search for majorana fermions in superconductors

Search for MajoranaFermions in SuperconductorsC.W.J. BeenakkerInstituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, The Netherlands; email:[email protected]

Annu. Rev. Condens. Matter Phys. 2013. 4:11336

TheAnnual Review of Condensed Matter Physics isonline at conmatphys.annualreviews.org

This articles doi:10.1146/annurev-conmatphys-030212-184337

Copyright 2013 by Annual Reviews.All rights reserved

Keywords

topological zero modes, midgap states, topological superconductors,topological insulators, non-Abelian anyons

Abstract

Majorana fermions (particles that are their own antiparticle) may ormay not exist in nature as elementary building blocks, but in con-densed matter they can be constructed out of electron and hole exci-tations. What is needed is a superconductor to hide the chargedifference and a topological (Berry) phase to eliminate the energy dif-ference from zero-point motion. A pair of widely separatedMajoranafermions, bound to magnetic or electrostatic defects, has non-Abelianexchange statistics. A qubit encoded in thisMajorana pair is expectedto have an unusually long coherence time. I discuss strategies to detectMajorana fermions in a topological superconductor, as well as possi-ble applications in a quantum computer. The status of the experimen-tal search is reviewed.

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1. WHAT ARE THEY?

AMajorana fermion is a hypothetical particle that is its own antiparticle. The search forMajoranafermions goes back to the early days of relativistic quantum mechanics.

1.1. Their Origin in Particle Physics

The notion of an antiparticle originated in 1930 with Paul Diracs interpretation of the negative-energy solutions of his relativistic wave equation for spin 1/2 particles. The positive-energysolutions describe electrons, and the negative-energy solutions correspond to particles with thesamemass and spin but opposite charge. The electron and its antiparticle, the positron, are relatedby a symmetry operation that takes the complex conjugate of the wave function. A particle andantiparticle can annihilate, producing a pair of photons. Whereas the photon (described by a realbosonic field) is its own antiparticle, Dirac fermions are described by complex fields with distinctparticles and antiparticles.

EttoreMajorana showed in 1937 that the complex Dirac equation can be separated into a pairof real wave equations, each of which describes a real fermionic field without the distinction ofparticleandantiparticle (1). The decomposition of a complexDirac fermion into a superposition oftwo real Majorana fermions is a formal step without physical consequences. But Majoranasuggested that neutral particlesmight be describedby a single real field and concluded thatthere isnow no need to assume the existence of antineutrons or antineutrinos.

We now know that the neutron and antineutron are distinct particles, but the neutrino andantineutrino couldwell be the same particle observed in different states ofmotion (2). It remains tobe seen whether the Majorana fermion will go the way of the magnetic monopole, as a mathe-matical possibility that is not realized by nature in an elementary particle.

1.2. Their Emergence in Superconductors

In condensedmatter we can build on what nature offers, by constructing quasiparticle excitationswith exotic properties out of simpler building blocks. This happened for magnetic monopoles (3),and it may happen for Majorana fermions (4). The strategy to use midgap excitations of a chiralp-wave superconductor goes back two decades (510) [with even earlier traces in the particlephysics literature (11)]. Recent developments in topological states of matter have brought thisprogram closer to realization (12, 13).

The electron and hole excitations of the superconductor play the role of particle and anti-particle. Electrons (filled states at energy E above the Fermi level) and holes (empty states at Ebelow the Fermi level) have opposite charge, but the charge difference of 2e can be absorbed asa Cooper pair in the superconducting condensate. At the Fermi level (E 0, in the middle of thesuperconducting gap), the eigenstates are charge-neutral superpositions of electrons and holes.

That themidgap excitations of a superconductor areMajorana fermions follows from electron-hole symmetry: The creation and annihilation operators g(E), g(E) for an excitation at energyE are related by

gE gE. 1:

At the Fermi level, g(0) [ g g, so the particle and antiparticle coincide. The anticommutationrelation for Majorana fermion operators has the unusual form

gngm gmgn 2dnm. 2:

The operators of two Majoranas anticommute, as with any pair of fermions, but the productg2n 1 does not vanish.

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As in the particle physics context, these are just formal manipulations if the state is degeneratebecause a Dirac fermion operator a 12g1 ig2 is fully equivalent to a pair of Majoranaoperators. Spin degeneracy, in particular, needs to be broken to realize an unpaired Majoranafermion. The early proposals (510) were based on an unconventional form of superconductivity,in which only a single spin band is involved. Such spin-triplet, p-wave pairing is fragile, easilydestroyed by disorder. Much of the recent excitement followed after Liang Fu and Charles Kaneshowed that conventional spin-singlet, s-wave superconductivity could be used, in combinationwith the strong spin-orbit coupling of a topological insulator (14).

The basic mechanism is illustrated in Figure 1. A three-dimensional (3D) topological insulatorhas an insulating bulk and ametallic surface (12, 13). The 2D surface electrons are massless Diracfermions, very much like in graphenebut without the spin and valley degeneracies of graphene.A superconductor deposited on the surface opens an excitation gap, which can be closed locallyby amagnetic field. Themagnetic field penetrates as anAbrikosov vortex, with subgap statesEn(n a)d, n 0, 61, 62, . . ., bound to the vortex core (15). (The level spacing d x D2/EF isdetermined by the superconducting gap D and the Fermi energy EF.) Electron-hole symmetryrestricts a to the values 0 or 1/2. For a 0, the zero modeE0 0 would be aMajorana fermion inview of Equation 1, but one would expect zero-point motion to enforce a 1/2.

E

rr

Superconductor

Majorana fermionbound to a vortex

||

E

3D topological insulator

Figure 1

Profile of the superconducting pair potential D(r) in an Abrikosov vortex (solid gray curves) and boundelectron-hole states in the vortex core (dasheddark blue lines). The left graph shows the usual sequence of levelsin an s-wave superconductor, arranged symmetrically around zero energy. The right graph shows the levelsequencewhen superconductivity is induced on the surface of a 3D topological insulator,with a nondegeneratestate at E 0. This midgap state is a Majorana fermion.

