having zero energy vortex modes in a topological superconductor to the trivial phase of the superconductor corresponds to a phase transition where the bulk energy gap of the BdG Hamiltonian in Eq. 74 closes at k = 0. In fact, this closure of the gap can be understood in terms of an application of Kitaev's topological invariant in Eq. 45 to the spinful superconductor, which is written as in particle-hole symmetric systems where "P f " stands for the Pfaffian. One key difference is that because of the different Nambu basis we use for spinful superconductors, the particlehole matrix in the Kitaev Hamiltonian τ x is replaced by τ x → σ y τ y . In addition, for the continuum system used to model systems such as the superconducting nanowire, the k = π term in Eq. 45 is trivial and may be dropped. By computing this Pfaffian for the 4 × 4 matrix using the standard definition 94 we get which is exactly the C 0 topological number (see Eq. 94) we obtained from studying vortices and edges.

D. Dimensional reduction to one dimensional Nanowires
Majorana modes can also be realized in the one dimensional analog of the semiconductorsuperconductor structure. This structure is experimentally simpler because one does not require a magnetic insulator to generate the Zeeman field. In the one dimensional case, the Zeeman field can be generated by a magnetic field parallel to the semiconductor superconductor wire. In this case, the Zeeman field cannot generate substantial undesired orbital effects that would suppress superconductivity, though a finite diameter of the wire can leave residual effects 95,96 .
Understanding the one dimensional nanowire does not require any new calculations. The BdG Hamiltonian Eq. 96 that we used to study the spectrum of domain wall states at k y = 0 is exactly the Hamiltonian for the semiconductor-superconductor nanowire heterostructure.
The end of the wire corresponds precisely to the edge of the system. The argument from the previous sub-section then tells us that a semiconductor nanowire will support topological Majorana modes for C 0 < 0. Additionally, the results from the sub-section on Josephson junctions implies that a π-phase Josephson junction will support a pair of such Majorana modes. We will discuss signatures of both the end Majorana modes as well as pair of Majorana modes in the following section.
One subtlety worth noting is that of the chiral symmetry (i.e. Eq. 82) of the BdG Hamiltonian for the superconducting semiconductor. This is because the exact form of the topological invariant depends on the symmetry class of the system. For example the Kitaev topological invariant Eq. 45 is only defined for one dimensional superconductors, which have particle-hole symmetry by definition and are classified as symmetry class D 97 . This invariant makes no assumption about the Hamiltonian being real, which happens to be the case for the specific superconducting semiconductor system being considered. In this case the reality of the BdG Hamiltonian leads to a chiral symmetry Eq. 82, which allows us to define a more refined topological invariant 69 in this new symmetry class called BDI. To define this operator we split the space into subspaces with projectors Σ ± of eigenvalues ±1 of the chiral symmetry operator S. This would imply that SΣ ± = ±Σ ± . The BdG Hamiltonian is now purely off-diagonal in this space so that Σ + H BdG (k)Σ + = Σ − H BdG (k)Σ − = 0. We can then define the off-diagonal part as a topological invariant characterizing the topological superconductor in symmetry class BDI 97 . This has interesting physical consequences such as it allows multi-channel generalizations of the superconductor semiconductor wires to have multiple Majorana modes that are protected as long as the chiral symmetry is preserved 69 .

IV. EXPERIMENTAL SIGNATURES
A. Transport signature As we saw from the schematic set-up in Fig. 4, both the zero-energy Majorana mode as well as the quasiparticle gap closing associated with the topological quantum phase transition may be probed by tunneling transport. Such transport involves transfer of electrons from a normal lead into the superconductor which is reflected back as an electron or hole according to the scattering matrix: The process of electron reflecting back as a hole from the normal-superconductor interface and transferring a Cooper pair to the superconductor is termed Andreev reflection.
Using the Blonder-Tinkham-Klapwijk 99 formalism, the conductance into the superconductor can be written in terms of elements of this scattering matrix as where [tr (..) for the trace associated with additional channels such as spin etc.] and N ch = 2 is the number of the electron mode in this single channel model, and the transmission. In the limit of a long wire, the transmission coefficient T ee = N − T eh so that the conductance becomes related to the Andreev reflection probability We can understand this conductance as resulting from the transfer of Cooper pairs from the normal lead to the superconductor.
