Majorana Fermions and Topological Insulators Charles Kane, - - PowerPoint PPT Presentation

Majorana Fermions and Topological Insulators

• I. Introduction: Topological insulators in 2D, 3D

Exotic surface phases :

• “Fractional” Integer Quantized Hall Effect
• Superconducting Proximity Effect
• II. Majorana Fermions on Topological Insulators
• A route to Topological Quantum Computing?
• A Z2 Interferometer for Majorana Fermions
• III. Non-Abelian Statistics in Three Dimensions

Thanks to Gene Mele, Liang Fu, Jeffrey Teo,

Charles Kane, University of Pennsylvania

Topological Insulators

Two dimensions: Quantum Spin Hall Insulator

Graphene

Kane, Mele ’05

HgCdTe quantum well

Bernevig, Hughes, Zhang ’06

Edge state transport experiments

Konig, et al. ‘07

G=2e2/h

Surface States probed by ARPES:

Bi1-x Sbx

Three dimensions: Strong Topological Insulator

Theory: Fu, Kane, Mele ’06, Moore, Balents ’06, Roy ‘06

Bi1-x Sbx

Fu, Kane ’07 (Th) Hsieh, et al ’07 (Exp)

Bi2 Se3, Bi2 Te3

Xia, et al ’09 (Exp+Th) Zhang, et al ’09 (Th) Hsieh, et al ’09 (Exp) Chen et al. ’09 (Exp)

Bulk - Boundary Correspondence

Equivalence classes of surface/edge: even or odd number

• f enclosed Dirac points

d=2 d=3 EF

Time Reversal Invariant 2 Topological Insulator

Time Reversal Symmetry : Kramers’ Theorem :

( )

1

( ) H H

−

Θ Θ = − k k

*

y

i ψ σ ψ Θ =

2

1 Θ = −

• All states doubly degenerate

2 : two ways to connect Kramers

pairs on surface

E

k=Λa k=Λb

E

k=Λa k=Λb

OR

2

• 2

2

3 ⊕

• ( )

d

H T ∈ k k

EF

k kF kx ky

(weak Topo. Ins.)

Unique Properties of Surface States

“Half” an ordinary 2DEG ; ¼ Graphene Spin polarized Fermi surface

• Charge Current ~ Spin Density
• Spin Current ~ Charge Density

π Berry’s phase

• Robust to disorder
• Weak Antilocalization
• Impossible to localize, Klein paradox

Exotic States when broken symmetry leads to surface energy gap:

• Quantum Hall state, magnetoelectric effect

Fu, Kane ’07; Qi, Hughes, Zhang ’08, Essin, Moore, Vanderbilt ‘09

• Superconducting state

Fu, Kane ‘08

• Excitonic Insulator

Seradjeh, Moore, Franz ‘09

EF

Isolated surface Dirac point on Bi2 Se3

D Hsieh, et al. Nature ‘09

Surface Quantum Hall Effect

2

1 2

xy

e n h σ

• =

+

• 2

2

xy

e h σ =

2

2

xy

e h σ =

B

ν=1 chiral edge state

Orbital QHE :

M M

E=0 Landau Level for Dirac fermions. “Fractional” IQHE

Anomalous QHE :

Induce a surface gap by depositing magnetic material

Chiral Edge State at Domain Wall : ∆M −∆M

†

( v )

z M

H i ψ σ µ σ ψ ∆ = − ∇ − +

• Mass due to Zeeman field

Egap = 2|∆M| EF

1 2

• 2
• 1

TI

2

sgn( ) 2

xy M

e h σ = ∆

2

2 e h +

2

2 e h −

Superconducting Proximity Effect

s wave superconductor Topological insulator

† † * †(

v )

S S

H i ψ σ µ ψ ψ ψ ψ ψ

↑ ↓ ↓ ↑

+ − − + ∆ ∇ ∆ =

• proximity induced superconductivity

at surface

• k

k

• Dirac point
• Half an ordinary superconductor
• Similar to 2D spinless px+ipy topological superconductor, except :
• Does not violate time reversal symmetry
• s-wave singlet superconductivity
• Required minus sign is provided by

π 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 γ0 “Half a State”

TI ∆ −∆ E

†

γ γ =

† E

γ

† E E

γ γ

− =

S.C. M QSHI Egap =2|m|

Domain wall bound state γ0

m<0 m>0

| | | |

S M

m = ∆ − ∆

Majorana Fermions

• Quasiparticles in fractional Quantum Hall effect at ν=5/2

Moore, Read ’91

• s-wave superconductor / Topological Insulator structure

Fu, Kane ‘08

• semiconductor - magnet - superconductor structures

Sau, Lutchyn, Tewari, Das Sarma ‘09

• .... among others
• 2 Majorana 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
• Adiabatic Braiding performs unitary operations
• Non Abelian Statistics

1 2

i γ γ Ψ = +

Potential Condensed Matter Hosts : Topological Quantum Computing Kitaev, 2003

Create Braid Measure (

)