115www.annualreviews.org Majorana Fermions in Superconductors

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Although a 1/2 indeed holds for the usual massive electrons and holes, 2D massless Diracfermions have a 0as discovered by Roman Jackiw and Paolo Rossi (11). The reader familiarwith graphene may recall the appearance of a Landau level at zero energy, signifying quantizationof cyclotronmotionwithout the usual 12Zvc offset from zero-point motion (16, 17). The absence ofa 12 d offset in an Abrikosov vortex has the same origin. Massless Dirac fermions have their spinpointing in the direction of motion. A closed orbit produces a phase shift of p from the 360rotation of the spin. This Berry phase adds to the phase shift of p in the Bohr-Sommerfeldquantization rule, converting destructive interference at E 0 into constructive interference andshifting the offset a from 1/2 to 0.

1.3. Their Potential for Quantum Computing

The idea to store quantum information in Majorana fermions originates from Alexei Kitaev (9).I illustrate the basic idea in Figure 2 in the context of a 2D topological insulator (18, 19), onedimension lower than in Figure 1. The massless Dirac fermions now propagate along a 1D edge

Majorana-boundstates

2D topological insulator

Edge state

E

|1

0

|0

2

0superconductor

Magnet ei 0

superconductor

Figure 2

Top view of a 2D topological insulator, contacted at the edge by two superconducting electrodes separated bya magnetic tunnel junction. A pair ofMajorana fermions is bound by the superconducting andmagnetic gaps.The tunnel splitting of the bound states depends on the superconducting phase differencef, as indicated in theplot [} cos(f/2)]. The crossing of the levels at f p is protected by quasiparticle parity conservation.

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state, again with the spin pointing in the direction of motion. (This is the helical edge state re-sponsible for the quantum spin Hall effect.) A Majorana fermion appears as a zero mode at theinterface between a superconductor (S) and a magnetic insulator (I).

Figure 2 shows two zero modes coupled by tunneling in an SIS junction, forming a two-levelsystem (a qubit). The two states j1and j0of the qubit are distinguished by the presence or absenceof an unpaired quasiparticle. For well-separated Majoranas, with an exponentially small tunnelsplitting, this is a nonlocal encoding of quantum information: Each zero mode by itself contains noinformation on the quasiparticle parity.

Dephasing of the qubit is avoided by hiding the phase in much the same way that one wouldhide the phase of a complex number by separately storing the real and imaginary parts. Thecomplex Dirac fermion operator a 12g1 ig2 of the qubit is split into two real Majoranafermion operators g1 and g2. The quasiparticle parity aa 121 ig1g2 is only accessible bya joint measurement on the two Majoranas.

Whereas twoMajoranas encode one qubit, 2nMajoranas encode the quantum information ofn qubits in 2n nearly degenerate states. Without these degeneracies, the adiabatic evolution ofa stateC along a closed loop in parameter spacewould simply amount tomultiplication by a phasefactor,C1 eiaC, but now the operation may result in multiplication by a unitary matrix,C1UC. Because matrix multiplications do not commute, the order of the operations matters. Thisproduces the non-Abelian statistics discovered by GregoryMoore and Nicholas Read (20), in thecontext of the fractional quantumHall effect, and byRead andDmitry Green (8), in the context ofp-wave superconductors.

The adiabatic interchange (braiding) of two Majorana bound states is a non-Abelian unitarytransformation of the form

C1expip

4sz

C, 3:

with sz, a Pauli matrix, acting on the qubit formed by the two interchanging Majoranas (21, 22).Two interchanges return the Majoranas to their starting position, but the final state iszC is ingeneral not equivalent to the initial state C.

An operation such as Equation 3 is called topological, because it is fully determined by thetopology of the braiding; in particular, the coefficient in the exponent is precisely p/4. This couldbe useful for a quantum computer, even though not all unitary operations can be performed by thebraiding of Majoranas (23, 24).

Before closing this section, I emphasize that the object exhibiting non-Abelian statistics is nottheMajorana fermion by itself, but theMajorana fermion bound to a topological defect (a vortexin Figure 1, the SI interface in Figure 2, or an e/4 quasiparticle in the fractional quantum Halleffect). The combined object is referred to as an Ising anyon in the literature on topologicalquantum computation (24). A free Majorana fermion (such as may be discovered in particlephysics) has ordinary fermionic statisticsit is not an Ising anyon. The same applies to unboundedMajorana fermions at the edge or on the surface of a topological superconductor.

In what follows I concentrate on the Majoranas bound to a topological defect, because of theirexotic statistics. I could have included the Ising anyons in the fractional quantum Hall effect, butbecausethat topic isalreadyverywell reviewed (25), I focus on the superconducting implementations.

2. HOW TO MAKE THEM

The route to Majorana fermions in superconductors can follow a great variety of pathways. Thegrowing list of proposals includes References 510, 14, 18, 19, 2659. There are so many ways to


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make Majorana fermions, because the requirements are so generic: Take a superconductor,remove degeneracies by breaking spin-rotation and time-reversal symmetries, and then close andreopen the excitation gap. As the gap goes through zero,Majorana fermions emerge as zeromodesbound to magnetic or electrostatic defects (60, 61). I summarize the main pathways, and refer thereader to a recent review (62) for a more detailed discussion.

2.1. Shockley Mechanism

From this general perspective, Majorana bound states can be understood as superconductingcounterparts of the Shockley states from surface physics (63, 64). The closing and reopening ofa band gap in a chain of atoms leaves behind a pair of states in the gap, bound to the end points ofthe chain (see Figure 3). Shockley states are unprotected and can be pushed out of the band gap bylocal perturbations. In a superconductor, in contrast, particle-hole symmetry requires the spec-trum to be6E symmetric, so an isolated bound state is constrained to lie at E 0 and cannot beremoved by any local perturbation (see Figure 4).

