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Topological Superconductors ISSP, The University of Tokyo, Masatoshi Sato 2 Outline 1. What is topological superconductor 2. Topological superconductors in various systems 3 What is topological superconductor ? Topological superconductors Bulk: gapped state with non-zero topological # Boundary: gapless state with Majorana condition 4 Bulk: gapped by the formation of Cooper pair In the ground state, the one-particle states below the fermi energy are fully occupied. 5 Topological # can be defined by the occupied wave function empty band occupied band Hilbert space of occupied state Entire momentum space Topological # = “winding number” 6 A change of the topological number = gap closing gap closing A discontinuous jump of the topological number Therefore, Vacuum ( or ordinary insulator) Topological SC Gapless edge state 7 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) 224511 . 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 1st Chern # 3D winding # (TKNN82, Kohmoto85) Z2 number (Kane-Mele 06, Qi et al (08)) (Schnyder et al (08)) 1+1D chiral edge mode 1+1D helical edge mode 2+1D helical surface fermion Sr2RuO4 Noncentosymmetric SC (MS-Fujimto(09)) 3He B 9 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 10 How about the Majorana condition ? The Majorana condition is imposed by superconductivity quasiparticle in Nambu rep. quasiparticle anti-quasiparticle Majorana condition [Wilczek , Nature (09)] 11 Topological superconductors Bulk: gapped state with non-zero topological # Bulk-edge Boundary: correspondence gapless Majorana fermion 12 A representative example of topological SC: Chiral p-wave SC in 2+1 dimensions BdG Hamiltonian [Read-Green (00)] spinless chiral p-wave SC with chiral p-wave 13 Topological number = 1st Chern number TKNN (82), Kohmoto(85) MS (09) 14 Edge state SC Fermi surface Spectrum 2 gapless edge modes (left-moving , right moving, on different sides on boundaries) Majorana fermion Bulk-edge correspondence 15 • There also exist a Majorana zero mode in a vortex We need a pair of the zero modes to define creation op. vortex 2 vortex 1 non-Abelian anyon topological quantum computer 16 Ex.) odd-parity color superconductor Y. Nishida, Phys. Rev. D81, 074004 (2010) color-flavor-locked phase two flavor pairing phase 17 For odd-parity pairing, the BdG Hamiltonian is 18 (A) With Fermi surface Topological SC • Gapless boundary state • Zero modes in a vortex (B) No Fermi surface Non-topological SC c.f.) MS, Phys. Rev. B79,214526 (2009) MS Phys. Rev. B81,220504(R) (2010) 19 Phase structure of odd-parity color superconductor Non-Topological SC Topological SC There must be topological phase transition. 20 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, 020401 (09) ; MS-Takahashi-Fujimoto, Phys. Rev. B82, 134521 (10) (Editor’s suggestion), J. Sau et al, PRL (10), J. Alicea PRB (10) 21 Majorana fermion in spin-singlet SC MS, Physics Letters B535 ,126 (03) ① 2+1 dim Dirac fermion + s-wave Cooper pair vortex Zero mode in a vortex [Jackiw-Rossi (81), Callan-Harvey(85)] With Majorana condition, non-Abelian anyon is realized [MS (03)] 22 On the surface of topological insulator Bi1-xSbx [Fu-Kane (08)] Hsieh et al., Nature (2008) Dirac fermion + s-wave SC S-wave SC Nishide et al., PRB (2010) Bi2Se3 Hsieh et al., Nature (2009) Topological insulator Spin-orbit interaction => topological insulator 23 2nd scheme of Majorana fermion in spin-singlet SC ② s-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 Gapless edge states x y Majorana fermion For a single chiral gapless edge state appears like p-wave SC ! Chern number nonzero Chern number 25 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