JournalofthePhysicalSocietyofJapan INVITEDREVIEWPAPERS Majorana Fermions and Topology in Superconductors MasatoshiSato1 andSatoshi Fujimoto2 6 1 0 1Yukawa InstituteforTheoreticalPhysics, Kyoto University,Kyoto 606-8502,Japan 2 2DepartmentofMaterialsEngineeringScience, Osaka University,Toyonaka, Osaka n u 560-8531,Japan J 2 Topological superconductors are novel classes of quantum condensed phases, characterized ] n by topologically nontrivial structures of Cooper pairing states. On the surfaces of samples o c and in vortex cores of topological superconductors, Majorana fermions, which are particles - r p identified with their own anti-particles, appear as Bogoliubov quasiparticles. The existence u s and stabilityofMajoranafermionsare ensured by bulktopologicalinvariantsconstrainedby . t a thesymmetriesofthesystems.Majoranafermionsintopologicalsuperconductorsobeyanew m - typeofquantumstatisticsreferredtoasnon-Abelianstatistics,whichisdistinctfromboseand d n fermistatistics,andcanbeutilizedforapplicationtotopologicalquantumcomputation.Also, o c Majorana fermions give rise to various exotic phenomena such as “fractionalization”, non- [ local correlation, and “teleportation”. A pedagogical review of these subjects is presented. 2 v We also discuss interaction effects on topological classification of superconductors, and the 6 2 basicpropertiesofWeyl superconductors. 7 2 0 KEYWORDS: topological superconductor, unconventional superconductivity, Majorana fermion, non- . 1 Abelianstatistics,Andreevboundstate,quantumcomputation,Weylsuperconductor 0 6 1 : v 1. Introduction i X Topology in condensed matter physics has a long history. It plays an important role in r a the classification of topological defects in condensed matter systems, such as vortices, dis- locations, and disclinations, i.e. non-trivial textures in real space configurations. In 1982, a milestonewas achieved by Thouless, Kohmoto, den Nijs and Nightingale, who found an in- timaterelation between topological invariantsand the Hall conductivityin the quantum Hall effect.1,2) This was the first example of topological nontriviality realized in bulk quantum solid state systems, which is the origin of the notion of topological phases. In contrast to topological textures in real space mentioned above, topological phases are characterized by nontrivialityin the Hilbert space of quantum states. In the last decade, remarkable advances have been achieved in this direction. After pioneering works by Haldane3) and Volovik4) for 1/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS the quantum Hall effect state and the superfluid 3He, respectively, Kane-Mele’s celebrated papers initiated the exploration of topological phases in band insulators.5,6) Since then, the notion of topological phases has been extended to various other systems, including super- conductors, magnets, and correlated electron systems.7–9) For topological superconductors, a nontrivial structure arises from the phase winding of superconducting order parameters in momentumspace.Thiscanberegardedasanaturalextensionofavortexofthesuperconduct- ingordertomomentumspace.Oneofthemostimportantconsequenceofsuchtopologically nontrivial structures in superconductors is the existence of Majorana fermions, which are zero-energyBogoliubovquasiparticles.10–13) Becauseofparticle-holesymmetryofsupercon- ducting states, a zero-energy single-particle state must be the equal-weight superposition of anelectronandahole.ThisimpliesthattheHermitianconjugateofthisstateisthesameasit- self;i.e.aparticleisidenticaltoan anti-particle,whichisasignatureofaMajoranafermion. In topological superconductors, the Majorana zero-energy state is realized as an Andreev boundstateatthesurfacesofsamplesandinvortexcores.Animportantpointhereisthatthe