The closing of the excitation gap, followed by its reopening with opposite sign, is a topologicalphase transition. The phase transition is called topological, as opposed to thermodynamic, be-cause the signQ 61 of the gap cannot be seen in thermodynamic properties. This so-called Z2topological quantum number counts the parity of the number ofMajorana fermions bound to thedefect. Only an odd number of Majoranas (Q 1) produces a stable zero mode. A defect withQ 1 is called topologically nontrivial, whereas for Q 1, it is called topologically trivial.

Because the Majorana fermions are constructed from ordinary Dirac electrons, an unpairedMajorana at a topologically nontrivial defect must have a counterpart somewhere else in thesystem. The two Majoranas are evident in Figures 2 and 4. In Figure 1 the second Majoranaextends along the outer perimeter of the superconductor. One could try to eliminate this secondMajorana by covering the entire topological insulator by a superconductor, but then the flux linewould intersect the superconductor at two points, producing again a pair of Majoranas.

Let us see how these topological phase transitions appear in some representative systems.

2.2. Chiral p-Wave Superconductors

The closing and reopening of the gap in Figure 4 is described by the Bogoliubov-De GennesHamiltonian of a 2D chiral p-wave superconductor:

H U p2=2m Dpx ipyDpx ipy

U p2=2m!. 4:

The diagonal elements give the electrostatic energy 6U(r) and kinetic energy 6p2/2m of elec-trons and holes. (Energies are measured relative to the Fermi level.) The off-diagonal elementscouple electrons and holes via the superconducting pair potential, which has the chiral p-waveorbital symmetry } px 6 ipy. (For equalspin triplet pairing, the spin degree of freedom can beomitted.)

The linedefect is constructedbychangingU from the background valueU0 toU0 dU in a stripofwidthW. As dU is varied,multiple closings and reopenings of the gap appear (see Figure 5). Thisis the 2Dgeneralization (6466) of the 1DKitaev chain (9). The gap closing is a result of destructiveinterference of transverse modes in the strip. Each new mode is associated with one closing-reopening of the gap, so that the defect is topologically nontrivial (Q1) for an odd number ofmodes and trivial (Q 1) for an even number of modes.

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Electrostatic line defects are one way of producing Majorana fermions in a chiral p-wave su-perconductor. Magnetic vortices are another way (6, 8, 6770); those defects are topologicallynontrivial forU< 0. Strontium ruthenate (Sr2RuO4) is a candidate p-wavematerial to observe thepredicted zero modes (10).

2.3. Topological Insulators

In a topological insulator, the closing and reopening of the band gap is a consequence of strongspin-orbit coupling, which inverts the order of conduction and valence bands (12, 13). The surfaceof a topological insulator supports nondegenerate, massless Dirac fermions, with Hamiltonian

a

E

v

x

0

Shockleystate

Shockleystate

a

Figure 3

Illustration of the Shockley mechanism for the formation of bound states at the end points of an atomic chain.The lower panel shows the potential profile along the chain and the upper panel shows the correspondingenergy levels as a function of the atomic separation a. The end states appear upon the closing and reopening ofthe band gap. Figure adapted from Reference 63.


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H0 vFp s U Msz. 5:

A 3D topological insulator, such as Bi2Se3 or Bi2Te3, has 2D Dirac fermions on the surface,whereas a 2D topological insulator, such as a HgTe/CdTe or InAs/GaSb quantum well, has 1DDirac fermions along the edge. The term p s represents pxsx pysy or pxsx for surface or edgeDirac fermions, respectively. The extra term Msz accounts for the exchange energy from amagnetic insulator.

As illustrated in Figures 1 and 2, both the surface and edge Dirac fermions can give rise toMajorana bound states (14). The Bogoliubov-De Gennes Hamiltonian that describes these zeromodes has the form

H H0 DDp syHp0sy

. 6:

The diagonal contains the electron and hole Dirac Hamiltonians (Equation 5), related to eachother by the time-reversal operation H01syHp0sy (which inverts p and s). The pair potentialD (which can be complex as a result of a magnetic vortex) is induced by s-wave superconductivity,so it is momentum independent.

Figure 6 illustrates the conversion of Dirac fermions intoMajorana fermions on the surface ofa 3D topological insulator (71, 72). Part of the surface is covered with a superconductor (M 0,

Majorana states

x

y

Defect potential

Ener

gy le

vels

5U0

U0

0

U0

0

Figure 4

Emergence of a pair of zero-energyMajorana states in a model calculation of a chiral p-wave superconductor containing an electrostaticline defect. The gap closes and reopens as the defect potential U0 dU is made more and more negative, at a fixed positive backgroundpotential U0. The inset shows the probability density of the zero mode in the 2D plane of the superconductor, with the line defect alongthe y-axis. Figure adapted from Reference 64.

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D 0) and part is covered with amagnetic insulator (D 0,M 0). Both D andM open a gap forthe surface states. The gap closes at theM D interface and also at theMM interface betweenopposite magnetic polarizations (fromM > 0 toM < 0). The 1D interface states propagate ina single direction only, much like the chiral edge states of the quantum Hall effect.

TheMM interface leaves electrons and holes uncoupled, so there are twomodes at the Fermilevel, one containing electrons and one containing holes. These are Dirac fermions, with distinctcreation and annihilation operators a and a. The electron-hole degeneracy is broken at theMDinterface, which supports only a single mode g g at the Fermi level. TheMMD tri-junctionsplits an electron or hole into two Majorana fermions:

a12g1 ig2, a

12g1 ig2. 7:

The inverse process, the fusion of two Majorana fermions into an electron or hole, happensafter the Majorana fermions have encircled the superconductor and picked up a relative phaseshift. At the Fermi level, this phase shift is entirely determined by the parity of the number n ofvortices in the superconductor. For even n, the fusion conserves the charge, whereas for odd n,a charge 2e is added as a Cooper pair to the superconductor (so an electron is converted into a hole

U0

+

U (m

2/2

)

U0+

U

U0

W

0 4 8 12

12

8

4

0

U0 (m2/ 2)

= +1

= 1

Figure 5

The green solid curves locate the closing of the excitation gap of an electrostatic line defect (width W 4Z/mD) in a chiral p-wavesuperconductor, described by the Hamiltonian (Equation 4). In the shaded regions, the defect is topologically nontrivial (Q 1), withMajorana states bound to the two ends. Figure adapted from Reference 64.


