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

mean field theory : ††††*   

H  1 †  Bogoliubov de Gennes 0 HH   H     BdG † Hamiltonian BdG 2  * k  H0

Intrinsic anti-unitary particle – hole symmetry

1 01   HH     x  * x  BdG BdG 10 2  1 E E Particle – hole redundancy k = k † same state EEEE      

1 Bloch - BdG Hamiltonians satisfy HHBdG()()kk    BdG  Topological classification problem similar to time reversal symmetry 1D 2 Topological Superconductor : n = 0,1 (Kitaev, 2000)

Bulk-Boundary correspondence : Discrete end state spectrum END

n=0 “trivial” n=1 “topological” E   Zero mode E  E E=0 † 0 0 EE00     -E   † EE Majorana fermion   bound state

† Majorana Fermion : Particle = Antiparticle 

   † ;     i   2 1 Real part of a Dirac fermion : 1 1 2 i †† 2 ii(    ) ;    1   2 i,2  j   ij “Half a state”

Two Majorana fermions define a single two level system occupied †  Hi2 0  1  2   0   empty 0  1 †  H0  HH   H   Mean Field:   BdG † BdG 2  * H0

Particle-hole symmetric spectrum of 1 body Hamiltonian HBdG uv()()rr* HE  nn BdG n n n nn nnv ()r * n un ()r Many body operators :

  †† dr u()()()() r  r  v r  r n n  n n  E 0 1 ††n n HE n  n†  n  n  n 2  nn0 En n

Zero Mode:  * dr u()()()() r  r  u*†† r  r  00uv00 0  0 0 0    Two zero modes :  i   0 0 0 E0  END END † Bogoliubov QP operator : 00   LR  i

2 Majorana bound states = 1 Dirac fermion bound state = 1 qubit Kitaev Model for 1D p wave superconductor

††††† H N  tcc()()i i1  cc i  1 i  cc i i   cc i i  1  cc i  1 i i   t c † k   ck c k H BdG () k † k ck

HBdG( k ) z (2 t cos k   )   x  sin k d ( k )   dx d(k) ||>2t : Strong pairing phase d trivial superconductor z

dx d(k) ||<2t : Weak pairing phase

topological superconductor dz

Similar to SSH model, except different symmetry : (,,)(,,)d d d  d  d d x y zkk x y z  Majorana Chain

H2 i t  t   cii12 i i  1 1i 2 i 2 2 i 1 i 1 i † ci c i 2 i 12 i  i †† For =t : nearest neighbor Majorana chain t ci c i1 c i  1 c i 2 it  1 i  2 i  1   2 i  1 i  1  †† t12, t 2 t ci c i1  c i  1 c i 2 i  1 i  2 i  1   2 i  1 i  1 

t1 t2 1i 2i t1>t2 trivial SC

t1

Unpaired Majorana Fermion at end 2D  topological superconductor (broken T symmetry)

Bulk-Boundary correspondence: n = # Chiral Majorana Fermion edge states

E T-SC  k Examples † k • Spinless p +ip superconductor (n=1) kk x y • Chiral triplet p wave superconductor

(eg Sr2RuO4) (n=2)

k 2 † ()k  k  ik Read Green model : H  ck c k  ()..k c k c k  c c  0  xy k 2m  Lattice BdG model : HBdG(kd ) z 2 t cos k x  cos k y      x sin k x   y sin k y   ( k )  

dx d(k) ||>4t : Strong pairing phase d trivial superconductor z Chern number 0 dy

dx d(k) ||<4t : Weak pairing phase dz Chern number 1 topological superconductor dy Majorana zero mode at a vortex

Boundary condition on fermion wavefunction

h 0 p1 p A (L ) ( 1) (0) 2e L   (x ) eiqm x ; q  2 m  1  p m L Hole in a topological superconductor threaded by flux

p even p odd E E 2v  ~ L q q

zero mode

2 Without the hole : Caroli, de Gennes, Matricon theory (‟64)  ~ EF Periodic Table of Topological Insulators and Superconductors Anti-Unitary Symmetries :

- Time Reversal : HH(kk ) 1   (  ) ; 2   1 1 2 - Particle - Hole : HH(kk )    (  ) ;    1

1 Unitary (chiral) symmetry : HH()()kk    ;    8 antiunitary symmetry classes

Complex K-theory

Altland- Zirnbauer Random Matrix Real Classes K-theory

Kitaev, 2008 Schnyder, Ryu, Furusaki, Ludwig 2008 Bott Periodicity d→d+8 Majorana Fermions and Topological Quantum Computing (Kitaev ‟03) The degenerate states associated with Majorana zero modes define a topologically protected quantum memory

• 2 Majorana separated bound states = 1 fermion  12  i - 2 degenerate states (full/empty) = 1 qubit • 2N separated Majoranas = N qubits • Quantum Information is stored non locally - Immune from local decoherence

Braiding performs unitary operations Measure  012 0 34 1 12 1 34  / 2 Non-Abelian statistics Interchange rule (Ivanov 03)

t ij Braid ji These operations, however, are not sufficient

to make a universal quantum computer Create 0012 34 Potential condensed matter hosts for Majorana bound states

• Quasiparticles in fractional Quantum Hall effect at n=5/2 Moore Read „91

• Unconventional superconductors

- Sr2RuO4 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

Hi† ( v   )  s wave superconductor ††* SS        Topological insulator proximity induced superconductivity at surface

• Half an ordinary superconductor

• Similar to spinless px+ipy superconductor, except : - Does not violate time reversal symmetry - s-wave singlet superconductivity -k ← Dirac point - Required minus sign is provided by  Berry‟s phase due to Dirac Point ↓ ↑ • Nontrivial ground state supports Majorana → fermion bound states at vortices k Majorana Bound States on Topological Insulators

1. h/2e vortex in 2D superconducting state E †  E h/2e  † 0 00 SC  † TI EE

Quasiparticle Bound state at E=0 Majorana Fermion 0 “Half a State”

2. Superconductor-magnet interface at edge of 2D QSHI

m | SM |  |  | M S.C. m>0

Egap =2|m| QSHI m<0

Domain wall bound state 0 1D Majorana Fermions on Topological Insulators 1. 1D Chiral Majorana mode at superconductor-magnet interface E

M SC kx

TI  † kk : “Half” a 1D chiral Dirac fermion Hi vF  x 2. S-TI-S Josephson Junction

 0       SC SC

TI

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

H  ivF L  x  L   R  x  R  i cos(  / 2)  L  R Manipulation of Majorana Fermions Control phases of S-TI-S Junctions   Majorana Tri-Junction : 2 1  present A storage register for Majoranas 0 

Create Braid Measure A pair of Majorana bound A single Majorana can be Fuse a pair of Majoranas. states can be created from moved between junctions. States |0,1> distinguished by the vacuum in a well defined Allows braiding of multiple • presence of quasiparticle. state |0>. Majoranas • supercurrent across line junction

E E E 0 1 0 0 0

0  0  0  Another route to the 2D p+ip superconductor Sau, Lutchyn, Tewari, Semiconductor - Magnet - Superconductor structure Das Sarma „09 E

Superconductor Rashba split 2DEG bands Semiconductor EF Magnetic Insulator k • Single Fermi circle with Berry phase  Zeeman splitting • 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.