Majorana Fermions and Topological Insulators

I. Topological Band Theory

- Integer Quantum Hall Effect - 2D Quantum Spin Hall Insulator - 3D Topological Insulator - Topological Superconductor

II. Majorana Fermions

- Superconducting Proximity Effect on Topological Insulators - A route to topological quantum computing?

Thanks to Gene Mele, Liang Fu, Jeffrey Teo

Charles L. Kane, University of Pennsylvania

The Insulating State

Covalent Insulator

Characterized by energy gap: absence of low energy electronic excitations

The vacuumAtomic Insulator

e.g. solid Ar

Dirac Vacuum

Egap ~ 10 eV

e.g. intrinsic semiconductor

Egap ~ 1 eV3p

4s

Silicon

Egap = 2 mec2 ~ 106 eV

electron

positron ~ hole

The Integer Quantum Hall State2D Cyclotron Motion, Landau Levels

IQHE without Landau Levels (Haldane PRL 1988)

Graphene in a periodic magnetic field B(r)

gap cE E

Hall Conductance xy = n e2/h

Band structureB(r) = 0Zero gap, Dirac point

B(r) ≠ 0Energy gapxy = e2/h

Egap

k

Topological Band TheoryThe distinction between a conventional insulator and the quantum Hall state

is a topological property of the manifold of occupied states

2( ) : ( ) Bloch Hamiltonans Brillouinwith

energy gap

zone H Tk

Trivial Insulator: n = 0 Quantum Hall state: xy = n e2/h

The TKNN invariant can only change at a phase transition where the energy gap goes to zero

Classified by the first Chern class (or TKNN invariant) (Thouless et al, 1984)

2Tr

2 T

in

F

The set of occupied Bloch wavefunctions defines a U(N)vector bundle over the torus.

1

( )N

i iu

k

ij i jA u du

d F A A A

Berry’s connection

Berry’s curvature

1st Chern class

Edge States

Gapless states must exist at the interface between different topological phases

IQHE staten=1

Egap

Domain wall bound state 0

Vacuumn=0

Edge states ~ skipping orbits

n=1 n=0

Band inversion – Dirac Equation

x

y

M<0

M>0

Gapless Chiral Fermions : E = v k

E

Haldane Modelky

Egap

KK’

Smooth transition : gap must pass through zero

Jackiw, Rebbi (1976)Su, Schrieffer, Heeger (1980)

Bulk – Edge Correspondence : n = # Chiral Edge Modes

Time Reversal Invariant 2 Topological Insulator

2 topological invariant (= 0,1) for 2D T-invariant band structures

=0 : Conventional Insulator =1 : Topological Insulator

Kramers degenerate at

time reversal invariant momenta

k* = k* + G

E

k*=0 k*=/a

E

k*=0 k*=/a

Edge States

1. Sz conserved : independent spin Chern integers : (due to time reversal) n n

is a property of bulk bandstructure. Easiest to compute if there is extra symmetry:

, mod 2n

2. Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch states

Quantum spin Hall Effect :J↑ J↓

E

Time Reversal Symmetry :

Kramers’ Theorem :

1 ( )H H k k *yi 2 1 All states doubly degenerate

2D Quantum Spin Hall Insulator

Theory: Bernevig, Hughes and Zhang, Science ’06Experiement: Konig et al. Science ‘07

HgTe

HgxCd1-xTe

HgxCd1-xTed

d < 6.3 nmNormal band order

d > 6.3 nm: Inverted band order

Conventional Insulator QSH Insulator

E

k

E

II. HgCdTe quantum wells

I. Graphene

• Intrinsic spin orbit interaction small (~10mK-1K) band gap

• Sz conserved : “| Haldane model |2”

• Edge states : G = 2 e2/h

Kane, Mele PRL ‘05

0 /a 2/a

↑

↓

↑

↑ ↓

↓

Eg

6 ~ s

8 ~ p 6 ~ s

8 ~ p Normal

Inverted

G ~ 2e2/h in QSHI

3D Topological InsulatorsThere are 4 surface Dirac Points due to Kramers degeneracy

Surface Brillouin Zone

2D Dirac Point

E

k=a k=b

E

k=a k=b

0 = 1 : Strong Topological Insulator

Fermi circle encloses odd number of Dirac points

Topological Metal :

1/4 graphene

Robust to disorder: impossible to localize

0 = 0 : Weak Topological Insulator

Fermi circle encloses even number of Dirac pointsRelated to layered 2D QSHI

OR

4

1 2

3

EF

How do the Dirac points connect? Determined by 4 bulk 2 topological invariants 0 ; (123)

kx

ky

Bi1-xSbx

Theory: Predict Bi1-xSbx is a topological insulator by exploiting inversion symmetry of pure Bi, Sb (Fu,Kane PRL’07)

Experiment: ARPES (Hsieh et al. Nature ’08)

• Bi1-x Sbx is a Strong Topological Insulator 0;(1,2,3) = 1;(111)

