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
自分の論文の引用

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.
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