Topology of Andreev bound state ISSP, The University of Tokyo, Masatoshi Sato 1.
-
date post
20-Dec-2015 -
Category
Documents
-
view
226 -
download
0
Transcript of Topology of Andreev bound state ISSP, The University of Tokyo, Masatoshi Sato 1.
Topology of Andreev bound state
ISSP, The University of Tokyo, Masatoshi Sato
1
• 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
2
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
3
Part I. Andreev bound state as Majorana fermions
4
5
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)]
6
• 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 .
8
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
10
[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.
11
Gapless edge statesx
y
a single chiral gapless edge state appears like p-wave SC !
Chern number
nonzero Chern number
For
12
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
14
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)]
15
y
x
dx2-y2 –wave gap function
dxy –wave gap function
Edge state
16
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
17
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
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)]
time-reversal invariance
Edge state
time-reversal invariance
19
[Yada et al. (10) ]
dxy+p-wave Rashba superconductor
The spin-orbit interaction is indispensable
Majorana fermion
No Majorana fermion
20
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
21
Part II. Topology of Andreev bound state
22
Chiral Edge state
Bulk-edge correspondence
Gapless state on boundary should correspond to bulk topological number
Chern # (=TKNN #)
23
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
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
B) Time-reversal symmetry
27
Nambu rep. of quasiparticle
C) Chiral symmetry
Combining PHS and TRS, one obtains
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
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)]
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
Periodicity of Brillouin zone
Integral along the momentum perpendicular to the surface
32
To consider the boundary, we introduce a confining potential V(x)
Superconductor vacuum
33
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
34
In the classical limit ,
Superconductor vacuum
Gap closing point
35
=> 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
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 ,
Because each zero energy ABS has the chirality
Index theorem(but
37
Now we adiabatically increase
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.
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)
40
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.
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
43
Thank you !
44
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
21
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