Majorana zero-energy state is protected by the bulk topological non-triviality of the Hilbert space(themomentumspace),andisnotaffectedbyextrinsicfactorssuchasconditionsofsur- faces, impurities,and crystal imperfection. Thisis in contrast to Andreev zero-energy bound staterealized at thesurface of d-wavesuperconductors, which is sensitiveto thedirection of thesurface. Majoranafermionsintopologicalsuperconductorsgiverisetovariousexoticphenomena. In this review, we discuss several representative phenomena: (i) non-Abelian statistics, (ii) “fractionalization” and the 4π-periodic Josephson effect, (iii) nonlocal correlation, and (iv) thermal responses. Non-Abelian statistics is a novel quantum statistics distinct from Fermi and Bose statistics.14–17) Thus, Majorana fermions in this context are not usual fermions. Also,theyaredifferentfromAbeliananyonsinthefractionalquantumHalleffect, whichare characterizedbyfractionalcharge.Themostimportantfeatureofnon-Abelianstatisticsisthat theexchangeoperations ofparticles are not commutative,and thestatedepends on theorder of the exchange operations. This peculiar property stems from the topological degeneracy associated with Majorana fermions. It has been proposed that particles obeying non-Abelian statistics can be utilized for the realization of fault-tolerant quantum computation, which is expectedtobedramaticallyrobustagainstdecoherencefromtheenvironment.Thisschemeis calledtopologicalquantumcomputation.18)Becauseofthepossibletechnologicalapplication in the future as well as the importance of its fundamental concept, the non-Abelian statistics is one of the most intriguing features of Majorana fermions. However, its experimental real- 2/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS izationanddetection arestillthemostimportantopen issue. Majorana fermions in topological superconductors behave as if they emerge from the splitting of electrons into two parts.19) In fact, in the second quantized language, an elec- tron field is complex, while a Majorana field is real, and two Majorana real fields can be combined into a complex fermion field for an electron (or a hole). Thus, the emergence of Majoranafermionsintopologicalsuperconductorscan beregardedas the”fractionalization” of electrons. Actually, the fractionalized character of Majorana fermions appears as the 4π- periodic Josephson effect.20,21) In this effect, the Josephson tunnel current mediated via two Majoranafermions carries charge e, not 2e: i.e., a Cooper pair with charge 2e splitsinto two CooperpairsofMajoranafermionswithchargee.Thefractionalizedcharacteralsoresultsin the non-local correlation of two Majorana fermions.22–25) Two spatially well separated Ma- jorana fermions exhibit a certain type of long-range correlation which is independent of the distance between them, and a phenomenon similar to teleportation can occur. These distinct features ofMajoranafermions,fractionalizationand non-localcorrelation, havenot yet been experimentallyestablished. As mentioned before, a Majorana fermion in topological superconductors is realized as theequal-weightsuperpositionofanelectronandahole.Thus,itstopologicalcharacterdoes notappearinchargetransport.However,sinceenergyisconserved,heattransportphenomena can be used for the characterization of topological responses. In fact, the quantum thermal Hall effect can occur in a topological superconductor with broken time reversal symmetry. The heat current is carried by Majorana surface states. At sufficiently low temperatures, the T-linear coefficient of thethermal Hall conductivityis quantized, reflecting thetotal number ofchiral Majoranamodescarrying theheat current. The organization of this paper is as