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Topological Superconductors ISSP, The University of Tokyo, Masatoshi Sato

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3 Outline 1.What is topological superconductor 2.Topological superconductors in various systems

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4 What is topological superconductor ? Topological superconductors Bulk: gapped state with non-zero topological # Boundary: gapless state with Majorana condition

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5 Bulk: gapped by the formation of Cooper pair In the ground state, the one-particle states below the fermi energy are fully occupied.

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6 Topological # can be defined by the occupied wave function Topological # = “winding number” Entire momentum space Hilbert space of occupied state empty band occupied band

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A change of the topological number = gap closing A discontinuous jump of the topological number Vacuum ( or ordinary insulator) Topological SC Gapless edge state 7 Therefore, gap closing

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Bulk-edge correspondence If bulk topological # of gapped system is non-trivial, there exist gapless states localized on the boundary. For rigorous proof, see MS et al, Phys. Rev. B83 (2011)

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different bulk topological # = different gapless boundary state 2+1D time-reversal breaking SC 2+1D time-reversal invariant SC 3+1D time-reversal invariant SC 1 st Chern # (TKNN82, Kohmoto85) Z 2 number (Kane-Mele 06, Qi et al (08)) 3D winding # (Schnyder et al (08)) 1+1D chiral edge mode 1+1D helical edge mode 2+1D helical surface fermion Sr 2 RuO 4 Noncentosymmetric SC (MS-Fujimto(09)) 3 He B 9

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10 The gapless boundary state = Majorana fermion Majorana Fermion Dirac fermion with Majorana condition 1.Dirac Hamiltonian 2.Majorana condition particle = antiparticle For the gapless boundary states, they naturally described by the Dirac Hamiltonian

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11 How about the Majorana condition ? The Majorana condition is imposed by superconductivity [ Wilczek, Nature (09) ] Majorana condition quasiparticleanti-quasiparticle quasiparticle in Nambu rep.

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12 Topological superconductors Bulk: gapped state with non-zero topological # Boundary: gapless Majorana fermion Bulk-edge correspondence

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A representative example of topological SC: Chiral p-wave SC in 2+1 dimensions 13 BdG Hamiltonian with chiral p-wave spinless chiral p-wave SC [Read-Green (00)]

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14 Topological number = 1 st Chern number TKNN (82), Kohmoto(85) MS (09)

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Fermi surface Spectrum 15 SC 2 gapless edge modes (left-moving, right moving, on different sides on boundaries) Edge state Bulk-edge correspondence Majorana fermion

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There also exist a Majorana zero mode in a vortex We need a pair of the zero modes to define creation op. vortex 1 vortex 2 non-Abelian anyon topological quantum computer 16

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Ex.) odd-parity color superconductor Y. Nishida, Phys. Rev. D81, (2010) color-flavor-locked phase two flavor pairing phase 17

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For odd-parity pairing, the BdG Hamiltonian is 18

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(B) Topological SC Non-topological SC Gapless boundary state Zero modes in a vortex (A) With Fermi surface No Fermi surface c.f.) MS, Phys. Rev. B79, (2009) MS Phys. Rev. B81,220504(R) (2010) 19

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20 Phase structure of odd-parity color superconductor Non-Topological SC Topological SC There must be topological phase transition.

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21 Until 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 ! A)MS, Physics Letters B535,126 (03), Fu-Kane PRL (08) B)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)

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22 Majorana fermion in spin-singlet SC ①2+1 dim Dirac fermion + s-wave Cooper pair Zero mode in a vortex With Majorana condition, non-Abelian anyon is realized [Jackiw-Rossi (81), Callan-Harvey(85)] [MS (03)] MS, Physics Letters B535,126 (03) vortex

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On the surface of topological insulator [Fu-Kane (08)] Spin-orbit interaction => topological insulator Topological insulator S-wave SC Dirac fermion+ s-wave SC Bi 2 Se 3 Bi 1-x Sb x 23 Hsieh et al., Nature (2008) Nishide et al., PRB (2010) Hsieh et al., Nature (2009)

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2nd scheme of Majorana fermion in spin-singlet SC ②s-wave SC with Rashba spin-orbit interaction [MS, Takahashi, Fujimoto PRL(09) PRB(10)] Rashba SO p-wave gap is induced by Rashba SO int. 24

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Gapless edge states x y a single chiral gapless edge state appears like p-wave SC ! Chern number nonzero Chern number For 25 Majorana fermion

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Summary Topological 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. 26

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