Introduction to Majorana fermions in condensed matter physics

Geneviève Fleury(1), Jian Li(2) and Markus Büttiker(2)

(1) CEA Saclay – Service de Physique de l’Etat Condensé

(2) University of Geneva – Département de Physique Théorique

Introduction to Majorana fermions

in condensed matter physics

WHAT IS A MAJORANA FERMION ?

Ettore Majorana

A Majorana fermion is a fermion

which is its own anti-particle

= †

(it has to be charge neutral)

E. Majorana, Nuovo Cimento 14, 171 (1937); F. Wilczek, Nature Physics 5, 614 (2009)

Quantum relativistic description of spin-1/2 particles

Dirac equation : (i - m) = 0

Complex solution

particle ≠ * antiparticle

There also exist real solution

particle = * antiparticle

Dirac (1928) Majorana (1937)

Majorana fermion

It is a spinor, solution of the Dirac equation for spin-1/2 particles

{i , j†} = ij : fermion anticommutation relation

Non-abelian statistics (promising for topological quantum computation)

² = 1 (instead of ĉ² = 0 for « normal » fermions)

A Majorana fermion is a fermion …

… but a very weird fermion …

WHY IS IT INTERESTING ?

… which has not yet been observed experimentally

Some candidates in particle physics : neutrinos, photinos, dark matter

New candidates in solid-state physics

…as excitations in topological superconductors

Qi & Zhang, Rev. Mod. Phys. 83, 1057 (2011)

Majorana fermions in condensed matter physics

• How to engineer p-wave superconductors (hosting Majoranas) ?

- With a topological insulator in proximity to a standard superconductor

- Topological insulator is not needed !

..Strong spin-orbit coupled semiconductors in a magnetic field are enough

• How to detect Majorana fermions ?

- First strategy : detection via the anomalous Josephson effect

- Second strategy : signature of Majoranas in interferometry experiments

• Why looking for Majoranas in p-wave superconductors?

- They are guaranteed by particle-hole symmetry in spinless superconductors

- They appear as egde states in superconductors with topological order

TOY MODEL OF A 1D SPINLESS P-WAVE SUPERCONDUCTOR

Kitaev (2001)

j j+1 j-1 1 N ………………… ……

p-wave pairing


Kitaev (2001)

j j+1 j-1 1 N ………………… ……

Change of

basis « Artificial »

Majorana fermions


Kitaev (2001)

j j+1 j-1 1 N ………………… ……

Majorana fermions have to come by pairs. They appear at the two ends of the wire.

No Majorana

Case 1 : = t =0

Two separated Majoranas

Case 2 : = 0 and = -t

Change of

basis « Artificial »

Majorana fermions

A QUICK REMINDER ABOUT THE QUANTUM HALL STATE

Number of edge states n

[ Quantized conductivity xy = n.(e²/h) ]

Thouless et al (TKNN), PRL 49, 405 (1982)

Bulk-edge correspondence

B

Fermi

level

Landau

levels Bulk topological (Chern) number n

Topologically robust to (small) perturbations

MAJORANA FERMIONS as EDGE STATES of a TOPOLOGICAL SC

2D p-wave (spinless) topological superconductor

Bulk superconducting gap

-

0

Qi et al, PRL 102, 187001 (2009)


ky

Energy


y

ky

Energy

Qi et al, PRL 102, 187001 (2009)


Bulk topological order

Edge states


ky

Energy

0= 0†


y

ky

Energy

Qi et al, PRL 102, 187001 (2009)

E= -E†

0= 0†

Chiral

Majorana

fermion

Chiral

Majorana

fermion


Bulk topological order

Edge states

= chiral Majorana

Guaranteed by

particle-hole symmetry

in spinless systems ( )

ENGINEERING SPINLESS P-WAVE SUPERCONDUCTORS

Majoranas should appear as midgap states in spinless p-wave superconductors

Read & Green, PRB 61, 10267 (2000); Ivanov, PRL 86, 268 (2001), Kitaev (2001) ~ 2000

PROBLEM : p-wave superconductors are rare in nature

Few candidates : Sr2RuO4 or He-3 superfluid phase

2008 - …

in proximity to a standard

superconductor

a material with

strong spin-orbit coupling

to avoid spin-degeneracy

Idea : building suitable Hamiltonians using …

Fu & Kane, PRL (2008) Sau et al, PRL (2010); J. Alicea, PRB (2010)

Oreg, Refael & Von Oppen, PRL(2010)

2D TIME-REVERSAL INVARIANT COUNTERPART OF THE QH STATE

Bernevig et al, Science 314, 1757 (2006); Hasan & Kane, Rev. Mod. Phys. 82, 3045 (2010)

2D time-reversal breaking

topological insulator

Intrinsic spin-orbit coupling

Quantum Hall state VS Quantum Spin Hall state

Fermi

level

2D time-reversal invariant

topological insulator

Insulating bulk

Helical edge states

B

Magnetic field

Insulating bulk

Chiral edge states

FROM 2D TO 3D TOPOLOGICAL INSULATORS

2D TI

Experimental realization :

