3What is topological superconductor OutlineWhat is topological superconductorTopological superconductors in various systems
4What is topological superconductor ? Topological superconductorsBulk:gapped state withnon-zero topological #Boundary:gapless state with Majorana condition
5Bulk: gapped by the formation of Cooper pair In the ground state, the one-particle states below the fermi energy are fully occupied.
6Topological # can be defined by the occupied wave function empty bandoccupied bandEntire momentum spaceHilbert space of occupied stateTopological # = “winding number”
7( or ordinary insulator) A change of the topological number = gap closinggap closingA discontinuous jump of the topological numberTherefore,Vacuum( or ordinary insulator)Topological SCGapless edge state
8Bulk-edge correspondence If bulk topological # of gapped system is non-trivial, there exist gapless states localized on the boundary.厳密な証明もできる [MS et al (11)]For rigorous proof , see MS et al, Phys. Rev. B83 (2011)
9different bulk topological # = different gapless boundary state 2+1D time-reversal breaking SC2+1D time-reversal invariant SC3+1D time-reversal invariant SC1st Chern #(TKNN82, Kohmoto85)Z2 number(Kane-Mele 06, Qi et al (08))3D winding #(Schnyder et al (08))1+1D chiral edge mode1+1D helical edge mode2+1D helical surface fermionSr2RuO4Noncentosymmetric SC (MS-Fujimto(09))3He B
10The gapless boundary state = Majorana fermion Dirac fermion with Majorana conditionDirac HamiltonianMajorana conditionBoundary state は Majorana fermionになることを書くparticle = antiparticleFor the gapless boundary states, they naturally described by the Dirac Hamiltonian
11How about the Majorana condition ? The Majorana condition is imposed by superconductivityquasiparticle in Nambu rep.quasiparticleanti-quasiparticleMajorana condition[Wilczek , Nature (09)]
12Topological superconductors Bulk-edge correspondenceBulk:gapped state withnon-zero topological #Boundary:gapless Majorana fermion
13A representative example of topological SC: Chiral p-wave SC in 2+1 dimensions[Read-Green (00)]BdG Hamiltonianspinless chiral p-wave SCwithchiral p-wave
14Topological number = 1st Chern number TKNN (82), Kohmoto(85)MS (09)
15Bulk-edge correspondence Edge stateSCFermi surface2 gapless edge modes (left-moving , right moving, on different sides on boundaries)SpectrumBoundary conditionを示す図がほしいMajorana fermionBulk-edge correspondence
16There also exist a Majorana zero mode in a vortex We need a pair of the zero modes to define creation op.vortex 2vortex 1non-Abelian anyontopological quantum computer
17Ex.) odd-parity color superconductor Y. Nishida, Phys. Rev. D81, (2010)color-flavor-locked phasetwo flavor pairing phase
18For odd-parity pairing, the BdG Hamiltonian is 自分の論文の引用
19(A) With Fermi surface Topological SC (B) No Fermi surface Gapless boundary stateZero modes in a vortex(B)No Fermi surface\muによる相転移を指摘Non-topological SCc.f.) MS, Phys. Rev. B79, (2009) MS Phys. Rev. B81,220504(R) (2010)
20Phase structure of odd-parity color superconductor Non-Topological SCTopological SCThere must be topological phase transition.
21Until recently, only spin-triplet SCs (or odd-parity SCs) had been known to be topological. Is it possible to realize topological SC in s-wave superconducting state?Yes !Odd-parity SC 以外ではtopological SC は可能かMS, Physics Letters B535 ,126 (03), Fu-Kane PRL (08)MS-Takahashi-Fujimoto ,Phys. Rev. Lett. 103, (09) ; MS-Takahashi-Fujimoto, Phys. Rev. B82, (10) (Editor’s suggestion), J. Sau et al, PRL (10), J. Alicea PRB (10)
22Majorana fermion in spin-singlet SC MS, Physics Letters B535 ,126 (03)2+1 dim Dirac fermion + s-wave Cooper pairvortexZero mode in a vortex[Jackiw-Rossi (81), Callan-Harvey(85)]With Majorana condition, non-Abelian anyon is realized[MS (03)]
23Topological insulator On the surface of topological insulator[Fu-Kane (08)] Bi1-xSbxHsieh et al., Nature (2008)Dirac fermion+ s-wave SCS-wave SCTopological insulatorNishide et al., PRB (2010)Bi2Se3Hsieh et al., Nature (2009)まずは、spin-singlet superconductorであっても、マヨラナフェルミオンが実現可能であることを説明したいと思います。Spin-orbit interaction=> topological insulator
242nd scheme of Majorana fermion in spin-singlet SC s-wave SC with Rashba spin-orbit interaction[MS, Takahashi, Fujimoto PRL(09) PRB(10)]Rashba SOp-wave gap is induced by Rashba SO int.
25a single chiral gapless edge state appears like p-wave SC ! Gapless edge statesxyMajorana fermionForvortexの話も書くa single chiral gapless edge state appears like p-wave SC !Chern numbernonzero Chern number
26SummaryTopological SCs are a new state of matter in condensed matter physics.Majorana fermions are naturally realized as gapless boundary states.Topological SCs are realized in spin-triplet (odd-parity) SCs, but with SO interaction, they can be realized in spin-singlet SC as well.文献を書く