Topological Superconductors

ISSP, The University of Tokyo, Masatoshi Sato

2

3

Outline

1. What is topological superconductor

2. 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

Topological # = “winding number”

Entire momentum space

Hilbert space of occupied state

empty band

occupied band

A change of the topological number = gap closing

A discontinuous jump of the topological number

Vacuum( or ordinary insulator) Topological SC

Gapless edge state 7

Therefore,

gap closing

Bulk-edge correspondence

If bulk topological # of gapped system is non-trivial, there exist gapless states localized on the boundary.

For rigorous proof , see MS et al, Phys. Rev. B83 (2011) 224511 .

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

Sr2RuO4Noncentosymmetric SC

(MS-Fujimto(09))3He B

9

10

The gapless boundary state = Majorana fermion

Majorana Fermion

Dirac fermion with Majorana condition

1. Dirac Hamiltonian

2. Majorana condition

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

[Wilczek , Nature (09)]

Majorana condition

quasiparticle anti-quasiparticlequasiparticle in Nambu rep.

12

Topological superconductors

Bulk:gapped state with non-zero topological #

Boundary:gapless Majorana fermion

Bulk-edge correspondence

A representative example of topological SC: Chiral p-wave SC in 2+1 dimensions

13

BdG Hamiltonian

with

chiral p-wave

spinless chiral p-wave SC

[Read-Green (00)]

14

Topological number = 1st Chern numberTKNN (82), Kohmoto(85)

MS (09)

Fermi surface

Spectrum

15

SC

2 gapless edge modes(left-moving , right moving, on different sides on boundaries)

Edge state

Bulk-edge correspondence

Majorana fermion

• There also exist a Majorana zero mode in a vortex

We need a pair of the zero modes to define creation op.

vortex 1vortex 2

non-Abelian anyon

topological quantum computer16

Ex.) odd-parity color superconductorY. Nishida, Phys. Rev. D81, 074004 (2010)

color-flavor-locked phase

two flavor pairing phase

17

For odd-parity pairing, the BdG Hamiltonian is

18

(B)

Topological SC

Non-topological SC

• Gapless boundary state• Zero modes in a vortex

(A) With Fermi surface

No Fermi surface

c.f.) MS, Phys. Rev. B79,214526 (2009) MS Phys. Rev. B81,220504(R) (2010) 19

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 !

A) MS, Physics Letters B535 ,126 (03), Fu-Kane PRL (08)

B) MS-Takahashi-Fujimoto ,Phys. Rev. Lett. 103, 020401 (09) ;MS-Takahashi-Fujimoto, Phys. Rev. B82, 134521 (10) (Editor’s suggestion),J. Sau et al, PRL (10), J. Alicea PRB (10)

22

Majorana fermion in spin-singlet SC

① 2+1 dim Dirac fermion + s-wave Cooper pair

Zero mode in a vortex

With Majorana condition, non-Abelian anyon is realized

[Jackiw-Rossi (81), Callan-Harvey(85)]

[MS (03)]

MS, Physics Letters B535 ,126 (03)

vortex

On the surface of topological insulator [Fu-Kane (08)]

Spin-orbit interaction => topological insulator

Topological insulator

S-wave SC

Dirac fermion + s-wave SC

Bi2Se3

Bi1-xSbx

23

Hsieh et al., Nature (2008)

Nishide et al., PRB (2010)

Hsieh et al., Nature (2009)

2nd scheme of Majorana fermion in spin-singlet SC

② 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.

24

Gapless edge statesx

y

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

Chern number

nonzero Chern number

For

25

Majorana fermion

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.

26