We can compute the conductance into the end of a semiconductor nanowire by first computing the scattering matrix S(E) numerically using a program such as KWANT 100 and then substituting the answer into Eq. 113. In addition to a discretized version of the BdG Hamiltonian in Eq. 96, to compute the scattering matrix S(E), we need to specify the normal lead and a tunnel barrier. We typically choose the normal lead Hamiltonian to be similar to Eq. 96 except that ∆ = 0 and the chemical potential µ lead is much higher than that in the semiconductor nanowire. The result shown in Fig. 5 shows a gap at small Zeeman potential, which closes as the Zeeman potential is increased and merges into a zero-energy peak which persists beyond the topological phase transition. The closure of the gap seen in the spectrum is consistent with the spectrum that we saw in Fig. 3 in the discussion of the topological quantum phase transition. Note that we do not typically see the reopening of the gap in these plots similar to experiments. Actually the measurement of a gap closure followed by the emergence of a zero-bias conductance peak qualitatively similar to that seen in Fig. 5, seen in experiments 101 a few years after the prediction has been one of the main motivating drivers of the field. However, several quantitative features are yet to be observed.
As we will discuss later these together with certain alternative scenarios that may arise in these systems have made the search for Majorana modes somewhat controversial.
One of the quantitative features in the conductance plot shown in Fig where W is an N ch × N matrix and N ch is the number of channels and N is the size of the BdG Hamiltonian of the nanowire. Since the conductance in the tunneling limit is not expected to depend on the details of the lead (which can be checked numerically), We can then write this scattering matrix in terms of the nanowire Green function g 0 (E) = (E − H) −1 by expanding as a formal power-series in W as By writing W = Γw, we can then formally resum the power-series to write S in terms of the local part of the Green function g l (E) = w † g 0 (E)w so that The local Green function g l (E) can be thought of the Green function at the end of the wire and is only an N ch × N ch matrix, where N ch = 2 in the simplest case of a spin-polarized lead. The two components of the channel are the particle and hole of the Nambu space. The zero-energy Majorana mode, which has a particle-hole symmetric wave-function, appears as a zero-energy pole of both g 0 (E) as well as g l (E). We can include this pole structure of g l (E), by approximating g l (E) near E ∼ 0 as Here we use a Nambu basis where the particle-hole symmetry of the Green function takes the form τ x g l (E)τ x = −g l (−E). Substituting g l (E) into Eq. 117 we get the amplitude of the electron-hole reflection amplitude to be where for the last step we took the limit of small a i.e. contribution from other states.
Note that r eh (E = 0) = −1, which is referred to the phenomenon of perfect Andreev reflection. This is a hall-mark signature of topological superconductors and indeed can be derived from the topological invariant 103 . In contrast, the non-topological superconducting case, which does not have any such zero energy pole can be understood from Eq. 118 by assuming u(E) ∝ E, is characterized by an Andreev reflection amplitude r eh (E ∼ 0) → 0 that vanishes. The reflection amplitude r eh determines the reflection probabilty T eh ii = |r eh | 2 , so that the conductance resonance for the Majorana mode is given by This is the standard form for the conductance resonance associated with tunneling into a Majorana zero mode. What is remarkable is that the height of the peak is quantized i.e. However, the total weight under the conductance peak, which is related to the current I M aj at bias voltages larger than the peak width Γu 2 is proportional to Γ i.e.
Since the current ultimately vanishes with Γ, this resolves the apparent paradox of the zero-bias conductance being independent of tunneling even as the tunneling rate Γ vanishes.
As already mentioned, the field received a large boost from early, though preliminary verification of the transport and Josephson predictions 101,[106][107][108][109] . Specifically, as we see in the recent conductance data seen in the left panel of Fig. 6, the conductance as a function of bias voltage V and applied Zeeman potential V Z , shows a gap at small Zeeman field, which closes at about a magnetic field of B ∼ 0.7T followed by the emergence of a zerobias conductance peak with conductance near the quantized value predicted for Majorana zero modes. This is qualitatively similar to the first measurements of the conductance into the device 101,106-109 , though improvements in device fabrication since then have significantly enhanced the quality of the features. Qualitatively similar results have been obtained by several groups since the first observation confirming that these results are quite reproducible 95,104,105,[110][111][112][113][114] . While these features are in qualitative agreement with the theory predictions seen in Fig. 5, there are discrepancies. The first discrepancy worth noting is that neither the theoretical conductance (left panel of Fig. 5) or the experimental measurement (left panel of Fig. 6) shows evidence of the bulk gap reopening expected from the right panel of Fig. 3.
While this is not technically a discrepancy in the sense that the theoretical conductance in most models also do not show this feature because of competition from the zero-bias peak it is an important feature to confirm. Secondly, the height of the conductance peak, although near the predicted value 104,112 , shows significant deviations both above and below the predicted value with changing parameters and therefore does not appear to be as robust as predicted by theory. We will elaborate on the implications of this discrepancy for the field in the last section of the chapter.