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and vice versa). The twoprocesses can be distinguished experimentally bymeasuring the current toground in the superconductor. Because each vortex binds oneMajorana fermion (seeFigure 1), thestructure of Figure 6 can be seen as an electrical interferometer in which mobile Majoranasmeasure the parity of the number of Majorana bound states.

2.4. Semiconductor Heterostructures

The gap inversion in a topological insulator happens without superconductivity. One might al-ternatively try to directly invert the superconducting gap. In an s-wave superconductor, a magneticfield closes the gap, but how is it reopened? A promising strategy is to rely on the competing effectsof a spin-polarizing Zeeman energy and a depolarizing spin-orbit coupling (28, 29, 73).

The 2D electron gas of a semiconductor heterostructure, such as an InAs quantum well, hasa strong spin-orbit coupling from theRashba effect. The orbital effect of a parallel magnetic field issuppressed, leaving only the spin-polarizing Zeeman effect. These two effects compete in theHamiltonian

H0 p2

2mU aso

Z

sxpy sypx

1

2geff mB Bsx. 8:

Characteristic length and energy scales are lso Z2/maso and Eso ma2so=Z2. Typical values inInAs are lso 100nm,Eso 0.1meV. TheZeeman energy isEZ 12geffmBB 1meVat amagneticfieldB 1T.A superconducting proximity effectwith a type-II superconductor such asNb is quitepossible at these field strengths. The pair potential D induced in the 2D electron gas then coupleselectrons and holes via the Bogoliubov-DeGennesHamiltonian (Equation 6) (nowwithH0 givenby Equation 8).

Majorana fermionsElectrons or holes

VM

M

Figure 6

Dirac-to-Majorana fermion converter on the surface of a 3D topological insulator. Arrows indicate thepropagating modes at the interface between a superconductor and a magnetic insulator and at the magnet-magnet interface. An electron (Dirac fermion) injected by the voltage source at the right is split into a pair ofMajorana fermions. These fuse at the left, either back into an electron or into a hole, depending onwhether thesuperconductor contains an evenor anoddnumberof vortices.The recombination as a hole adds aCooper pairto the superconductor, which can be detected in the current to ground. Figure adapted from Reference 72.

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As discovered in References 31 and 32, the resulting band gap in a nanowire geometry closesand reopens upon variation of electron density (through a variation of U) or magnetic field(see Figure 7). Majorana bound states at the two ends of the nanowire alternately appear anddisappear at each of these topological phase transitions.

3. HOW TO DETECT THEM

Majorana fermions modify the transport properties and the thermodynamic properties of thesuperconductor, providing ways to detect them. I summarize some of the signature effects ofMajorana fermions; others can be found in References 19, 71, 72, and 7485.

3.1. Half-Integer Conductance Quantization

Tunneling spectroscopy is a direct method for detecting a Majorana bound state (86, 87, 88):Resonant tunneling into the midgap state produces a conductance of 2e2/h, whereas without thisstate the conductance vanishes. A complication in the interpretation of tunneling spectroscopy isthat the zero-bias peak may be obscured by resonances from subgap states at nonzero energy(89, 90). A ballistic point contact provides a more distinctive signature of the topologicallynontrivial phase (91), through the half-integer conductance plateaus shown in Figure 8.

0.0510

0

1010

0

1010

0

1010

0

10Trivial Transitional Nontrivial

= 5 = 5 = 7.625 = 7.625

= 0 = 0 = 11 = 11

= 1.223 = 1.223 = 15.9 = 15.9

= 4 = 4 = 17 = 17

= 5 = 7.625

= 0 = 11

= 1.223 = 15.9

= 4 = 17

0.05 0.05klSO

E/E S

O

0.05 0.05 0.05klSO

Figure 7

Closing and reopening of the band gap in the Hamiltonian (Equation 8) of a 2D semiconducting nanowire(widthW lso) on a superconducting substrate in a parallelmagnetic field (D10Eso,Ez10.5Eso). The eightpanels show the excitation energy near the Fermi level (E 0) as a function of the wave vector k alongthe nanowire, for different values of the chemical potential m[ U (listed in units of the spin-orbit couplingenergy Eso). Blue or tan shading indicates that the system is in a topologically trivial or nontrivial phase,respectively. The topological phase transition occurs in the unshadedpanels. The nanowire supportsMajoranabound states in the tan-shaded panels. Data supplied by M. Wimmer, unpublished data.


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Both the tunneling and ballistic conductances can be understood from the general relation (92)between the conductanceG of a normal-superconductor (NS) junction and the Andreev reflectioneigenvalues Rn:

G 2e2

h

XnRn. 9:

TheRns represent the probability for Andreev reflection in the nth eigenmode at the Fermi level.The factor of two is not due to spin (which is included in the sum over n), but due to the fact thatAndreev reflection of an electron into a hole doubles the current.

There is no time-reversal symmetry, so Kramers degeneracy does not apply. Still, particle-holesymmetry requires that anyRn is twofold degenerate [Bri degeneracy (93)]with two exceptions:Rn 0 andRn 1may be nondegenerate. The nondegenerate Andreev reflection eigenvalue fromaMajorana bound state is pinned to unity, contributing to the conductance a quantized amount of2e2/h. All other fully Andreev-reflected modes are twofold degenerate and contribute 4e2/h. Theresulting conductance plateaus therefore appear at integer or half-integer multiples of 4e2/h,depending on whether the superconductor is topologically trivial or nontrivial.