follows. In Sect. 2, we introduce the basic concepts of topological superconductors, and in Sect. 3 we discuss the classification of topological phasesofsuperconductorsbasedonthesymmetryofsystems,andpresenttopologicalinvari- ants characterizing distincttopological phases. We also discuss theclassification of topolog- ical defects such as a vortex which plays an important role in superconductors. In Sect. 4, we explain the fundamental properties of topological superconductors, particularly focusing onthecasewithbrokentime-reversalsymmetry,forwhichtypicalfeaturessuchasMajorana surface states emerge. We also present various scenarios for the realization of topological superconductors in materials. In Sect. 5, we overview candidate materials of topological su- perconductors,mentioningthecurrentstateofexperimentalresearches.FromSect.6toSect. 9, we discuss exotic phenomena caused by Majorana fermions. In Sect. 6, we review the 3/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS Table I. VariousAndreevboundstatesandcorrespondingtopologicalnumbers. Energydispersion Chiral Helical Conical Flat (E =ck ) (E = ck ) (E = c k2+k2) (E =0) y ± y ± x y q Topologicalnumber TKNNnumber Kane-Mele’sZ number 3Dwindingnumber 1Dwindingnumber 2 Relatedmaterials Sr RuO NoncentrosymmetricSCs 3He-B High-T cuprates 2 4 c (a) E (b) E (c) (d) E ky ky E ky ky kx Fig. 1. (Coloronline)Edgeandsurfacestatesintopologicalsuperconductors.(a)ChiralMajoranaedgemode. (b)HelicalMajoranaedgemode.(c)HelicalsurfaceMajoranafermion.(d)Flatedgemode. non-Abelian statistics of Majorana fermions in topological superconductors. We introduce the basic ideas of the non-Abelian statistics of Majorana fermions, and discuss possible ex- perimental detection schemes for this intriguing phenomenon. In Sect. 7, we consider the 4π-periodic Josephson effect, and in Sect. 8, we overview the non-local correlation effects of Majorana fermions such as ”teleportation”. In Sect. 9, the thermal responses of Majorana fermions are discussed. In Sect. 10, we present an elementary introduction to topological quantum computation utilizing Majorana fermions. In Sect. 11, the recent development of our understanding of interaction effects in topological superconductors is reviewed. In Sect. 12,weoverviewthebasicpropertiesofWeylsuperconductivity,whichisanothertopological phase of superconductors, characterized by the existence of Weyl fermions as bulk gapless quasiparticles.ThechiralanomalyassociatedwithWeylfermionsgivesrisetovariousexotic phenomena. 2. TopologicalSuperconductors 2.1 Andreevboundstates Throughthestudyofunconventionalsuperconductivity,ithasbeenrecognizedthatsome unconventional superconductors may support surface bound states called Andreev bound state.26) For instance, chiral p-wave superconductors, which are considered to be realized in Sr RuO , host Andreev bound states with a linear dispersion, as illustrated in Fig. 1(a). Be- 2 4 foretherecentstudyoftopologicalsuperconductors,however,suchAndreevboundstateshad 4/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS been understood, on a case-by-case basis, as a result of interference between quasiparticles due to the Andreev reflection. Recent development of topological superconductors provides a unified topological viewpoint for these Andreev bound states, which has been obtained by using analogy between the Andreev bound states and edge states in quantum Hall effects.27) Indeed, it has been known that quantum Hall states also have gapless edge states similar to theAndreev boundstatesinchiral p-wavesuperconductors.4,10) For quantum Hall states, the existence of edge states has been understood as a conse- quence of the intrinsic topology of the system.28) First, for a bulk quantum Hall