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Topological Superconductors

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Presentation on theme: "Topological Superconductors"— Presentation transcript:

1 Topological Superconductors
ISSP, The University of Tokyo, Masatoshi Sato


3 What is topological superconductor
Outline What is topological superconductor Topological superconductors in various systems

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

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

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

7 ( or ordinary insulator)
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

8 Bulk-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)

9 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 # (TKNN82, Kohmoto85) Z2 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 Sr2RuO4 Noncentosymmetric SC (MS-Fujimto(09)) 3He B

10 The gapless boundary state = Majorana fermion
Dirac fermion with Majorana condition Dirac Hamiltonian Majorana condition Boundary state は Majorana fermionになることを書く particle = antiparticle For the gapless boundary states, they naturally described by the Dirac Hamiltonian

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

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

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

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

15 Bulk-edge correspondence
Edge state SC Fermi surface 2 gapless edge modes (left-moving , right moving, on different sides on boundaries) Spectrum Boundary conditionを示す図がほしい Majorana fermion Bulk-edge correspondence

16 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

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

18 For odd-parity pairing, the BdG Hamiltonian is

19 (A) With Fermi surface Topological SC (B) No Fermi surface
Gapless boundary state Zero modes in a vortex (B) No Fermi surface \muによる相転移を指摘 Non-topological SC c.f.) MS, Phys. Rev. B79, (2009) MS Phys. Rev. B81,220504(R) (2010)

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

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 ! 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)

22 Majorana fermion in spin-singlet SC
MS, Physics Letters B535 ,126 (03) 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)]

23 Topological insulator
On the surface of topological insulator [Fu-Kane (08)]  Bi1-xSbx Hsieh et al., Nature (2008) Dirac fermion + s-wave SC S-wave SC Topological insulator Nishide et al., PRB (2010) Bi2Se3 Hsieh 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

24 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.

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

26 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. 文献を書く

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