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15 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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Topological Superconductors

ISSP, The University of Tokyo,
Masatoshi Sato

2

3

Outline


1.
What is topological superconductor


2.
T
opological 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

e
mpty 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

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Therefore,

g
ap
closing

B
ulk
-
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

1
st

Chern

#

(TKNN82,

Kohmoto85)

Z
2

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+1
D
helical

surface fermion





Sr
2
RuO
4

Noncentosymmetric

SC
(MS
-
Fujimto
(09))

3
He B

9

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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

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How about the
Majorana

condition ?


The
Majorana

condition

is imposed by superconductivity

[
Wilczek

, Nature (09)
]

Majorana

condition

quasiparticle

anti
-
quasiparticle

quasiparticle

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)]

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Topological number = 1
st

Chern

number

TKNN (82),
Kohmoto
(85)

MS (09)

Fermi

surface

Spectrum

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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 1

vortex 2

non
-
Abelian

anyon


topological quantum computer

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Ex.) odd
-
parity color superconductor

Y. 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)

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Phase structure of odd
-
parity color superconductor

Non
-
Topological
SC

Topological SC

There must be topological phase transition.

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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)

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Majorana

fermion

in
spin
-
singlet SC


2+1 dim

Dirac fermion + s
-
wave Cooper pair

Z
ero
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

Bi
2
Se
3


Bi
1
-
x
Sb
x

23

Hsieh et al., Nature (2008)

Nishide

et
al., PRB (2010)

Hsieh et al., Nature (2009)

2
nd 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.

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Gapless edge states

x

y

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

Chern number

nonzero
Chern

number

For

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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.

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