B. Bulk gap closure
The conductance from the end shown in Fig. 5 only shows us an apparent gap closure but not the gap reopening expected from the right panel of Fig. 3. Additionally, even the gap closure feature is typically expected to be obscured by the presence of Andreev bound states associated with complicated end potentials that we will discuss later. Instead, we can consider a more direct measure of the gapless states at the phase transition by studying transport through such states. Such an experiment can be performed by adding another lead on the right end R of the semiconductor wire, in addition to the lead L at the left end. Since the superconductor has to be grounded this is referred to as a three-terminal configuration.
The scattering matrix S must now be doubled to include scattering from both leads where each block S ij has the particle-hole structure of the scattering matrix in Eq. 110 from the previous sub-section. We can characterize the transport properties of such a threeterminal device by a conductance matrix where I L (I R ) is the current entering the left (right) normal lead from the scattering region, and V L (V R ). In the limit of a long wire, G LL and G RR are the local conductances at each end that we discussed in the last sub-section and are plotted in the left panel of Fig. 5. The transport properties across the wire would be measured from the nonlocal conductances (G LR and G RL ). These non-nonlocal conductances can be written in terms of the transmission probabilities defined in Eq. 112 and are therefore expecteed to vanish for a gapped wire (away from the topological phase transition).
The appearance of a finite non-local conductance near zero bias would signal such a bulk gap closure. This expectation is verified 115 is verified by the numerical results for the nonlocal conductance shown in the middle panels Fig. 3. These results were obtained for a scattering matrix S computed using KWANT 100 using the BdG Hamiltonian Eq. 74 for a nanowire. An unfortunate complication of these results is that the non-local conductance actually, even at the topological phase transition, vanishes at exactly zero bias as a consequence of particle-hole symmetry 116 . While the vanishing at the critical point is linear in voltage as shown in the right panel of Fig. 3 as opposed to a gapped signal away from the phase transition, introduction of disorder etc might make the bulk gap closure difficult to identify in a definitive way. It is worth mentioning that an advantge of the scattering matrix formalism is that we do not need the bulk Hamiltonian or the topological condition Eq. 94, which are limited to clean single-band systems, to identify the topology of the nanowire. We can compute the topological invariant T V L directly from the scattering matrix S LL (E = 0) from the scattering matrix topological invariant The numerical result for this invariant is shown in the inset of the right panel of Fig. 3 and confirms that the topological invariant T V L vanishes at the quantum critical point.
The vanishing of T V L at the topological quantum critical point, suggests a zero-mode in the scattering matrix S LL (E = 0). This means that there is a mode that suffers no reflection when incident from the left lead L and must thus be completely transmitted. This seems to contradict the linear vanishing of the non-local conductance G RL that we discussed in the last paragraph. As will become clear when we discuss the teleportation process through a This entropy transfer at the bulk gap closure contributes to the heat conductance which can also be computed from the transmission probabilities T ab ij computed from the scattering matrix S. The perfect transmission of quasiparticles at the bulk gap closure associated with the topological quantum phase transition appears as a quantized peak in thermal conductance with height κ 0 = π 2 k 2 B τ /6h at temperature τ happens at TQPT 117-119 (h for the Planck constant and k B for the Boltzmann constant).
The non-local conductance, for which theoretical results were shown in the middle panel of Fig. 5, has also been measured in recent experiments 105 . The results, though not as extensively as the end conductance, shown in the right panel of Fig. 6 indicate the existence of a gap at low magnetic fields consistent with the local conductance. In this plot, we infer a gap from the range of bias voltage over which the conductance vanishes. Increasing a magnetic field appears to suppress this gap near the critical value where a zero bias conductance peak appears. However, the data is not definitive about a re-entrant gap beyond the critical value, where only a slight suppression of non-local conductance is seen.

C. Fractional Josephson effect
As we saw earlier from the spectrum of Josephson junction in topological superconductors (i.e. Eq. 101), a π phase difference leads to a degenerate pair of states, which splits in energy as the phase difference φ deviates from π. Formally Eq. 101 contains a wave-vector k y . However, we saw in the dimensional reduction argument to one dimensional nanowire (see subsection III D) that the results for the nanowire can be obtained by setting k y = 0 in the domain wall results for the two dimensional superconductor. A detailed numerical calculation of the spectrum for a Josephson junction using the BdG Hamiltonian of a nanowire (i.e. Eq. 74), which is shown in panel (a) of Fig. 7, confirms the existence of the pair of zero-energy bound states at phase φ = π on in the topological superconducting (i.e. topologically nontrivial) phase. The topologically trivial phase shows a gapped spectrum as a function of phase φ as expected.
