Topology of Andreev bound state
ISSP, The University of Tokyo,
Masatoshi Sato
1
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
2
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
3
Part I. Andreev bound state as Majorana fermions
4
What is Majorana fermion
Majorana Fermion
Dirac fermion with Majorana condition
1.
Dirac Hamiltonian
2.
Majorana condition
particle = antiparticle
• Originally, elementary particles.
• But now, it can be realized in superconductors.
5
chiral p+ip–wave 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
B
1dim (gapless) Dirac fermion
• Majorana condition is imposed by superconductivity
TKNN # = 1
6
• Majorana zero mode in a vortex
creation = annihilation ?
We need a pair of the vortices to define creation op.
vortex 2
vortex 1
non-Abelian anyon
topological quantum computer
7
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 .
8
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) ..
2. Majorana fermion is possible in nodal superconductor
MS-Fujimoto PRL (10)
3. Time-reversal invariant Majorana fermion
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
① 2+1 dim odd # of Dirac fermions + s-wave Cooper pair
Majorana zero mode on a vortex
[MS (03)]
Non-Abelian statistics of Axion string
[MS (03)]
On the surface of topological insulator
[Fu-Kane (08)]
Bi2Se3
Bi1-xSbx
Spin-orbit interaction
=> topological insulator
10
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.
11
Gapless edge states
x
y
Majorana
fermion
For
a single chiral gapless edge state appears like p-wave SC !
Chern number
nonzero Chern number
Similar to quantum
Hall state
12
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 ….
13
Majorana fermion in nodal superconductor
Model: 2d Rashba d-wave superconductor
[MS, Fujimoto (10)]
Rashba SO
Zeeman
dx2-y2 –wave gap function
dxy –wave gap function
14
Edge state
dx2-y2 –wave gap function
x
y
dxy –wave gap function
15
Majorana zero mode on a vortex
•Zero mode satisfies Majorana condition!
Non-Abelian anyon
•The zero mode is stable against nodal excitations
1 zero mode
on a vortex
4 gapless
mode from
gap-node
From the particle-hole symmetry, the modes become massive in pair.
Thus at least one Majorana zero mode survives on a vortex
16
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
+ nodal excitation
Topologically stable Majorana fermion
There exist an even number of gapless
Majorana fermions
+ nodal excitation
No stable Majorana fermion
17
How to realize our model ?
2dim seminconductor on high-Tc Sc
Zeeman field
(a) Side View
(b) Top View
dxy-wave SC
Semi Conductor
d-wave SC
dx2-y2-wave SC
18
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
19
dxy+p-wave Rashba superconductor
Majorana fermion
[Yada et al. (10) ]
The spin-orbit interaction is
indispensable
No Majorana
fermion
20
Summary (Part I)
With SO interaction, various superconductors become
topological superconductors
1. Majorana fermion in spin singlet superconductor
2. Majorana fermion in nodal superconductor
3. Time-reversal invariant Majorana fermion
21
Part II. Topology of Andreev bound state
22
Bulk-edge correspondence
Gapless state on boundary should correspond to
bulk topological number
Chern # (=TKNN #)
Chiral Edge state
23
different type ABS = different topological #
chiral
Chern #
helical
Cone
Z2 number
3D winding
number
(TKNN (82))
(Kane-Mele (06))
Sr2RuO4
Noncentosymmetric SC
(MS-Fujimto(09))
(Schnyder et al (08))
3He
B
24
Which topological # is responsible for Majorana
fermion with flat band ?
?
25
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 !
26
Symmetry of the system
A) Particle-hole symmetry
Nambu rep. of quasiparticle
B) Time-reversal symmetry
27
Combining PHS and TRS, one obtains
C) Chiral symmetry
c.f.) chiral symmetry of Dirac operator
28
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
Indeed , for ABS, we obtain the generalized index theorem
Number of flat ABS with chirality +1
Superconductor
Number of flat ABS with chirality -1
ABS
Generalized index theorem
[MS et al (11)]
30
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
31
Topological number
Integral along the momentum
perpendicular to the surface
Periodicity of Brillouin zone
32
To consider the boundary, we introduce a confining potential V(x)
vacuum
Superconductor
33
Strategy
1. Introduce an adiabatic parameter 𝑎 in the Planck’s constant
original value of
Planck’s constant
2. Prove the index theorem in the semiclassical limit
3. Adiabatically increase the parameter 𝑎 as 𝑎 → 1
34
In the classical limit 𝑎 = 0,
vacuum
Superconductor
Gap closing point
=> zero energy ABS
35
Around the gap closing point,
Replacing 𝑘𝑥 with −𝑖ℏ𝜕𝑥 in the above, we can perform the semiclassical quantization , and construct the zero energy ABS explicitly.
From the explicit form of the obtained solution, we can determine its
chirality as
36
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 𝑊(𝒌∥ ) ,
Because each zero energy ABS has the chirality Γ = sgn[𝜂𝑖 ]
Index theorem
(but ℏ ≪ 1)
37
Now we adiabatically increase 𝑎 → 1 (ℏ → ℏ0 )
 𝑛+ − 𝑛− is adiabatic invariance
Non-zero mode
should be paired
Thus, the index theorem holds exactly
38
dxy+p-wave SC
Thus, the existence of Majorana fermion with flat dispersion is
ensured by the index theorem
39
remark
• It is well known that dxy-wave SC has similar ABSs with flat dispersion.
S.Kashiwaya, Y.Tanaka (00)
In this case, we can show that 𝑊 𝑘𝑦 = ±2. Thus, it can be explain by the
generalized index theorem, but it is not a single Majorana fermion
40
Summary
Majorana fermions are possible in various superconductors
other than chiral spin-triplet SC if we take into accout the spinorbit 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.
41
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)
42
Thank you !
43
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.
The Chern number
44
On the other hand, the parity of the Chern number does not
depend on the perturbation
particle-hole symmetry
T-invariant
momentum
ky

3
1
all states contribute
4
2

kx
Non-centrosymmetric Superconductors
(Possible candidate of helical superconductor)
CePt3Si
LaAlO3/SrTiO3 interface
Bauer-Sigrist et al.
H 0   ck ( k   (k )  )ck
k
(k )  (k ) Space-inversion
Mixture of spin singlet
and triplet pairings
Possible helical
superconductivity
M. Reyren et al 2007