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Topological Superconductors. ISSP, The University of Tokyo, Masatoshi Sato. Outline. What is topological superconductor T opological superconductors in various systems. What is topological superconductor ?. Topological superconductors . Bulk : gapped state with
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Topological Superconductors ISSP, The University of Tokyo, Masatoshi Sato
Outline What is topological superconductor Topological superconductors in various systems
What is topological superconductor ? Topological superconductors Bulk: gapped state with non-zero topological # Boundary: gapless state with Majorana condition
Bulk: gapped by the formation of Cooper pair In the ground state, the one-particle states below the fermi energy are fully occupied.
Topological # can be defined by the occupied wave function empty band occupied band Hilbert space of occupied state Entire momentum space Topological #= “winding number”
A change of the topological number = gap closing gap closing A discontinuousjump of the topological number Therefore, Vacuum ( or ordinary insulator) Topological SC Gapless edge state
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
The gapless boundary state = Majorana fermion MajoranaFermion Dirac fermion with Majorana condition Dirac Hamiltonian Majorana condition particle = antiparticle For the gapless boundary states, they naturally described by the Dirac Hamiltonian
How about the Majorana condition ? The Majorana conditionis imposed by superconductivity quasiparticle in Nambu rep. quasiparticle anti-quasiparticle Majorana condition [Wilczek , Nature (09)]
Topological superconductors Bulk-edge correspondence Boundary: gaplessMajoranafermion Bulk: gapped state with non-zero topological #
A representative example of topological SC: Chiral p-wave SC in 2+1 dimensions [Read-Green (00)] BdG Hamiltonian spinlesschiral p-wave SC with chiral p-wave
Topological number = 1stChernnumber TKNN (82), Kohmoto(85) MS (09)
Edge state SC Fermisurface 2 gapless edge modes(left-moving , right moving, on different sides on boundaries) Spectrum Majorana fermion Bulk-edge correspondence
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-Abeliananyon topological quantum computer
Ex.) odd-parity color superconductor Y. Nishida, Phys. Rev. D81, 074004 (2010) color-flavor-locked phase two flavor pairing phase
(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)
Phase structure of odd-parity color superconductor Non-Topological SC Topological SC There must be topological phase transition.
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 ! MS, Physics Letters B535 ,126 (03), Fu-Kane PRL (08) 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)
Majoranafermionin spin-singlet SC MS, Physics Letters B535 ,126 (03) 2+1 dim Dirac fermion + s-wave Cooper pair vortex Zero mode ina vortex [Jackiw-Rossi (81), Callan-Harvey(85)] With Majorana condition, non-Abeliananyon is realized [MS (03)]
On the surface of topological insulator [Fu-Kane (08)] Bi1-xSbx Hsieh et al., Nature (2008) + s-wave SC Dirac fermion Topological insulator S-wave SC Nishide et al., PRB (2010) Bi2Se3 Hsieh et al., Nature (2009) Spin-orbit interaction => topological insulator
2nd scheme of Majorana fermion in spin-singlet SC 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.
Gapless edge states x y Majoranafermion For a single chiral gapless edge state appears like p-wave SC ! Chern number nonzero Chernnumber
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.