1. Topology of Andreev bound state
ISSP, The University of Tokyo, Masatoshi Sato

2. In collaboration with Satoshi Fujimoto, Kyoto University
Yoshiro Takahashi, Kyoto University
Yukio Tanaka, Nagoya University
Keiji Yada, Nagoya University
Akihiro Ii, Nagoya University
Takehito Yokoyama, Tokyo Institute for Technology

3. Outline Andreev bound state
“Edge (or Surface) state” of superconductors
Part I. Andreev bound state as Majorana fermions
Part II. Topology of Andreev bound states with
flat dispersion

4. Part I. Andreev bound state as Majorana fermions

5. What is Majorana fermion
Dirac fermion with Majorana condition
Dirac Hamiltonian
Majorana condition
particle = antiparticle
Originally, elementary particles.
But now, it can be realized in superconductors.

6. chiral p+ip–wave SC
[Read-Green (00), Ivanov (01)]
analogues to quantum Hall state = Dirac fermion on the edge
[Volovik (97), Goryo-Ishikawa(99),Furusaki et al. (01)]
chiral edge state B
1dim (gapless) Dirac fermion
Majorana condition is imposed by superconductivity
TKNN # = 1

7. Majorana zero mode in a vortex
creation = annihilation ?
We need a pair of the vortices to define creation op.
vortex 2
vortex 1
non-Abelian anyon
topological quantum computer

8. uniqueness of chiral p-wave superconductor
spin-triplet Cooper pair
full gap unconventional superconductor
no time-reversal symmetry
Question: Which property is essential for Majorana fermion ?
Answer: None of the above .

9. Spin-orbit interaction is indispensable !
Majorana fermion is possible in spin singlet superconductor
MS, Physics Letters B (03), Fu-Kane PRL (08),
MS-Takahashi-Fujimoto PRL (09) PRB (10), J.Sau et al PRL (10), Alicea PRB(10) ..
Majorana fermion is possible in nodal superconductor
MS-Fujimoto PRL (10)
Time-reversal invariant Majorana fermion
Tanaka-Mizuno-Yokoyama-Yada-MS, PRL (10)
MS-Tanaka-Yada-Yokoyama, PRB (11)
Spin-orbit interaction is indispensable !

10. Majorana fermion in spin-singlet SC
2+1 dim odd # of Dirac fermions + s-wave Cooper pair
Majorana zero mode on a vortex
[MS (03)]
Non-Abelian statistics of Axion string
[MS (03)]
On the surface of topological insulator
[Fu-Kane (08)]
 Bi1-xSbx
Bi2Se3
Spin-orbit interaction
=> topological insulator

11. Majorana fermion in spin-singlet SC (contd.)
s-wave SC with Rashba spin-orbit interaction
[MS, Takahashi, Fujimoto (09,10)]
Rashba SO
p-wave gap is induced by Rashba SO int.

12. a single chiral gapless edge state appears like p-wave SC !
Gapless edge states x y
Majorana fermion
For
a single chiral gapless edge state appears like p-wave SC !
Chern number
Similar to quantum Hall state
nonzero Chern number

13. strong magnetic field is needed
a) s-wave superfluid with laser generated Rashba SO coupling
[Sato-Takahashi-Fujimoto PRL(09)]
b) semiconductor-superconductor interface
[J.Sau et al. PRL(10)
J. Alicea, PRB(10)]
c) semiconductor nanowire on superconductors ….

14. Majorana fermion in nodal superconductor
[MS, Fujimoto (10)]
Model: 2d Rashba d-wave superconductor
Rashba SO
Zeeman
dx2-y2 –wave gap function
dxy –wave gap function

15. Edge state
dx2-y2 –wave gap function x y
dxy –wave gap function

16. Majorana zero mode on a vortex
Zero mode satisfies Majorana condition!
Non-Abelian anyon
The zero mode is stable against nodal excitations
1 zero mode on a vortex
4 gapless mode from gap-node
From the particle-hole symmetry, the modes become massive in pair.
Thus at least one Majorana zero mode survives on a vortex

17. The non-Abelian topological phase in nodal SCs is characterized by the parity of the Chern number
There exist an odd number of gapless Majorana fermions
There exist an even number of gapless Majorana fermions
+ nodal excitation
+ nodal excitation
No stable Majorana fermion
Topologically stable Majorana fermion

18. How to realize our model ?
2dim seminconductor on high-Tc Sc
(a) Side View
(b) Top View
dxy-wave SC
Zeeman field
Semi Conductor
d-wave SC
dx2-y2-wave SC

19. Time-reversal invariant Majorana fermion
[Tanaka-Mizuno-Yokoyama-Yada-MS PRL(10)
Yada-MS-Tanaka-Yokoyama PRB(10)
MS-Tanaka-Yada-Yokoyama PRB (11)]
Edge state
time-reversal invariance
time-reversal invariance

20. dxy+p-wave Rashba superconductor
Majorana fermion
[Yada et al. (10) ]
The spin-orbit interaction is indispensable
No Majorana fermion

21. Summary (Part I)
With SO interaction, various superconductors become topological superconductors
Majorana fermion in spin singlet superconductor
Majorana fermion in nodal superconductor
Time-reversal invariant Majorana fermion

22. Part II. Topology of Andreev bound state

23. Bulk-edge correspondence
Gapless state on boundary should correspond to bulk topological number
Chern # (=TKNN #)
Chiral Edge state

24. different type ABS = different topological #
chiral
helical
Cone
Chern # (TKNN (82))
Z2 number (Kane-Mele (06))
3D winding number (Schnyder et al (08))
Sr2RuO4
Noncentosymmetric SC (MS-Fujimto(09))
3He B

25. Which topological # is responsible for Majorana fermion with flat band ?

26. The Majorana fermion preserves the time-reversal invariance, but without Kramers degeneracy
Chern number = 0
Z2 number = trivial
3D winding number = 0
All of these topological number cannot explain the Majorana fermion with flat dispersion !