Theplateausat (n 1/2)3 4e2/h are reminiscent of the quantumHall plateaus in graphene, andboth originate froma zeromode, but the sensitivity to disorder is entirely different. The topologicalquantum numberQ2Z for the quantumHall effect andQ2Z2 for a topological superconductor.The corresponding topological protection against disorder extends to all plateaus for the quantumHall effect but only to the lowest n 0 plateau for the topological superconductor.

Normal

Nontrivial

Trivial

4

3

2

1

1/2

3/2

5/2

7/2S

N

60 80 100 120402000

1

2

3

4

EF VQPC (ESO)

G (4

e2 /

h)

Figure 8

(Solid curves) Conductance of a ballistic normal-superconductor (NS) junction, with the superconductor ina topologically trivial or nontrivial phase. The dashed gray curve indicates an entirely normal system. The dataare calculated from the model Hamiltonian (Equation 8). The point contact width is varied by varying thepotentialVQPC inside the constriction at constant Fermi energyEF. The dashedhorizontal lines indicate the shiftfrom integer to half-integer conductance plateaus upon transition from the topologically trivial to nontrivialphase. Figure adapted from Reference 91.

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It may appear paradoxical (94, 95) to have an electrical current flowing through a singleMajorana bound state, given that one Majorana fermion operator g represents only half of anelectronic state. However, the Hermitian operator i(a a)g is a local coupling of Dirac andMajorana operators (87), so electrical conduction can be a fully local processinvolving only oneof the two spatially separated Majorana fermions.

3.2. Nonlocal Tunneling

Nonlocal conduction involving both Majorana fermions becomes possible if there is a couplingbetween them (19, 88, 96101). The coupling term has the generic form iEMg1g2 , with eigenvalues6EM. The energy EMmay be a tunnel coupling due to overlap of wave functions, in which case itdecays exponentially } ed=j0 with the ratio of the separation d of the Majoranas and thesuperconducting coherence length j0. If the superconductor is electrically isolated (not grounded)and of small capacitance C, then the charging energy EMx e

2/C provides a Coulomb couplingeven without overlap of wave functions. (Recall that the two states of a pair of Majoranas aredistinguished by the presence or absence of an unpaired quasiparticle; see Section 1.3.)

Nonlocal tunneling processes appear if the level splitting EM is large compared to the levelbroadening G1, G2. For a grounded superconductor the nonlocality takes the form of nonlocalAndreev reflection, which amounts to a splitting of a Cooper pair by the two Majorana boundstates (19) (seeFigure 9). TheCooper pair splitting can be detected in a noisemeasurement througha positive cross-correlation of the currents I1 and I2 to the left and right of the superconductor.

For an electrically isolated superconducting island, any charge transfer onto the island is for-bidden by the charging energy, so there can be no Andreev reflection. An electron incident on oneside of the island is either reflected to the same side or transmitted, still as an electron, to the other

Electron

HoleHole

magnetEM

+EM

EM

1I1 I2magnet

2superconductor

E

0

Majorana-bound state

Majorana-bound state

Figure 9

Majorana bound states (red) at the edge of a 2D topological insulator (cf. Figure 2), split into a pair of levels at6EM by a nonzero overlap.The levels are broadened due to a tunnel coupling G1, G2 through the magnet to the outside edge state. An electron incident from theleft on the grounded superconductor can be Andreev reflected as a hole, either locally (to the left) or nonlocally (to the right). NonlocalAndreev reflection is equivalent to the splitting of a Cooper pair by the two Majoranas. For G1, G2 EM, local Andreev reflection issuppressed. Figure adapted from Reference 19.


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side. The nonlocality (96) now appears in the ratio of the reflection and transmission probabilitieson resonance:

R=T G1 G22=G1 G22, 10:

which is independent of the size of the island.Nomatter how far the twoMajoranas are separated,the charging energy couples them into a single electronic level. For identical tunnel couplings G1G2 in particular, the electron is resonantly transmitted through the island with unit probability.

3.3. 4p-Periodic Josephson Effect

So far, I have discussed signatures ofMajoranas in the electrical conduction out of equilibrium, inresponse to a voltage difference between the superconductor and a normal-metal electrode. Inequilibrium, an electrical current (supercurrent) can flow between two superconductors in theabsence of any applied voltage. This familiar DC Josephson effect (102) acquires a new twist(9, 18, 31, 32, 103112) if the junction between the superconductors contains Majoranafermions, as in Figure 2.

Quite generally, the supercurrent IJ is given by the derivative

IJ 2eZ

dEdf

11:

of the Josephson junctions energy E, with respect to the superconducting phase difference f.Although in the conventional Josephson effect only Cooper pairs can tunnel (with probabilityt 1), Majorana fermions enable the tunneling of single electrons (with a larger probabilityt

p). The switch from 2e to e as the unit of transferred charge between the superconductors

amounts to a doubling of the fundamental periodicity of the Josephson energy, from E } cos fto E } cos (f/2).

If the superconductors form a ring, enclosing a fluxF, the period of the flux dependence of thesupercurrent IJ doubles from 2p to 4p as a function of the Aharonov-Bohm phase 2eF/Z. This isthe 4p-periodic Josephson effect (9, 103). As a function of the enclosed flux, IJ has the same h/eperiodicity as the persistent current IN through a normal-metal ring (radius L), but the size de-pendence is entirely different: Whereas IN decays as 1/L or faster, IJ has the L-independence ofa supercurrent.

Because the twobranchesof theE-f relation differ by one unpaired quasiparticle (see Figure 2),external tunneling events that change the quasiparticle parity (so-called quasiparticle poisoning)restore the conventional 2p-periodicity (18). In a closed system, the 4p-periodicity is thermo-dynamically stable, provided that the entire ring is in a topologically nontrivial state (to preventquantum phase slips) (108).

3.4. Thermal Metal-Insulator Transition

Collective properties of Majorana fermions can be detected in the thermal conductance. Super-conductors are thermal insulators, because the excitation gapD suppresses the energy transport byquasiparticle excitations at low temperaturesT0D/kB. Disorder can create states in the gap, butthese are typically localized. However, the Majorana midgap states in a topological supercon-ductor can give rise to extended states, given that they are all resonant at the Fermi level. Thistransforms a thermal insulator into a thermal metal (113, 114).