• 5 surface state bands cross EF

between and M

ARPES Experiment : Y. Xia et al., Nature Phys. (2009).Band Theory : H. Zhang et. al, Nature Phys. (2009).Bi2 Se3

• 0;(1,2,3) = 1;(000) : Band inversion at

• Energy gap: ~ .3 eV : A room temperature topological insulator

• Simple surface state structure : Similar to graphene, except only a single Dirac point

EF

Control EF on surface byexposing to NO2

Topological Superconductor, Majorana Fermions

††( ) k

k k BdGk k

cH c c H k

c

Particle-Hole symmetry :

Quasiparticle redundancy :

Discrete end state spectrum :

1( ) ( )BdG BdGH k H k † E E E E

0

*0

BdG

HH

H

0

E

E

-E0

E=0

Bogoliubov de GennesHamiltonian

=0“trivial”

=1“topological”

†0 0

Majorana Fermionbound state

“half a state”

1D 2 Topological Superconductor : = 0,1

END

†E E

E

(Kitaev, 2000)

BCS mean field theory : † † † † *

Periodic Table of Topological Insulators and Superconductors

Anti-Unitary Symmetries : - Time Reversal : - Particle - Hole :

Unitary (chiral) symmetry :

1( ) ( ) 12 ; H H k k

1( ) ( ) 12 ; H H k k

1( ) ( ) ; H H k k

RealK-theory

ComplexK-theory

Bott Periodicity

Altland-ZirnbauerRandom MatrixClasses

Kitaev, 2008Schnyder, Ryu, Furusaki, Ludwig 2008

Majorana Fermion : spin 1/2 particle = antiparticle ( † )

Particle Physics : • Neutrino (maybe) Allows neutrinoless double -decay.

Condensed matter physics : Possible due to pair condensation

• Quasiparticles in fractional Quantum Hall effect at =5/2• h/4e vortices in p-wave superconductor Sr2RuO4

• s-wave superconductor/ Topological Insulator ... among others

Current Status : NOT OBSERVED

† † 0

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

• 2N separated Majoranas = N qubits

• Quantum Information is stored non locally - Immune from local sources of decoherence

• Adiabatic Braiding performs unitary operations - Non Abelian Statistics a ab bU

1 2i

Potential Hosts :

Topological Quantum Computing Kitaev, 2003

Proximity effects : Engineering exotic gappedstates on topological insulator surfaces

†0 ( v )H i

† † *

Dirac Surface States :

Protected by Symmetry

1. Magnetic : (Broken Time Reversal Symmetry)

• Orbital Magnetic field :

• Zeeman magnetic field :

• Half Integer quantized Hall effect :

2. Superconducting : (Broken U(1) Gauge Symmetry)

proximity induced superconductivity

p p eA †

zM 21

( )2xy

en

h

• S-wave superconductor

• Resembles spinless p+ip superconductor

• Supports Majorana fermion excitations

Fu,Kane PRL 07Qi, Hughes, Zhang PRB (08)

Fu,Kane PRL 08

M. ↑

T.I.

S.C.

T.I.

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

TI

0

E

†0 0

S.C.M

QSHIEgap =2|m|

Domain wall bound state 0

m<0

m>0

| | | |S Mm

1D Majorana Fermions on Topological Insulators

2. S-TI-S Josephson Junction

SC

TI

SC

SCM

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

TI

kx

E

†k k : “Half” a 1D chiral Dirac fermion Fv xH i

0

Gapless non-chiral Majorana fermion for phase difference

cos( / 2)Fv L x L R x R L RH i i

Manipulation of Majorana FermionsControl phases of S-TI-S Junctions

12

0

Majorana present

Tri-Junction : A storage register for Majoranas

CreateA pair of Majorana boundstates can be created from the vacuum in a well definedstate |0>.

BraidA single Majorana can bemoved between junctions.Allows braiding of multipleMajoranas

MeasureFuse a pair of Majoranas.

States |0,1> distinguished by• presence of quasiparticle.• supercurrent across line junction

E

00

1E

E

0

0

0

0

0

0

0

A Z2 Interferometer for Majorana Fermions

A Signature for Neutral Majorana Fermions Probed with Charge Transport

2

hNe

e e

e h

1

2

1

2 2

N even

N odd

†1 2

1 2

c i

c i

• Chiral electrons on magnetic domain wall split into a pair of chiral Majorana fermions

• “Z2 Aharonov Bohm phase” converts an electron into a hole

• dID/dVs changes sign when N is odd.

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

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

• Experimental Challenges - Charge and Spin transport Measurements on topological insulators

- Superconducting structures : - Create, Detect Majorana bound states - Magnetic structures : - Create chiral edge states, chiral Majorana edge states - Majorana interferometer

• Theoretical Challenges

- Further manifestations of Majorana fermions and non-Abelian states - Effects of disorder and interactions on surface states