Topology of Andreev bound state

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Topology of Andreev bound state

• ISSP, The University of Tokyo, Masatoshi Sato

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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

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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
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Part I. Andreev bound state as Majorana fermions
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What is Majorana fermion
Majorana Fermion
Dirac fermion with Majorana condition

1. Dirac Hamiltonian

1. Majorana condition

particle antiparticle

• Originally, elementary particles.
• But now, it can be realized in superconductors.
  

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chiral pipwave 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
1dim (gapless) Dirac fermion

• Majorana condition is imposed by
  superconductivity

TKNN 1
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• Majorana zero mode in a vortex

creation annihilation ?
We need a pair of the vortices to define
creation op.
vortex 2
vortex 1
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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 .
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1. 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) ..

1. Majorana fermion is possible in nodal
   superconductor

MS-Fujimoto PRL (10)

1. Time-reversal invariant Majorana fermion

Tanaka-Mizuno-Yokoyama-Yada-MS, PRL
(10) MS-Tanaka-Yada-Yokoyama, PRB (11)
Spin-orbit interaction is indispensable !
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Majorana fermion in spin-singlet SC

1. 21 dim odd of Dirac fermions s-wave Cooper
   pair

Majorana zero mode on a vortex
MS (03)
MS (03)
Non-Abelian statistics of Axion string
On the surface of topological insulator
Fu-Kane (08)
 Bi1-xSbx
Bi2Se3
Spin-orbit interaction gt topological insulator
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Majorana fermion in spin-singlet SC (contd.)

1. s-wave SC with Rashba spin-orbit interaction

MS, Takahashi, Fujimoto (09,10)
Rashba SO
p-wave gap is induced by Rashba SO int.
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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
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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 .
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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
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Edge state
dx2-y2 wave gap function
x
y
dxy wave gap function
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Majorana zero mode on a vortex
Non-Abelian anyon

• The zero mode is stable against nodal excitations

4 gapless mode from gap-node
1 zero mode on a vortex
From the particle-hole symmetry, the modes become
massive in pair. Thus at least one Majorana zero
mode survives on a vortex
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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
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How to realize our model ?
2dim seminconductor on high-Tc Sc
(a) Side View
(b) Top View
dxy-wave SC
Zeeman field
dx2-y2-wave SC
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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
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dxyp-wave Rashba superconductor
Majorana fermion
Yada et al. (10)
The spin-orbit interaction is indispensable
No Majorana fermion
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Summary (Part I)
With SO interaction, various superconductors
become topological superconductors

1. Majorana fermion in spin singlet superconductor

1. Majorana fermion in nodal superconductor

1. Time-reversal invariant Majorana fermion

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Part II. Topology of Andreev bound state
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Bulk-edge correspondence
Gapless state on boundary should correspond to
bulk topological number
Chern (TKNN )
Chiral Edge state
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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
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Which topological is responsible for Majorana
fermion with flat band ?
?
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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 !
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Symmetry of the system

1. Particle-hole symmetry

Nambu rep. of quasiparticle

1. Time-reversal symmetry

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Combining PHS and TRS, one obtains

1. Chiral symmetry

c.f.) chiral symmetry of Dirac operator
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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
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Indeed , for ABS, we obtain the generalized
index theorem
Superconductor
Number of flat ABS with chirality 1
Number of flat ABS with chirality -1
ABS
Generalized index theorem
MS et al (11)
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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
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Topological number
Integral along the momentum perpendicular to the
surface
Periodicity of Brillouin zone
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To consider the boundary, we introduce a
confining potential V(x)
Superconductor
vacuum
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Strategy
 
original value of Plancks constant

1. Prove the index theorem in the semiclassical
   limit

 
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Superconductor
vacuum
Gap closing point
gt zero energy ABS
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Around the gap closing point,
 
From the explicit form of the obtained solution,
we can determine its chirality as
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We also calculate the contribution of the
gap-closing point to topological ,
 
 
 
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Non-zero mode should be paired
Thus, the index theorem holds exactly
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dxyp-wave SC
Thus, the existence of Majorana fermion with flat
dispersion is ensured by the index theorem
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remark

• It is well known that dxy-wave SC has similar
  ABSs with flat dispersion.

S.Kashiwaya, Y.Tanaka (00)
 
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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.

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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, 094504 (2009),
• Non-Abelian Topological Order in s-wave
  Superfluids of Ultracold Fermionic Atoms, by MS,
  Y. Takahashi, S. Fujimoto, PRL 103, 020401
  (2009),
• Non-Abelian Topological Orders and Majorna
  Fermions in Spin-Singlet Superconductors, by MS,
  Y. Takahashi, S.Fujimoto, PRB 82, 134521 (2010)
  (Editors suggestion)
• Existence of Majorana fermions and topological
  order in nodal superconductors with spin-orbit
  interactions in external magnetic field,
  PRL105,217001 (2010)
• Anomalous Andreev bound state in
  Noncentrosymmetric superconductors, by Y. Tanaka,
  Mizuno, T. Yokoyama, K. Yada, MS, PRL105, 097002
  (2010)
• Surface density of states and topological edge
  states in noncentrosymmetric superconductors by
  K. Yada, MS, Y. Tanaka, T. Yokoyama, PRB83,
  064505 (2011)
• Topology of Andreev bound state with flat
  dispersion, MS, Y. Tanaka, K. Yada, T. Yokoyama,
  PRB 83, 224511 (2011)

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Thank you !
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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.
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On the other hand, the parity of the Chern number
does not depend on the perturbation
particle-hole symmetry
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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