Topological Matter

• 2D topological insulators (quantum spin-Hall effect)

• 2D topological superconductors (Majorana fermions)

• 1D topological superconductors

Topological Matter

• 2D topological insulators (quantum spin-Hall effect)

• 2D topological superconductors (Majorana fermions)

• 1D topological superconductors

E

EFnormal semiconductor,

or vacuumtopological

insulator

topological insulators (parity inversion)

E

EFodd parity(p-band)

even parity(s-band)

normal semiconductorgraphene :(

E

EFeven parity

(s-band)

odd parity(p-band)

HgTe/CdTe or InAs/GaSb quantum wells (2D),

Bi2Se3, Bi2Te3, SmB6 (3D)+ many, many more candidates

needed for parity inversion: strong spin-orbit coupling (heavy atoms)

E

EF

¿topology?

E

EF

© Justin Wells (Aarhus)

¿topology?

© Ju

stin

Wel

ls (

Aar

hus)

Hong Kong

China

E

EF

vacuum inverted band gapsemiconductor

topological insulator:insulating bulk + gapless surface

topological consequenceof band inversion

only a “bulk” inversion can remove the crossing (“topological protection”)

Stockholm, 3 September 1967

2D topological insulator(HgTe quantum well)

2D topological insulator

edg

e st

ate

s

ordinary insulator

x

y

© M

olen

kam

p

quantum spin Hall effect – zero-field analogue of thequantum Hall effect

¿quantized conductance?

valence band

conduction band

valence band

momentum

ener

gy

conduction band

topological insulator ordinary insulator

surface states

band gap

0

momentum

0

Fu, Kane & Mele | Moore & Balents | S.C. Zhang et al.

Kramers degeneracy protects the crossing at zero momentum

• 2D topological insulators (quantum spin-Hall effect)

• 2D topological superconductors (Majorana fermions)

• 1D topological superconductors

Topological Matter

Majorana fermions might emerge as midgap states in a superconductor

Majorana fermion at E=0(p-wave order needed to remove zero-point motion)

Volovik (1999)

Bogoliubov meets Majorana

Bogoliubov quasiparticle

superposition of electron and hole excitations in a superconductor

spinless bound state at zero energy is a Majorana fermion

particle = antiparticleobstacle: zero-point motion

bound states in a vortex core

however, zero-point motion prevents a bound state at E=0

Fu & Kane (2008): use Berry phase of massless electrons to eliminate the ½ phase shift

so Majorana fermion at E=0

electron-hole symmetry:

Bpseudospin rotates 2π ➟ Berry phase shift of π ➟ Landau levels shifted

2005: electrons in a carbon monolayer are massless (like photons) and have a “pseudospin”

pointing in the direction of motion

the problem with graphene

• each Majorana fermion is four-fold degenerate (2x spin + 2x valley)

• nonlocality broken if spins and/or valleys can mix

for full protection from decoherence, we need massless electrons without spin or valley degeneracy

3D topological insulators to the rescue

3D topological insulator(Bi2Se3, Bi2Te3)

massless Dirac Hamiltonian

all degeneracies removed

proximity to a superconductor transforms the surface of a 3D topological insulator into a 2D topological superconductor, with Majorana fermions in vortex cores

insulating bulk, metallic surface

Hsi

eh, H

asan

(200

9)

• 2D topological insulators (quantum spin-Hall effect)

• 2D topological superconductors (Majorana fermions)

• 1D topological superconductors

Topological Matter

1D topological superconductor(InSb nanowire)

in a superconductor, electron-hole symmetry would pin these states at E=0

Shockley states

Majorana fermions!

gap closes and reopens upon formation of a line defect, leaving behind a pair of bound states at the end points

Majorana-Shockley statesin a p-wave superconductor

p-wave from s-wave

© Jay Sau

Lutchyn, Sau & Das Sarma

Oreg, Refael & Von Oppen

Nb or Al superconductor

InAs nanowireMagnetic field

s-wave proximity effectRashba spin-orbit couplingZeeman spin polarization

+p-wave superconductor

�0.04 �0.02 0 0.02 0.04klso

�20

�10

0

10

20

E/E

so

µ = �5

−300 −200 −100 0 100 200 300Vbias [ µV ]

0

0.5

1

1.5

G[2e2/h]

model calculation: Wimmer & Akhmerov

superconductor (Nb)

semiconductor (InSb)

barrierbound state

experiment:Mourik, Zuo, Frolov, Plissard, Bakkers & Kouwenhoven(Science, April 2012)