12 34 12 34

0 0 1 1 / 2 +

t

12 34

0 0

1D Majorana Fermions on Topological Insulators

• 2. S-TI-S Josephson Junction

SC TI SC φ = π φ ≠ π SC M

• 1. 1D Chiral Majorana mode at superconductor-magnet interface

TI kx E

† k k

γ γ − =

: “Half” a 1D chiral Dirac fermion

F

v

x

H i γ γ = − ∂

• φ

Gapless non-chiral Majorana fermion for phase difference φ = π

( )

cos( / 2)

F

v

L x L R x R L R

H i i γ γ γ γ φ γ γ = − ∂ − ∂ + ∆

Manipulation of Majorana Fermions

Control phases of S-TI-S Junctions

φ1 φ2

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 φ−π

1

E φ−π E φ−π

A Z2 Interferometer for Majorana Fermions

A signature for neutral Majorana fermions probed with charge transport

2 h N e Φ =

e e e h

γ1 γ2 γ1 γ2 −γ2

N even N odd

† 1 2 1 2

c i c i γ γ γ γ = − = +

• Chiral electrons on magnetic domain wall split

into a pair of chiral Majorana fermions

• “Z2 Aharonov Bohm phase” converts an

electron into a hole

Fu and Kane, PRL ‘09 Akhmerov, Nilsson, Beenakker, PRL ‘09

N G

Majorana Fermions in Three Dimensions

Majorana bound states arise as solutions to three dimensional BdG theories

Qi, Hughes, Zhang Model for edge of 3D topological insulator

( ) ( )

( ) Re ( ) Im ( )

z x z x y

H i m τ µ σ µ τ τ

• =

− ∇ + + ∆ + ∆

• r

r r

• ( )

H i n γ = − ⋅∇ + Γ⋅ r

• m < 0

Topological Insulator m > 0 Trivial Insulator

(γ1,γ2,γ3), (Γ1,Γ2,Γ3) : 8x8 Dirac matrices

( ) ( )

1 2 3

( ) , , Re( ),Im( ), n n n n m = = ∆ ∆ r

• Superconducting pairing

at surface

“hedgehog” configuration

Minimal O(3) n-vector model :

( ) n r

• Majorana bound state

Topological Classification of Defects

• Adiabatic approximation:

away from defect H varies slowly

( , ) H H = k r

• Particle-Hole symmetry (Class D) :

3

T ∈ k

2

S ∈ r

1

( , ) ( , ) H H

−

− = −Ξ Ξ k r k r

crystal momentum enclosing surface in real space

• 2 Topological Invariant : signals an enclosed zero mode

3 2

3 2 5

( , ) mod 2

T S d

d Q µ

×

= k r A F

Q5 = Chern Simons 5 form Analogous to Qi,Hughes Zhang formula

• O(3) n-vector model :

µ = Hedgehog number mod 2

2

S

r

Non-Abelian Exchange Statistics in 3D

Exchange a pair of hedgehogs: 2π rotation : Wavefunction of Majorana bound state changes sign Interchange rule: Ising Anyons

1 2 2 1

γ γ γ γ → → −

1 2

4 12 i

T e

π γ γ

=

[ ]

( )

1 2

(3) O π =

1

( ) n n = r

• 2

( ) n n = r

• Nayak, Wilczek ’96

Ivanov ‘01

Braidless Operation

In 3D, braids can be contracted to zero. There therefore must exist non trivial operations on stationary Majorana states. Gedanken expt: Spherical TI’s coated with superconductor, 4 vortices 1 3 2 4

12 34

+

• +

Braidless Operation

In 3D, braids can be contracted to zero. There therefore must exist non trivial operations on stationary Majorana states. Gedanken expt: Spherical TI’s coated with superconductor, 4 vortices

12 34 12 34

0 0 1 1

2 13 1 3 :

T γ γ

→

=

Braid 3 around 1

• r

Advance phase

2 ϕ ϕ π → +

1 2 3 4

ϕ

Braidless Operation

In 3D, braids can be contracted to zero. There therefore must exist non trivial operations on stationary Majorana states. Gedanken expt: Spherical TI’s coated with superconductor, 4 vortices

12 34 12 34

0 0 1 1

2 13 1 3 :

T γ γ

→

=

Braid 3 around 1

• r

Advance phase

2 ϕ ϕ π → +

1 2 3 4

2 ϕ π +

Fractional Josephson Effect

e

• Fu, Kane ’08

Kitaev ’01 Kwon, Sengupta, Yakovenko ‘04

γ1 γ2 γ4 γ3

12 34

0 0

12 34

1 1

• 4π perioidicity of E(φ) protected by local conservation
• f fermion parity.
• AC Josephson effect with half the usual frequency: f = eV/h

Conclusion

• A new electronic phase of matter has been predicted and observed
• 2D : Quantum spin Hall insulator in HgCdTe QW’s
• 3D : Strong topological insulator in Bi1-xSbx , Bi2Se3, Bi2Te3
• Superconductor/Topological Insulator structures host Majorana Fermions
• A Platform for Topological Quantum Computation
• Ising non-Abelian statistics allowed in 3D
• braidless operations
• Experimental Challenges
• Charge and spin transport measurements on topological insulators
• Superconducting structures :
• Create, Detect Majorana bound states
• Magnetic structures :
• Create chiral Dirac edge states, chiral Majorana edge states
• Majorana interferometer