The thermal metal-insulator transition is called a class D Anderson transition, in referenceto a classification of disordered systems in terms of the presence or absence of time-reversal,

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spin-rotation, and particle-hole symmetry (115117). Class D has only particle-hole symmetry.The chiral p-wave superconductor is a two-dimensional system in class D. Its thermal transportproperties in the absence ofMajorana fermions are similar to the electrical transport properties inthe quantumHall effect (class A; all symmetries broken). The bulk is insulatingwhile the boundarysupports chiral (unidirectional) edge states that give rise to the thermal quantumHall effect (7, 8).The thermal analog of the conductance quantum e2/h is G0 p2k2BT0=6h.

The correspondence between thermal and electrical quantum Hall effect breaks down in thepresence ofMajorana fermions. Their collective effect is illustrated inFigure 10, obtained from theHamiltonian (Equation 4) of a chiral p-wave superconductor (64, 118). [Similar results have beenobtained in other models of Majorana fermions (119121).] A randomly varying electrostaticpotential creates a random arrangement of Majorana midgap states, via the Shockley mechanismof Figure 4. The states are slightly displaced from E 0 by the overlap of wave functions. The

0 0.05 0.10 0.05 0.100.2

0.3

0.4

0.5

0.6

0.7

E/U

(E)

a

2/

U

ln|E |

ln

L

1040.2

0.4

0.6

0.8

1.0

103 102

E/U

a2/U

10 1 1000

0.4

0.8

1.2

a b1.4

101

M E T A L

I N S U L A T O R

L/

/G

0

Figure 10

(Main panel and inset b) Average density of states r in a model calculation of a chiral p-wave superconductor with electrostatic disorder(64). The Hamiltonian (Equation 4) is discretized on a lattice of size 400 a3 400 a, and the potential fluctuates randomly from site to site(root-mean-square DU). Majorana fermions produce a midgap peak in the density of states. (Inset a) Average thermal conductivity s ina strip geometry of length L and width W 5L, for the same Hamiltonian (Equation 4) but calculated with a different method ofdiscretization (118). Data points of different color correspond to different disorder strengths DU and different scattering lengths j. Uponincreasing disorder, a transition from insulating tometallic scaling is observed. In themetallic phase, the conductivity and density of stateshave a logarithmic dependence on, respectively, system size and energy.


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resulting density of states has the logarithmic profile } ln jEj, responsible for the logarithmic sizedependence of the thermal conductivity (7):

s G0=p lnL constant. 12:

The thermal metal-insulator transition of Figure 10 has no electrical analog.

4. HOW TO USE THEM

FindingMajorana fermions in a superconductor is rewarding in and of itself. These particlesmightalso provide a fundamentally new way to store and manipulate quantum information, withpossible applications in a quantum computer.

4.1. Topological Qubits

In Section 1.3, we considered a qubit formed out of a pair ofMajorana fermions. The two states j0and j1of this elementary qubit differ by quasiparticle parity, which prevents the creation of a co-herent superposition. For a quantum computation we combine two elementary qubits into a singlelogical qubit, consisting of fourMajorana fermions (24). Without loss of generality one can assumethat the joint quasiparticle parity is even. The two states of the logical qubit are then encoded as j00and j11. These two stateshave the samequasiparticleparity, so coherent superpositionsareallowed.

An arbitrary state jCof the logical qubit has the form

jC aj00 bj11, jaj2 jbj2 1: 13:

Pauli matrices in the computational basis j00, j11 are bilinear combinations of Majoranaoperators:

sx ig2g3, sy ig1g3, sz ig1g2. 14:

It is said that the qubit (Equation 13) is topologically protected from decoherence by the envi-ronment (9), because the bit-flip or phase-shift errors produced by the Paulimatrices (Equation 14)can only appear if there is a coupling between pairs of Majorana fermions. The two types ofcoupling were discussed in Section 3.2: tunnel coupling when theMajoranas are separated by lessthan a coherence length and Coulomb coupling when the Majoranas are on a superconductingisland of small capacitance.

The topological protection relies on the presence of a nonzero gap for quasiparticle excitations.Subgap excitations may exchange a quasiparticle with a Majorana fermion, provoking a bit-fliperror. Error correction is possible if the subgap excitations remain bound to the Majorana fer-mion; in particular, subgap excitations in a vortex core are not a source of decoherence (122124).The topological protection does not apply if the superconductor is contacted by a gapless metal,allowing for the exchange of unpaired electrons (quasiparticle poisoning) (125, 126).

4.2. Readout

To read out a topological qubit one needs to remove the topological protection by coupling theMajorana fermions and then measure the quasiparticle parity. Tunnel coupling is one option. Forexample in the geometry of Figure 2, the quasiparticle parity of two Majorana fermions can bemeasured by the difference in tunnel splitting (18). The alternative Coulomb coupling allows thejoint readout ofmore than a single qubit (127129). I concentrate on this second option, given thattwo-qubit readout is required for quantum computations (24).

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Consider a superconducting island (charge Q, superconducting phase f) containing 2NMajorana fermions. Cooper pairs can enter and leave the island via a Josephson junction. Thereadout operation amounts to a measurement ofQmodulo 2e. The conventional even-odd parityeffect of the superconducting ground state does not apply here, because there is no energy cost ofD for an unpaired electron in amidgap state. Indeed, the quasiparticle parityP does not enter in theHamiltonian directly but as a constraint on the eigenstates (96, 130):

cf eipPcf 2p, P 12 12iNg1g2g2N . 15:

This constraint enforces that the eigenvalues of the charge operatorQ2ei d/df are even or oddmultiples of e when P equals 0 or 1, respectively.

