# Majorana Fermions and

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15 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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Topological Superconductors,
Majorana Fermions and
Topological Quantum Computation
1. Bogoliubov de Gennes Theory
2. Majorana bound states, Kitaev model
3. Topological superconductor
4. Periodic Table of topological insulators and superconductors
5. Topological quantum computation
6. Proximity effect devices
BCS Theory of Superconductivity

1
2
BdG
H H

 
  
 

 

k
Intrinsic anti
-
unitary particle

hole symmetry
1
BdG BdG
H H

   

E E E E
   
 
   

0
*
0
BdG
H
H
H
 

 

 
 
 
Bogoliubov de Gennes
Hamiltonian
mean field theory :
† † † † *

        
Bloch
-
BdG Hamiltonians satisfy
Topological classification problem similar to time reversal symmetry
*
x
 
 
Particle

hole redundancy
2
1
  
0 1
1 0
x

 

 
 
=
E
k
E
k
1
( ) ( )
BdG BdG
H H

    
k k
same state
Bulk
-
Boundary correspondence :
Discrete end state spectrum
0

E
E
-
E
0

E=0
n
=0 “trivial”
n
=1 “topological”

0 0
E E

 
   
Majorana fermion
bound state
1D

2
Topological Superconductor :
n
= 0,1
END

E E

  
E

(Kitaev, 2000)
Majorana Fermion : Particle = Antiparticle
Real part of a Dirac fermion :
“Half a state”
Two
Majorana fermions define a
single
two level system
Zero mode

1 1 2
† †
2 1 2
;
( ) ;
i
i i
  
  
   
     

,2
i j ij
  

0 1 2 0
2
H i
 
   

0
empty
occupied
2
1
i

 

1
2
BdG
H H

 
  
 

 

*
*
( ) ( )
( )
( )

n n
n n
n
n
u v
v
u
 
 
 
 
 
 
 
 
 
r r
r
r
n n
 
 
 
BdG n n n
H E
 
 
 

† †
( ) ( ) ( ) ( )
n n n n
d u v
 
      

r r r r r

† †

0
1
2
0
n
n
n n n n n
n n
n
n
E
H E
E
 

 
     
 
 
 

 
 
 
Particle
-
hole symmetric spectrum of
1 body
Hamiltonian H
BdG
Many body operators :
Zero Mode:
0 0
 
 
*
0 0
u v

* † †
0 0 0 0
( ) ( ) ( ) ( )
d u u
 
    

r r r r r
Two zero modes :
0 0 0
i
  
  
 
0
E

 
Bogoliubov QP operator :

0 0
L R
i
 
 
    
2 Majorana bound states = 1 Dirac fermion bound state = 1 qubit
Mean Field:
0
*
0
BdG
H
H
H
 

 

 
 
 
END
END
Kitaev Model for 1D p wave superconductor
† † † † †
1 1 1 1
( ) ( )
i i i i i i i i i i
i
H N t c c c c c c c c c c
 
   
     

( ) (2 cos ) sin ( )
BdG z x
H k t k k k
   
     
d

( )
k
k k BdG
k
k
c
c c H k
c

 

 
 

||
>2t : Strong pairing phase
trivial superconductor
d
(k)
d
(k)
d
z
d
x
d
z
d
x
||
<2t : Weak pairing phase
topological superconductor
Similar to SSH model, except different symmetry :
(,,) (,,)
x y z x y z
k k
d d d d d d

  

t

Majorana Chain
1 2
i i i
c i
 
 

1 2
† †
1 1 1 2 1 2 1 1
† †
1 1 1 2 1 2 1 1
2
2
2
i i i i
i i i i i i i i
i i i i i i i i
c c i
t c c c c it
c c c c i
 
 
 
   
   

  
    

1 1 2 2 2 1 1
2
i i i i
i
H i t t
 

 

1 2
, 2
t t t

 
t
1
>t
2
trivial SC
t
1
<t
2
topological SC
t
1
t
2

1i

2i
Unpaired Majorana Fermion at end
For

=t : nearest neighbor Majorana chain
2D

topological superconductor (broken T symmetry)
Bulk
-
Boundary correspondence: n = # Chiral Majorana Fermion edge states
k
E

k k
 

k

Examples

Spinless p
x
+ip
y
superconductor (n=1)

Chiral triplet p wave superconductor
(eg Sr
2
RuO
4
) (n=2)

2

( )..
2
H c c c c c c
m

 
    
 
 

k k k k
k
k
k

0
( )
x y
k ik
   
k
Lattice BdG model :

( ) 2 cos cos sin sin ( )
BdG z x y x x y y
H t k k k k k
    
 
      
 
k d
||
>4t : Strong pairing phase
trivial superconductor
d
(
k
)
d
z
d
x
||
<4t : Weak pairing phase
topological superconductor
d
y
d
z
d
x
d
y
d
(
k
)
Chern number 0
Chern number 1
T
-
SC
Majorana zero mode at a vortex
2
h
p
e