state with- out a boundary, the Hall conductance σ is given by a topological number ν called xy TKNN theThouless-Kohmoto-denNijs- Nightingale(TKNN) number(or the first Chern numberin mathematics),1,2) e2 σ = ν , (1) xy TKNN h where e is the unit charge of an electron and h is the Planck constant. On the other hand, if thequantumHallstatehasa boundary,theHallconductanceis givenby e2 σ = N, (2) xy h with N the number of edge states on the boundary.27) Because these two different equations expressthesamequantumHalleffect,thebulktopologicalnumberν shouldbethesame TKNN as N, ν = N. (3) TKNN Thus, we can understand the existence of edge states, i.e., a nonzero value of N, as being a result of a nonzero value of ν . Generally, such a relation between bulk topological TKNN numbersandgaplessboundarystatesis called “bulk-boundarycorrespondence”. Remembering the similarity between Andreev bound states and edge states in quantum Hall states, one can naturally expect a similar correspondence for the Andreev bound states. Actually, even for chiral p-wave superconductors, we can define the TKNN number for the bulksystemswithoutboundaries,whichyields ν = 1 (or ν = 2 ifthespindegrees TKNN TKNN | | | | offreedomaretakenintoaccount).SimilarlytoquantumHallstates,theexistenceoftheedge statein chiral p-wave superconductorscan beexplainedbytheintrinsictopologyofthebulk systems.4,10) Depending on the symmetryof the system, Andreev bound states may have different en- ergy dispersions from that in Fig. 1(a). From the viewpoint of the bulk-boundary correspon- dence, these differences result from differences in the corresponding topological numbers. 5/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS Forinstance,Andreevboundstateswithaflatdispersionexistonthe(110)surfaceofhigh-T c cuprates(Fig.1(d)).29–31) Inthiscase,thecorrespondingtopologicalnumberisnottheTKNN number but the one-dimensional (1D) winding number.32) Furthermore, on surfaces of the superfluid 3He-B, Andreev bound states appear with conical dispersion (Fig. 1(c)),33) whose corresponding topological number is the three-dimensional (3D) winding number.34–36) We summarize the relation between various Andreev bound states and the corresponding topo- logicalnumbersinTableI. Superconductorswithnonzerobulktopologicalnumbersarecalledtopologicalsupercon- ductors.The existenceof gaplessAndreev bound states on theirsurfaces is one ofthe pieces ofdirect evidencefortopologicalsuperconductivity. 2.2 Majoranafermions As mentioned above, there is a similarity between Andreev bound states and edge states inquantumHallstates.What isthedifferencebetween them? ThemostimportantdifferenceisthatAndreevboundstatesdonotcarry definitecharges, whereas edge states in quantum Hall states do. In superconducting states, an electron can becomeaholebytheformationofaCooperpair(seeFig.2).Therefore,fermionicexcitations in the superconducting state are naturally expressed as a superposition of an electron c (x) σ andaholec (x), †σ c (x) ↑   SinceanelectronandaholehaveaΨn(oxp)p=ositeccc†↓†↑↓c(((hxxxa)))rge., thequasiparticlecannothaveadefin(i4te) charge. Thewavefunctionin Eq.(4)satisfies Ψ (x) = τ Ψ(x), (5) ∗ x whereτ is thePaulimatrixin theNambuspace. This equationimpliesthat thequasiparticle x Fig. 2. Anelectron(blackcircle)becomesahole(whitecircle)bytheformationofaCooperpair. 6/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS Table II. Symmetry of superconductors (SCs) and the relevant topological numbers. The third to fifth columnsindicate the absence (0) or presence ( 1) of time-reversalsymmetry (TRS), particle-hole symmetry ± (PHS),andchiralsymmetry(CS),respectively,where denotesthesignof 2and 2.Thetopologicalinvari- ± T C ants,Z(γgeom),Z(TKNN),Z(γgeom/2),Z(3dW),andZ(1dW)aregivenbyEqs.