Topology of Andreev bound state

ISSP, The University of Tokyo, Masatoshi Sato

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• 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

In collaboration with

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Outline

Part I. Andreev bound state as Majorana fermions

Part II. Topology of Andreev bound states with flat dispersion

“Edge (or Surface) state” of superconductors

Andreev bound state

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Part I. Andreev bound state as Majorana fermions

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Majorana Fermion

Dirac fermion with Majorana condition

1. Dirac Hamiltonian

2. Majorana condition

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

particle = antiparticle

What is Majorana fermion

chiral p+ip–wave SC

• analogues to quantum Hall state = Dirac fermion on the edge

[Read-Green (00), Ivanov (01)]

chiral edge state

1dim (gapless) Dirac fermionB

TKNN # = 1

• Majorana condition is imposed by superconductivity

[Volovik (97), Goryo-Ishikawa(99),Furusaki et al. (01)]

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• Majorana zero mode in a vortex

creation = annihilation ?

We need a pair of the vortices to define creation op.

vortex 1vortex 2

non-Abelian anyon

topological quantum computer7

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

2. Majorana fermion is possible in nodal superconductor

3. Time-reversal invariant Majorana fermion

•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) ..

MS-Fujimoto PRL (10)

Tanaka-Mizuno-Yokoyama-Yada-MS, PRL (10)MS-Tanaka-Yada-Yokoyama, PRB (11)

Spin-orbit interaction is indispensable !9

Majorana fermion in spin-singlet SC

[MS (03)]

① 2+1 dim odd # of Dirac fermions + s-wave Cooper pair

Non-Abelian statistics of Axion string

On the surface of topological insulator

Majorana zero mode on a vortex

[Fu-Kane (08)]

Bi2Se3 Bi1-xSbx

Spin-orbit interaction => topological insulator

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[MS (03)]

Majorana fermion in spin-singlet SC (contd.)

② 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 statesx

y

a single chiral gapless edge state appears like p-wave SC !

Chern number

nonzero Chern number

For

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Similar to quantum Hall state

Majorana fermion

a) s-wave superfluid with laser generated Rashba SO coupling

b) semiconductor-superconductor interface

[Sato-Takahashi-Fujimoto PRL(09)]

[J.Sau et al. PRL(10) J. Alicea, PRB(10)]

strong magnetic field is needed

c) semiconductor nanowire on superconductors ….13

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Model: 2d Rashba d-wave superconductor

dx2-y2 –wave gap function dxy –wave gap function

Rashba SO

Zeeman

Majorana fermion in nodal superconductor[MS, Fujimoto (10)]

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y

x

dx2-y2 –wave gap function

dxy –wave gap function

Edge state

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Majorana zero mode on a vortex

•Zero mode satisfies Majorana condition!

•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

Non-Abelian anyon

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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

Topologically stable Majorana fermion No stable Majorana fermion

+ nodal excitation + nodal excitation

Semi Conductor

d-wave SC

Zeem

an fi

eld dxy-wave SC

dx2-y2-wave SC

(a) Side View (b) Top View

How to realize our model ?

2dim seminconductor on high-Tc 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)]

time-reversal invariance

Edge state

time-reversal invariance

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[Yada et al. (10) ]

dxy+p-wave Rashba superconductor

The spin-orbit interaction is indispensable

Majorana fermion

No Majorana fermion

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1. Majorana fermion in spin singlet superconductor

2. Majorana fermion in nodal superconductor

3. Time-reversal invariant Majorana fermion

Summary (Part I)With SO interaction, various superconductors become topological superconductors

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Part II. Topology of Andreev bound state

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Chiral Edge state

Bulk-edge correspondence

Gapless state on boundary should correspond to bulk topological number

Chern # (=TKNN #)

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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))

Sr2RuO4Noncentosymmetric 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

A) Particle-hole symmetry

B) Time-reversal symmetry

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Nambu rep. of quasiparticle

C) Chiral symmetry

Combining PHS and TRS, one obtains

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 #29

Number of flat ABS with chirality +1

Number of flat ABS with chirality -1

ABS

Superconductor

Indeed , for ABS, we obtain the generalized index theorem

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

Periodicity of Brillouin zone

Integral along the momentum perpendicular to the surface

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To consider the boundary, we introduce a confining potential V(x)

Superconductor vacuum

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Strategy

1. Introduce an adiabatic parameter in the Planck’s constant

2. Prove the index theorem in the semiclassical limit

original value of Planck’s constant

3. Adiabatically increase the parameter as

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In the classical limit ,

Superconductor vacuum

Gap closing point

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=> zero energy ABS

Around the gap closing point,

Replacing with 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

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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 ,

Because each zero energy ABS has the chirality

Index theorem(but

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Now we adiabatically increase

is adiabatic invariance

Non-zero mode should be paired

Thus, the index theorem holds exactly

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dxy+p-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.

In this case, we can show that . Thus, it can be explain by the generalized index theorem, but it is not a single Majorana fermion

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) (Editor’s 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

perturbation

Formally, it seems that the Chern number can be defined after removing the gap node by perturbation

However, the resultant Chern number depends on the perturbation.

The Chern number

On the other hand, the parity of the Chern number does not depend on the perturbation

particle-hole symmetry

xk

yk

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3 4

T-invariant momentum

all states contribute

M. Reyren et al 2007

Non-centrosymmetric Superconductors (Possible candidate of helical superconductor)

CePt3Si LaAlO3/SrTiO3 interface

Bauer-Sigrist et al.

kkkk ckcH ))((0

)()( kk Space-inversion

Mixture of spin singlet and triplet pairings

Possible helical superconductivity