While the value of phase φ where the zero-energy crossing occurs is not protected against changes in detail of the Hamiltonian of the junction, the crossing itself is protected by particle-hole symmetry. Let us start by writing the dispersion of the Andreev bound states (i.e. Eq. 101) near phase φ ∼ π as where the index ± refer to the two branches of the spectrum for the topologically nontrivial phase seen in Fig. 7(a).  Fig. 7(a). An odd number of such crossings is uniquely associated with topological superconducting phase based on the invariant in Eq. 106.
A physical consequence of the change of the Pfaffian from a change in flux is the change in ground state fermion parity 19,29? associated with such a flux in the set-up shown in Fig. 8. To understand the connection between the Pfaffian and ground state fermion parity, note that the two branches of the spectrum that are seen near the zero-energy crossing in Fig. 7(a) are related by particle-hole symmetry that transforms E → −E and also transforms the creation operator ψ † for the state to ψ † → ψ. Thus, we can combine these two branches in terms of the creation operator to Noting that the fermion operator is continuous across φ ∼ π i.e. ψ † φ∼π ∼ ψ † φ , the crossing of the pair of Andreev bound states seen in Fig. 7(a) is really a zero crossing of the energy ζ(φ − π) of the fermion state with creation operator ψ † . Once the energy of such a state goes from being negative to positive, we can lower the energy of the system by emptying the The superconducting ring used in the setup in Fig. 8 to detect the fermion parity change is practically challenging to construct. Alternatively, we can consider a case where the JJ is isolated from external leads so that the fermion parity of the system remains fixed as one changes the phase φ by 2π. This necessarily forces a topological superconducting system with a changing ground state fermion parity to enter an excited state when the phase changes by 2π. Introducing a second change of phase φ by 2π restores the system to the ground state. This leads to a 4π periodicity of the supercurrent of the form where I 2π is a conventional contribution to the supercurrent from topologically trivial channels and I 4π is the amplitude of the 4π periodic component from the topologically superconducting channel. Such a 4π periodic supercurrent that arises from the change in quasiparticle occupation in the junction is termed the fractional Josephson effect 122 and is a hall-mark of topological superconductivity. The supercurrent for the topologically nontrivial phase in Fig. 7(b) is consistent with the form of I JJ in the limit where I 4π I 2π .
The 4π periodic supercurrent I JJ (φ) associated with the fractional Josephson effect may be measured from the radiation of a voltage biased Josephson junction 123 . Applying a voltage V , the phase across the Josephson junction varies as φ = V t, which leads to an ac Josephson supercurrent I JJ (φ = V t) = I 2π sin V t + I 4π sin V t/2. The radiation from the ac current for the topologically trivial phase at zero magnetic field, which is shown in the top left panel of Fig. 9, is peaked at a frequency that has a conventional slope with respect to the applied voltage across the junction. In contrast the frequency of the radiation in the topologically non-trivial phase at large Zeeman field has slope with voltage, which is half of the conventional case. Being able to detect the radiation from the Josephson junction in a single nanowire requires sensitive on-chip detector technology. Because of this, the first measurements of the fractional Josephson effect was based on the Shapiro voltage steps where a microwave irradiated Josephson junction is used to generate finite voltage steps 125 . The origin of the voltage steps can be understood by considering a phase of the form φ = V t + φ ac sin (ωt), where one can check that a dc current is supported in the limit of ω and V becoming commensurate. Similar to the half frequency radiation from the fractional Josephson effect, the voltage steps V (relative to ω) in a potentially topologically non-trivial superconducting phase was observed to be larger by a factor of 2 124 as seen in the right panel of Fig. 9. Since these measurements involve measurement of DC voltages, they preceded the radiation measurements and were observed in 2012 124 about the same time as the zero-bias conductance peak measurements. In fact both these effects have also been claimed to be observed in other systems as well 126,127 . However, the constraint of a fixed fermion parity, which is required for the validity of the fractional Josephson effect (i.e. Eq. 129), requires a dynamical measurement on a time-scale shorter than the quasiparticle

D. Teleportation
The fact that a pair of Majorana modes γ 1,2 can be used to construct a single fermion mode c † = γ 1 + iγ 2 leads to a unique transport property of Coulomb blockaded Majorana wires 132 when in the set-up shown in Fig. 10. Before considering the effect of Coulomb blockade on the central wire, let us first assume that the pair of end Majorana modes γ 1,2 have a small splitting . The Hamiltonian for such a system is where t α is the coupling to the lead fermions d † The teleportation process can be used to measure a Majorana qubit, which is one of the key ingredients of the braiding protocols discussed earlier in subsection II E. To understand how we would use H C to measure a Majorana qubit, imagine that the superconducting island in Fig. 10 had an additional semiconductor wire with its own set of Majorana modes γ 3,4 . The parity of the extra pair of Majorana modes would affect the fermion parity of the island, which ultimately affects the sign of C . Since the Coulomb blockade can be chosen to favor a particular total fermion parity iγ 1 γ 2 γ 3 γ 4 , we can view the sign of C as being set by the fermion parity of the Majorana wire iγ 1 γ 2 . Thus, the effective tunneling between the leads is modified to The sign of this effective tunneling amplitude reflects the phase of the transmitted electron, which can be measured by interference 136 . Measurement of the interference phase would be equivalent to the measurement of the fermion parity of the Majorana qubit.