(17),(19),(20),(23),and(38),respectively. 2 2 Z(KM)istheKane-Mele’sZ number. 2 2 AZclass TRS PHS CS d =1 d =2 d =3 SpinfulorSpinlessSC D 0 +1 0 Z(γgeom) Z(TKNN) 0 2 SpinfulSCwithTRS DIII -1 +1 1 Z(γgeom/2) Z(KM) Z(3dW) 2 2 SpinfulSCwithSU(2)-SRS C 0 -1 0 0 2Z(TKNN) 0 SpinfulSCwithSU(2)-SRS+TRS CI +1 -1 1 0 0 2Z(3dW) SpinlessSCwithTRS BDI +1 +1 1 Z(1dW) 0 0 Ψ(x) is essentially the same as its antiparticle Ψ (x). Such a self-conjugate property is not ∗ seen in ordinary fermions. This self-conjugate condition is called the Majorana condition becauseaclassoffermionsnamed Majoranafermionssatisfiesthiscondition. Majorana fermions are Dirac fermions satisfying the self-conjugate property. Originally, they were introduced as an elementary particle.37) As discussed in the above, all quasipar- ticles in superconductors, i.e., even those in conventional s-wave superconductors, satisfy the Majorana condition. Furthermore, surface Andreev bound states have linear dispersions, andthustheireffectiveHamiltonianis givenby themasslessDiracHamiltonian.Hence, An- dreev bound states can naturally be considered as Majorana fermions in condensed matter physics.10,38) 3. Symmetry andTopology In this section, we discuss the classification of topological superconductors based on the symmetryofsystems.35,39–41)Aswillbeseenbelow,particle-holesymmetryandtime-reversal symmetry play crucial roles. We also present topological invariants which characterize dis- tincttopologicalphases.In Sects.3.2 and3.3,wediscussgenericspinfulcases, andconsider the case with spin-rotation symmetry in Sect. 3.4 and the spinless (or fully spin-polarized) case in Sect. 3.5. The case with additional crystal symmetry such as mirror reflection sym- metryisdiscussedinSects. 3.7and 3.8. 7/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS 3.1 BdG Hamiltonian Generally,theHamiltonianofelectronsin superconductorsisgivenby 1 = (k)c c + ∆ (k)c c +h.c. , (6) H Eαβ †kα kβ 2 αβ †kα †kβ Xαβk Xαβk (cid:16) − (cid:17) where c (c ) is the annihilation (creation) operator of an electron with momentum k, and kα †kα thesuffixαlabelsinternaldegreesoffreedomforthefermionsuchasspin,orbit,andsoforth. The first term on the right-hand side is the Hamiltonian of electrons in the normal state, and thesecondtermappearsinthesuperconductingstatebecauseoftheformationofCooperpairs withagapfunction∆ (k).Heretheanticommutationrelationofc yields∆(k) = ∆t( k). αβ †kα − − TheaboveHamiltonianis calledtheBogoliubov-deGennes (BdG)Hamiltonian. TheBdG Hamiltoniancan bewrittenin thefollowingmatrixform: 1 = Ψ (k) (k)Ψ(k), (7) † H 2 H X k with (k) ∆(k) c kα (k) = E , Ψ(k) = , (8)     where the summaHtionovert∆he†(ikn)dex−αEti(s−ikm)pliαcβitin Eq. (7).Wce†−hkαave neglected the constant term caused by the anticommutation relation between c and c since it merely shifts the kα †kα originoftheenergy. PerformingtheFouriertransformation, 1 Ψ(k) = eikxΨ(x), (k) = eikx (x), (9) √V Xx H Xx H wealsohavetheBdG Hamiltonianin thecoordinatespace, 1 = Ψ (x) (x y)Ψ(y) (10) † H 2 H − Xxy Thisformindicatesthatfermionicexcitationsofasuperconductoraregivenasasuperposition ofanelectronandhole,Ψ(x)inEq.(4),asmentionedinSect.2.2.Thequasi-particlespectrum inthesuperconductingstateisobtainedbysolvingtheeigenequationoftheBdGHamiltonian (k)u (k) = E (k)u (k) . (11) n n n H | i | i 3.2 Particle-holesymmetry(charge-conjugationsymmetry) As discussed in Sect.2.2, the wave function Ψ(x) satisfies the self-conjugate condition in Eq.(5). Correspondingly, the BdG Hamiltonian in Eq.