The parity constraint (Equation 15) modifies the energy of f1f 2p quantum phase slips ofthe superconducting island, induced by the nonzero charging energy EC e2/2C. The P de-pendence can be measured spectroscopically in a SQUID geometry (127) or in a Cooper pair box(129). I show the latter geometry in Figure 11.

The magnitude of the P-dependent energy shift DE in the Cooper pair box is exponentiallysensitive to the ratio of Josephson and charging energies:

DE 2P 1U, UxZvpEJ

qeZvp=EJ . 16:

(The frequency vp 8ECEJ

p=Z is the Josephson plasma frequency.) By varying the flux F

through a split Josephson junction, the Josephson energy EJ } cos(eF/Z) becomes tunable. In thetransmon design of the Yale group, a variation of EJ/EC over two orders of magnitude has been

Topological qubit

Cooper pair box

Superconducting transmission line

Majoranafermions

Figure 11

Read out of a topological qubit in a Cooper pair box. Two superconducting islands (brown), connected bya split Josephson junction (Xmarks), form theCooper pair box.The topological qubit is formedby twopairs ofMajorana fermions (red dots), at the end points of two undepleted segments (dark blue) of a semiconductornanowire (gray-shaded ribbon indicates the depleted region). A magnetic flux F enclosed by the Josephsonjunction controls the charge sensitivity of the Cooper pair box. To read out the topological qubit, one pair ofMajorana fermions is moved onto the other island. Depending on the quasiparticle parity, the resonancefrequency in a superconducting transmission line (light blue) enclosing the Cooper pair box is shifted upwardor downward by the amount given in Equation 16. Figure adapted from Reference 129.


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realized (131). The Coulomb coupling U of the Majorana fermions can therefore be switched onand off by varying the flux.

4.3. Braiding

In the 2D geometry of Figure 1 the Majorana bound states can be exchanged by moving thevortices around (14). TheMajorana fermions in the 1D geometry of Figure 11 are separated byinsulating regions on a single nanowire, so they cannot be exchanged [at least not without

3

3

a

b

21

213

3

1

0

1 2

2

31 12 23

1111

22

222

11

333

333333

Figure 12

(b) Three Cooper pair boxes connected at a tri-junction via three overlappingMajorana fermions (red circles) [which effectively producea single zeromode g0 31=2g01 g02 g03 at the center]. (a) Schematic of the three steps of the braiding operation. The fourMajoranasof the tri-junction (the three outerMajoranas g1, g2, g3 and the effective central Majorana g0) are represented by circles and the couplingUk is represented by lines (solid in the on state, dashed in the off state). Gray circles indicate strongly coupled Majoranas and coloredcircles those with a vanishingly small coupling. The small diagram above each arrow shows an intermediate stage, with one Majoranadelocalized over three coupled sites. The three steps together exchange theMajoranas 1 and 2, which is a non-Abelian braiding operation.Figure adapted from Reference 136.

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rotating the wire itself (132)]. The exchange of Majorana fermions, called braiding, is neededto demonstrate their non-Abelian statistics (8). It is also an essential ingredient of a topo-logically protected quantum computation (23). To be able to exchange the Majoranas, onecan use a second nanowire, running parallel to the first and connected to it by side branches(133, 134).

TheminimalHamiltonian that can describe the braiding contains threeMajorana fermions g1,g2, g3, coupled to a fourth one g0:

H X3k1

Ukig0gk. 17:

bb

a

0 200 400 200

0.1

0.2

0.3

0.4

0.5

V (V)

1 m

NN

BInSb

SS

dI/d

V (2

e2/h

)

400

Figure 13

(a) InSbwire between a normal-metal (N) and a superconducting electrode (S). A barrier gate creates a confinedregion (red) at the interface with the superconductor. Other gates are used to locally vary the electron density.A magnetic field B is applied parallel to the wire. (b) Differential conductance at 60 mK for B incrementingfrom 0 to 490mT in 10mT steps. (Traces are offset for clarity, except for the lowest trace at B 0.) The peaksat 6250meV correspond to the gap induced in the wire by the superconducting proximity effect. UponincreasingB, a peakdevelops at zero voltage, signaling the appearanceof aMajorana zeromode in the confinedregion. Figure adapted from Reference 149.


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The three parameters Uk 0 can describe tunnel coupling (135) (tunable by a gate voltage) orCoulombcoupling (136) (tunable by the flux through a Josephson junction). A tri-junction of threeCooper pair boxes that is described by this Hamiltonian is shown in Figure 12.

The braiding operation consists of three steps, denotedO31, O12, and O23. At the beginning andat the end of each step, two of the couplings are off and one coupling is on. The stepOkk0 consists ofthe sequence {k,k} {on ,off }1{on, on}1 {off, on}. The effect of this sequence is to transfer theuncoupled Majorana gk01 gk. (The minus sign conserves the quasiparticle parity.) The resultafter the three steps shown in Figure 12 is that the Majoranas at sites 1 and 2 are switched, witha difference in sign, g21 g1, g11g2. The corresponding adiabatic time evolution operator inthe Heisenberg representation gk1 UgkU is given by

U 12

p 1 g1g2 expp4g1g2

exp

ip

4sz

. 18:

This is the operator of Equation 3, representing a non-Abelian exchange operation.

5. OUTLOOK ON THE EXPERIMENTAL PROGRESS

Aswe have seen in Section 2, there is no shortage of proposals for superconducting structures thatshould bind a Majorana zero mode to a magnetic vortex or electrostatic defect. This gives much

0 100 200 300 200 1000

0.5

1.0

1.5

V (V)

dI/d

V (2

e2/h

)

300

Figure 14

Model calculation for the experiment of Figure 13, based on the Hamiltonian (Equation 8). The nanowire hasfour orbital modes, with the highest mode producing the Majorana bound state. The closing of the gap inthe highest mode is indicated by the red dashed lines. (This precursor of the topological phase transition is toosmall to be visible in the experimental data.) Figure supplied byM.Wimmer & A.R. Akhmerov (unpublisheddata).