Boundary condition on fermion wavefunction
1
( ) ( 1) (0)
p
L
 

 
0
L
A

( ) ; 2 1
m
iq x
m
x e q m p
L

   
p even
p odd
zero mode
Hole in a topological superconductor threaded by flux
2 v
~
L

E
E
q
q
Without the hole : Caroli, de Gennes, Matricon theory (‟64)
2
~
F
E

Periodic Table of Topological Insulators and Superconductors
Anti
-
Unitary Symmetries :
-
Time Reversal :
-
Particle
-
Hole :
Unitary (chiral) symmetry :
1
( ) ( ) 1
2
;
H H

       
k k
1
( ) ( ) 1
2
;
H H

       
k k
1
( ) ( )
H H

    
k k
;
Real
K
-
theory
Complex
K
-
theory
Bott Periodicity d
→d+8
Altland
-
Zirnbauer
Random
Matrix
Classes
Kitaev, 2008
Schnyder, Ryu, Furusaki, Ludwig 2008
8 antiunitary symmetry classes

2 Majorana separated bound states = 1 fermion
-
2 degenerate states (full/empty) = 1 qubit

2N separated Majoranas = N qubits

Quantum Information is stored non locally
-
Immune from local decoherence
1 2
i
 
 
Create
Braid
Measure

12 34 12 34
0 0 1 1/2

t
12 34
0 0
Majorana Fermions and Topological Quantum Computing
The degenerate states associated with Majorana zero modes
define a topologically protected quantum memory
(Kitaev ‟03)
Braiding performs unitary operations
Non
-
Abelian statistics
Interchange rule
(Ivanov 03)
i j
j i
 
 


These operations, however, are not sufficient
to make a universal quantum computer
Potential condensed matter hosts for Majorana bound states

Quasiparticles in fractional Quantum Hall effect at
n
=5/2

Unconventional superconductors
-
Sr
2
RuO
4
Das Sarma, Nayak, Tewari „06
-
Fermionic atoms near feshbach resonance
Gurarie „05

Proximity Effect Devices using ordinary s wave superconductors
-
Topological Insulator devices
Fu, Kane „08
-
Semiconductor/Magnet devices
Sau, Lutchyn, Tewari, Das Sarma ‟09, Lee ‟09, ...

.... among others
Current Status :
Not Observed
Superconducting Proximity Effect
s wave superconductor
Topological insulator
† † *

( v )
S S
H i
  
 
   

 


proximity induced superconductivity
at surface
-
k
k

Dirac point

Half an ordinary superconductor

Similar to spinless p
x
+ip
y
superconductor, except :
-
Does not violate time reversal symmetry
-
s
-
wave singlet superconductivity
-
Required minus sign is provided by

Berry‟s phase due to Dirac Point

Nontrivial ground state supports Majorana
fermion bound states at vortices
Majorana Bound States on Topological Insulators
SC
h/2e
1. h/2e vortex in 2D superconducting state
2. Superconductor
-
magnet interface at edge of 2D QSHI
Quasiparticle Bound state at E=0 Majorana Fermion

0
“Half a State”
TI
0

E

0 0
 

E

E E
 

S.C.
M
QSHI
E
gap
=2|m|
Domain wall bound state

0
m<0
m>0
| | | |
S M
m
   
1D Majorana Fermions on Topological Insulators
2. S
-
TI
-
S Josephson Junction
SC
TI
SC
  

SC
M
1. 1D Chiral Majorana mode at superconductor
-
magnet interface
TI
k
x
E

k k
 

: “Half” a 1D chiral Dirac fermion
F
v
x
H i
 
  

0
Gapless non
-
chiral Majorana fermion for phase difference
  

cos(/2)
F
v
L x L R x R L R
H i i
     
      
Manipulation of Majorana Fermions
Control phases of S
-
TI
-
S Junctions

1

2
0
Majorana
present

Tri
-
Junction :
A storage register for Majoranas
Create
A pair of Majorana bound
states can be created from
the vacuum in a well defined
state |0>.
Braid
A single Majorana can be
moved between junctions.
Allows braiding of multiple
Majoranas
Measure
Fuse a pair of Majoranas.
States |0,1> distinguished by

presence of quasiparticle.

supercurrent across line
junction
E

0
0
1
E

E

0
0
0
0
0
0
0
Another route to the 2D p+ip superconductor
Semiconductor
-
Magnet
-
Superconductor structure
Sau, Lutchyn, Tewari,
Das Sarma „09
Magnetic Insulator
Superconductor
Semiconductor
Rashba split
2DEG bands
Zeeman splitting
E
k
E
F

Single Fermi circle with Berry phase

Topological superconductor with Majorana edge states
and Majorana bound states at vortices.

Variants
:
-
use applied magnetic field to lift Kramers degeneracy (Alicea ‟10)
-
Use 1D quantum wire (eg InAs ). A route to 1D p wave superconductor
with Majorana end states. (Oreg, von Oppen, Alicea, Fisher ‟10

Challenge : requres very low electron density
→ high purity.