Interestingly, preliminary experimental evidence for teleportation has been seen in recent experiments as seen in Fig. 11. The conductance shown in the left panel of Fig. 11, shows that conductance through such a wire in the Coulomb blockade regime which is e periodic in gate voltage. This is characteristic of transport through an electronic bound state in the wire. A relatively trivial possibility for the origin of such a state, which is difficult to rule out experimentally 121 , is a conventional bound state in the middle of a short wire. The signal from such a state in the case of a long wire would be negligible and can only arise if there are a pair of bound states similar to the Majorana states seen in Fig. 10. However, the same signal can arise from Andreev bound states (ABSs) at the ends of the wire that we will discuss later 137 . The non-local conductance from ABSs is expected to be different from those arising from Majorana modes i.e. the conductance from ABSs would not show interference as seen in the right panel of Fig. 11. However, as mentioned, it is difficult to rule out the short wire scenario without more careful experiments such as those that have been proposed 121 . resulted in substantial enhancement in the height of the zero bias conductance peak from about ∼ 0.1 2e 2 h in the first generation Majorana devices 101 to zero bias conductance peak height approaching and exceeding 2e 2 h in the recent experiments 104,112,114 . However, despite the claims of several breakthroughs, the field has perennially seemed to remain on the cusp of a confirmatory evidence of MZM that has not yet been materialized in experiments.
The main obstacle to a confirmatory evidence of MZM is that ends of a nanowire are often locations where robust zero energy states are induced by various non-topological effects that have little or no connection to MZMs. To distinguish such robust zero energy states of non-topological origin from topological MZMs, experiments investigating the quantization of the zero bias peak heights at value 2e 2 h , and the persistence of this quantized peak height with variation in the experimental parameters (the so-called quantized conductance plateau), have recently attracted vigorous attention. While ballistic Andreev reflection 99 from an ordinary zero mode leads to a conductance quantization with zero bias peak height 4e 2 h , MZMs should lead to a quantized peak height of 2e 2 h as they effectively behave as "half a fermion". Moreover, topological MZMs being insensitive to weak perturbations, the persistence of the ZBP height with variations in the tunnel gate potential and applied magnetic field are taken as spectacular transport evidence unique to MZMs.
In recent experiments 104,112,114 the height of the zero bias conductance peak has indeed been observed to approach or exceed ∼ 2e 2 h . However, a quantized conductance plateau for a convincing range of tunnel barrier or magnetic field is yet to be realized. Unfortunately, quantized conductance peak alone at isolated points in the parameter space cannot be taken as confirmatory evidence for MZMs. Furthermore, the absence of a convincing quantization plateau around the points in parameter space with ZBP height ∼ 2e 2 h may be an indication that the robust zero energy states in these systems may in fact have originated from either, (a) disorder-induced weak anti-localization that leads to robust class-D conductance peaks with peak height between 0 and 2e 2 h , 139-142 or (b) partially separated Andreev bound states (ps-ABS) 143 , also known as quasi-Majorana modes 144,145 , whose low bias conductance signature depends on the overlap of the wave functions of component Majorana bound states of a conventional fermionic state. The observation of zero bias conductance peaks with peak height ∼ 2e 2 h only at isolated points in the parameter space and absence of convincing quantized plateaus around them may be an indication of significant residual disorder in the hybrid nanowires. With sustained improvements in material parameters by reducing disorder and interface inhomogeneity, we hope that quantized conductance plateau with peak