27. Symmetry of the system Particle-hole symmetry Time-reversal symmetry
Nambu rep. of quasiparticle
Time-reversal symmetry

28. Combining PHS and TRS, one obtains
Chiral symmetry
c.f.) chiral symmetry of Dirac operator

29. The chiral symmetry is very suggestive
The chiral symmetry is very suggestive. For Dirac operators, its zero modes can be explained by the well-known index theorem.
Number of zero mode with chirality +1
Number of zero mode with chirality -1
2nd Chern #= instanton #

30. Indeed , for ABS, we obtain the generalized index theorem
Number of flat ABS with chirality +1
Superconductor
Number of flat ABS with chirality -1
ABS
Generalized index theorem
[MS et al (11)]

31. Atiya-Singer index theorem Our generalized index theorem
Dirac operator
General BdG Hamiltonian with TRS
Topology in the coordinate space
Topology in the momentum space
Zero mode localized on soliton in the bulk
Zero mode localized on boundary

32. Topological number
Integral along the momentum perpendicular to the surface
Periodicity of Brillouin zone

33. To consider the boundary, we introduce a confining potential V(x)
vacuum
Superconductor

34. Introduce an adiabatic parameter 𝑎 in the Planck’s constant
Strategy
Introduce an adiabatic parameter 𝑎 in the Planck’s constant
original value of Planck’s constant
Prove the index theorem in the semiclassical limit
Adiabatically increase the parameter 𝑎 as 𝑎→1

35. In the classical limit 𝑎=0,
vacuum
Superconductor
Gap closing point
=> zero energy ABS

36. Around the gap closing point,
Replacing 𝑘 𝑥 with −𝑖ℏ 𝜕 𝑥 in the above, we can perform the semi-classical quantization , and construct the zero energy ABS explicitly.
From the explicit form of the obtained solution, we can determine its chirality as

37. We also calculate the contribution of the gap-closing point to topological # ,
The total contribution of such gap-closing points should be the same as the topological number 𝑊( 𝒌 ∥ ) ,
Because each zero energy ABS has the chirality Γ=sgn[ 𝜂 𝑖 ]
Index theorem
(but ℏ≪1)

38. Now we adiabatically increase 𝑎 →1 (ℏ→ ℏ 0 )
𝑛 + − 𝑛 − is adiabatic invariance
Non-zero mode should be paired
Thus, the index theorem holds exactly

39. dxy+p-wave SC
Thus, the existence of Majorana fermion with flat dispersion is ensured by the index theorem

40. remark
It is well known that dxy-wave SC has similar ABSs with flat dispersion.
S.Kashiwaya, Y.Tanaka (00)
In this case, we can show that 𝑊 𝑘 𝑦 =±2. Thus, it can be explain by the generalized index theorem, but it is not a single Majorana fermion

41. Summary
Majorana fermions are possible in various superconductors other than chiral spin-triplet SC if we take into accout the spin-orbit interctions.
Generalized index theorem, from which ABS with flat dispersion can be expalined, is proved.
Our strategy to prove the index theorem is general, and it gives a general framework to prove the bulk-edge correspondence.

42. Reference
Non-Abelian statistics of axion strings, by MS, Phys. Lett. B575, 126(2003),
Topological Phases of Noncentrosymmetric Superconductors: Edge States, Majorana Fermions, and the Non-Abelian statistics, by MS, S. Fujimoto, PRB79, (2009),
Non-Abelian Topological Order in s-wave Superfluids of Ultracold Fermionic Atoms, by MS, Y. Takahashi, S. Fujimoto, PRL 103, (2009),
Non-Abelian Topological Orders and Majorna Fermions in Spin-Singlet Superconductors, by MS, Y. Takahashi, S.Fujimoto, PRB 82, (2010) (Editor’s suggestion)
Existence of Majorana fermions and topological order in nodal superconductors with spin-orbit interactions in external magnetic field, PRL105, (2010)
Anomalous Andreev bound state in Noncentrosymmetric superconductors, by Y. Tanaka, Mizuno, T. Yokoyama, K. Yada, MS, PRL105, (2010)
Surface density of states and topological edge states in noncentrosymmetric superconductors by K. Yada, MS, Y. Tanaka, T. Yokoyama, PRB83, (2011)
Topology of Andreev bound state with flat dispersion, MS, Y. Tanaka, K. Yada, T. Yokoyama, PRB 83, (2011)

43. Thank you !

44. The parity of the Chern number is well-defined although the Chern number itself is not
Formally, it seems that the Chern number can be defined after removing the gap node by perturbation
perturbation
However, the resultant Chern number depends on the perturbation.
The Chern number

45. On the other hand, the parity of the Chern number does not depend on the perturbation
particle-hole symmetry
T-invariant momentum
all states contribute

46. Non-centrosymmetric Superconductors
(Possible candidate of helical superconductor)
CePt3Si
LaAlO3/SrTiO3 interface
Bauer-Sigrist et al.
Space-inversion
Mixture of spin singlet
and triplet pairings
Possible helical
superconductivity
M. Reyren et al 2007