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hope for a variety of experimental demonstrations in the coming years. There has already beenremarkable progress.

A Josephson effect at the surface of a 3D topological insulator with superconducting electrodeshas been observed in BiSb alloys (137), and in crystalline Bi2Se3 (138141) and Bi2Te3 (142, 143).These experiments, and related Andreev conductance measurements (144147), all involve wideelectrodes with a macroscopic number of occupied modes at the Fermi level. Although theJosephson effect and theAndreev conductance show interesting andunusual features, these cannotbe readily attributed to the single Majorana zero mode (typically only one out of 105 modes).Vortices (as in Figure 1) or other means of confinement would be needed to produce a Majoranabound state.

The edge of a 2D topological insulator provides a single-mode conductor that could supportspatially separated Majorana bound states, as in Figure 2. A superconducting proximity effecthas been observed in an InAs/GaSb quantum well (148), and HgTe/CdTe would be a promisingsystemif theMajorana can be confined to the superconducting interface by amagnetic insulatoror magnetic field.

As ofMarch 2012, when this review was written, semiconductor nanowires had come furthestin the realization of Majorana fermions, following the proposals of Lutchyn et al. (31) and Oreget al. (32). Convincing evidence for aMajorana zero mode in an InSb nanowire has been reportedby Kouwenhoven and his group (149; see Figures 13 and 14). These developments give hope thatthe rich variety of unusual properties of Majorana fermions, reviewed in this article, will soon beobserved experimentally.

DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or financial holdings thatmightbe perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

My own research on Majorana fermions was done in collaboration with A.R. Akhmerov,M. Burrello, T-P. Choy, J.P. Dahlhaus, J.M. Edge, F. Hassler, B. van Heck, C-Y. Hou,M.V. Medvedyeva, J. Nilsson, J. Tworzydo, and M. Wimmer. Support by the DutchScience Foundation NWO/FOM and by an ERC Advanced Investigator Grant is gratefullyacknowledged.

LITERATURE CITED

1. Majorana E. 1937. Nuovo Cimento 14:17184. Transl. L Maiani, 1981, in Soryushiron Kenkyu63:14962 (from Italian)

2. Wilczek F. 2009. Nat. Phys. 5:614183. Gingras MJP. 2009. Science 326:375764. Service RF. 2011. Science 332:193955. Kopnin NB, Salomaa MM. 1991. Phys. Rev. B 44:9667776. Volovik G. 1999. JETP Lett. 70:609147. Senthil T, Fisher MPA. 2000. Phys. Rev. B 61:9690988. Read N, Green D. 2000. Phys. Rev. B 61:10267979. Kitaev AYu. 2001. Phys. Usp. 44(Suppl.):13136

10. Das Sarma S, Nayak C, Tewari S. 2006. Phys. Rev. B 73:220502R11. Jackiw R, Rossi P. 1981. Nucl. Phys. B 190:68191


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Annual Review of

Condensed Matter

Physics

Volume 4, 2013 Contents

Why I Havent RetiredTheodore H. Geballe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Quantum Control over Single Spins in DiamondV.V. Dobrovitski, G.D. Fuchs, A.L. Falk, C. Santori,and D.D. Awschalom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Prospects for Spin-Based Quantum Computing in Quantum DotsChristoph Kloeffel and Daniel Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Quantum Interfaces Between Atomic and Solid-State SystemsNikos Daniilidis and Hartmut Hffner . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Search for Majorana Fermions in SuperconductorsC.W.J. Beenakker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Strong Correlations from Hunds CouplingAntoine Georges, Luca de Medici, and Jernej Mravlje . . . . . . . . . . . . . . . 137

Bridging Lattice-Scale Physics and Continuum Field Theory withQuantum Monte Carlo SimulationsRibhu K. Kaul, Roger G. Melko, and Anders W. Sandvik . . . . . . . . . . . . 179

Colloidal Particles: Crystals, Glasses, and GelsPeter J. Lu ( ) and David A. Weitz . . . . . . . . . . . . . . . . . . . . . . . 217

Fluctuations, Linear Response, and Currents in Out-of-Equilibrium SystemsS. Ciliberto, R. Gomez-Solano, and A. Petrosyan . . . . . . . . . . . . . . . . . . . 235

Glass Transition Thermodynamics and KineticsFrank H. Stillinger and Pablo G. Debenedetti . . . . . . . . . . . . . . . . . . . . . 263

Statistical Mechanics of Modularity and Horizontal Gene TransferMichael W. Deem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

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Physics of Cardiac ArrhythmogenesisAlain Karma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Statistical Physics of T-Cell Development and Pathogen SpecificityAndrej Kosmrlj, Mehran Kardar, and Arup K. Chakraborty . . . . . . . . . . 339

Errata

Anonline log of corrections toAnnual Review of CondensedMatter Physics articlesmay be found at http://conmatphys.annualreviews.org/errata.shtml

Contents vii

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Annual Reviews OnlineSearch Annual ReviewsAnnual Review of Condensed Matter PhysicsOnlineMost Downloaded Condensed Matter PhysicsReviews Most Cited Condensed Matter PhysicsReviews Annual Review of Condensed Matter PhysicsErrata View Current Editorial Committee

All Articles in the Annual Review of Condensed Matter Physics, Vol. 4 Why I Havent RetiredQuantum Control over Single Spins in DiamondProspects for Spin-Based Quantum Computing in Quantum DotsQuantum Interfaces Between Atomic and Solid-State SystemsSearch for Majorana Fermions in SuperconductorsStrong Correlations from Hunds CouplingBridging Lattice-Scale Physics and Continuum Field Theory with Quantum Monte Carlo SimulationsColloidal Particles: Crystals, Glasses, and GelsFluctuations, Linear Response, and Currents in Out-of-Equilibrium SystemsGlass Transition Thermodynamics and KineticsStatistical Mechanics of Modularity and Horizontal Gene TransferPhysics of Cardiac ArrhythmogenesisStatistical Physics of T-Cell Development and Pathogen Specificity

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