Topological Superconductors
ISSP, The University of Tokyo,
Masatoshi Sato
2
Outline
1. What is topological superconductor
2. Topological superconductors in various
systems
3
What is topological superconductor ?
Topological superconductors
Bulk:
gapped state with
non-zero topological #
Boundary:
gapless state with
Majorana condition
4
Bulk: gapped by the formation of Cooper pair
In the ground state, the one-particle states below the
fermi energy are fully occupied.
5
Topological # can be defined by the occupied wave
function
empty band
occupied band
Hilbert
space of
occupied
state
Entire
momentum
space
Topological # = “winding number”
6
A change of the topological number = gap closing
gap
closing
A discontinuous jump of the topological number
Therefore,
Vacuum
( or ordinary insulator)
Topological SC
Gapless edge state
7
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 #
3D winding #
(TKNN82, Kohmoto85)
Z2 number
(Kane-Mele 06, Qi et al (08))
(Schnyder et al (08))
1+1D chiral
edge mode
1+1D helical
edge mode
2+1D helical
surface fermion
Sr2RuO4
Noncentosymmetric SC
(MS-Fujimto(09))
3He
B
9
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
10
How about the Majorana condition ?
The Majorana condition is imposed by superconductivity
quasiparticle in Nambu rep.
quasiparticle
anti-quasiparticle
Majorana condition
[Wilczek , Nature (09)]
11
Topological superconductors
Bulk:
gapped state with
non-zero topological #
Bulk-edge
Boundary:
correspondence
gapless Majorana
fermion
12
A representative example of topological SC:
Chiral p-wave SC in 2+1 dimensions
BdG Hamiltonian
[Read-Green (00)]
spinless chiral p-wave SC
with
chiral p-wave
13
Topological number = 1st Chern number
TKNN (82), Kohmoto(85)
MS (09)
14
Edge state
SC
Fermi surface
Spectrum
2 gapless edge modes
(left-moving , right
moving,
on different sides on
boundaries)
Majorana fermion
Bulk-edge
correspondence
15
• There also exist a Majorana zero mode in a vortex
We need a pair of the zero modes to define creation op.
vortex 2
vortex 1
non-Abelian anyon
topological quantum computer
16
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
(A)
With Fermi surface
Topological SC
• Gapless boundary state
• Zero modes in a vortex
(B)
No Fermi surface
Non-topological SC
c.f.) MS, Phys. Rev. B79,214526 (2009)
MS Phys. Rev. B81,220504(R) (2010)
19
Phase structure of odd-parity color superconductor
Non-Topological
SC
Topological SC
There must be topological phase transition.
20
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)
21
Majorana fermion in spin-singlet SC
MS, Physics Letters B535 ,126 (03)
① 2+1 dim Dirac fermion + s-wave Cooper pair
vortex
Zero mode in a vortex
[Jackiw-Rossi (81), Callan-Harvey(85)]
With Majorana condition, non-Abelian anyon is realized
[MS (03)]
22
On the surface of topological insulator
Bi1-xSbx
[Fu-Kane (08)]
Hsieh et al., Nature (2008)
Dirac fermion + s-wave SC
S-wave SC
Nishide et al., PRB (2010)
Bi2Se3
Hsieh et al., Nature (2009)
Topological insulator
Spin-orbit interaction
=> topological insulator
23
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 states
x
y
Majorana
fermion
For
a single chiral gapless edge state appears like p-wave SC !
Chern number
nonzero Chern number
25
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