(10) should be invariant under the replacement of Ψ (x) and Ψ(k) with their conjugates Ψt(x)τt and τ Ψ (x), respectively. Be- † x x ∗ 8/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS cause transformsas H 1 Ψt(x)τt (x y)τ Ψ (y) H → 2 xH − x ∗ Xx,y 1 1 = Ψ (x) τt (y x)τ tΨ(y)+ tr (x) −2 † xH − x 2 H Xx,y (cid:2) (cid:3) Xx (12) underthereplacement, thisrequirement imposesaconstrainton (x), H τt (x)τ = t( x). (13) xH x −H − Here note that the last term in Eq.(12) vanishes if (x) satisfies Eq.(13). In the momentum H space, theconstraintisgivenas τt (k)τ = t( k). (14) xH x −H − Usingthehermiticityof (k), Eqs.(13)and (14)arealso writtenas H (x) 1 = (x), (k) 1 = ( k), (15) − − CH C −H CH C −H − where = τ K withthecomplexconjugateoperator K. Theconstrainton theBdG Hamilto- x C nian in Eq. (15) is called as the particle-hole symmetry. We can verify directly that (k) in H Eq.(8)actuallyhastheparticle-holesymmetry.Thissymmetryisoneofthefundamentalsym- metries in the Altland-Zirnbauer classification scheme,41) and systems supporting particle- holesymmetryare classified as classD. From the particle-hole symmetry, one can show that a positive energy state is always pairedwithanegativeenergystate.Indeed, fromasolution u (k) ofEq.(11)withapositive n | i (negative)energy E (k) > 0 [E (k) < 0],onecan obtainasolutionwithmomentum k as n n − u ( k) = u (k) , (16) n n | − − i C| i whichhas anegative(positive)energy E (k). n − In one dimension, the paired structure of the spectrum enables us to define a topologi- cal number.42) The paired structure implies that the gauge field constructed from negative- energy states, ( )(k) = i u (k)∂ u (k) , is not independent of that constructed A− En(k)<0h n | k n i from positive-energy states,P (+)(k) = i u (k)∂ u (k) , which leads to the relation A En(k)>0h n | k n i ( )(k) = (+)( k). Therefore, the geomePtrical phase of A( )(k) along the 1D Brillouin zone − − A A − (BZ), γ = dk ( )(k), (17) geom − I A BZ 9/81 J.Phys. Soc. Jpn. INVITEDREVIEWPAPERS isrecast into 1 γ = dk (+)(k)+ ( )(k) geom − 2 I A A BZ h i i = dk u (k)∂ u (k) n k n 2 I h | i BZ Xn i = dktr U 1(k)∂ U(k) − k 2 I BZ h i i = dk∂ ln[detU(k)], (18) k 2 I BZ where U (k) is given by the m-th component of u (k) . The U(1) phase of detU(k) con- mn n | i tributes to the line integral, and thus the uniqueness of detU(k) in the BZ yields γ = πN geom with an integer N. Finally, taking into account the 2π ambiguityof γ , we havetwo phys- geom ically different values of γ , i.e., γ = 0 and π, which define the Z topological in- geom geom 2 variant for 1D superconductors. The system is topologically trivial (non-trivial) if γ = 0 geom (γ = π). geom As mentioned in Sect.2.1, in two dimensions, class D superconductors are topologically characterized by theTKNNinteger, 1 ν = d2k ( )(k), (19) TKNN 2π Z Fxy− BZ where ( )(k) is thefield strengthof thegaugefield ( )(k) = i u (k)∂ u (k) . On Fxy− Aµ− En(k)<0h n | kµ n i theotherhand,notopologicallynontrivialclassDsuperconductoPrexistsinthreedimensions. 3.3 Time-reversalsymmetryandchiralsymmetry Similarly to topologicalinsulators, the presence of time-reversalsymmetry also enriches thetopologicalstructureinsuperconductors.Thequasiparticlestate u(k) hasaKramerspart- | i ner u( k) withtime-reversaloperator , andwecan definevarioustopologicalnumbers, T| − i T using the Kramers degeneracy. The Kane-Mele’s Z invariant for quantum spin Hall states 2 alsocharacterizesthetopologyoftwo-dimensional(2D)time-reversalinvariantsuperconduc- tors,43–45) but the coexistence of particle-hole symmetry gives other topological numbers as well. In one dimension, we can generalize the Z topological invariant introduced in Sect. 2 3.2.46) Although γ in Eq.(17) itself is always trivial because each component of the geom Kramers doublet equally contributes to γ , one can avoid it by taking only one state geom for each Kramers pair to calculate the geometric phase. Namely, for the Kramers pair 10/81