HgTe/CdTe quantum well

Experimental realization :

Bi1-xSbx, Bi2Se3, Bi2Te3

Massless Dirac fermions in 1D

H0 = ħvF pxz

Massless Dirac fermions in 2D

« 1/4 - graphene »

H0 = ħvF (pxx + pyy)

Edge states

3D TI

Surface states

2D Topological insulator 3D Topological insulator

Topologically

protected crossing

Basis :

ENGINEERING an EFFECTIVE P-SC with a STANDARD SC and a TI

Fu & Kane, PRL 100, 096407 (2008)

Surface state of a TI in proximity to a standard SC = 2D topological superconductor

p-wave pairing

Change of basis :

with

s-wave pairing

Superconducting proximity effect : Volkov et al, Physica C 242, 261 (95)

MF IN {TI+SC} STRUCTURES : VARIOUS GEOMETRIES

Fu & Kane, PRL 100, 096407 (2008)

2D TI

Supra Ferro

Localized MF

0= 0†

E = 0

Energy

Localized MF

0= 0†

3D TI

SC

E = 0

Energy

Chiral MF

E= -E†

ky

Energy

0= 0†

InAs

Magnetic

insulator M

(M B// + Dresselhaus SOC)

Zeeman spin-splitting

MAPPING BETWEEN INAS AND TOPOLOGICAL INSULATOR

Sau et al, PRL 104, 040502 (2010), PRB 82, 214509 (2010)

J. Alicea, PRB 81, 125318 (2010), Oreg, Refael & Von Oppen, 105, 177002 (2010)

Semiconductor

Topological

insulator

InAs

Semicond. with strong

spin-orbit coupling

Rashba spin-splitting

Around the Fermi level and for large enough magnetic field :

InAs chiral edge states of a topological insulator

But there is « no topology » in InAs semiconductor

No Majorana No Majorana One Majorana

in the vortex core

TOPOLOGICAL PHASE TRANSITION IN INAS

Trivial phase

No magnetic

field B

Trivial phase Phase

transition

Topological

phase

SC

M M

InAs

Chiral Majorana fermion

at an interface

ky

InAs

SC

+ B field

One Majorana bound state

in the vortex core

E = 0

MF IN {INAS+SC} STRUCTURES : VARIOUS GEOMETRIES

InAs nanowire on SC

B

Two Majorana bound states

at the ends of the nanowire

E = 0

SIGNATURE OF MAJORANA : ANOMALOUS JOSEPHSON EFFECT

One crossing, robust

One pair of Majorana for =

Josephson

current

I(=) = 0

2-periodicity

Josephson

current

I(=) max

4-periodicity

Trivial phase

Topological phase

= 0 = 0 ≠ 0 ≠ 0 = 0 ≠ 0

weak B strong B

p-SC: Kwon, Sengupta & Yakovenko, Low Temp. Phys. 30, 613 (2004)

TI+SC : Fu & Kane, PRB 79, 161408 (2009), Ioselevich & Feigel’man, arXiv:1012.0407 (2010)

InAs+SC : Lutchyn, Sau & Das Sarma, PRL 105, 077001 (2010)

Two crossings, not robust

NO Majorana

Theory of Majorana interferometry in Majorana basis : Fleury, Li & Büttiker (writing in progress)

SIGNATURE OF MAJORANA in INTERFEROMETRY EXPERIMENTS - 1 :

FABRY-PEROT INTERFEROMETER

Phase acquired by around the loop :

= kL + + n

Berry phase

(spin)

Berry phase

(n vortices)




is only coupled to L1 and R1

L2 and R2 are completely reflected

(true at low energy : E 0)

Great simplification

Change of basis

« Artificial » Majorana fermions in the normal leads ( = L,R)

Phase acquired by around the loop :

= kL + + n

Berry phase

(spin)

Berry phase

(n vortices)

Law et al, PRL 103, 237001 (2009); In time-reserval inv. TSC : Béri, arXiv:1102.4541 (2011)


Current is given by the interference of (a pair of) Majorana fermions

(even though each individual of them does not carry charge)

Average current in the right lead :

IR = IRR + IRL

IRR

IRL



… because two Majoranas coming from different leads are

charge neutral and not phase-coherent

IRL = 0 , ie the left lead does not inject current into the right one …

Two Majoranas

coming from the same lead are

phase-coherent and can interfere



Zero-frequency current-current cross-correlator :

Fabry-Perot

for electrons

Fabry-Perot

for Majoranas

Cross-correlation is given by the interference of 2 pairs of Majorana fermions only

Thermal noise

Exchange noise Shot noise


MACH-ZEHNDER INTERFEROMETER

TI+SC : Fu & Kane, PRL 102, 216403 (2009), Akhmerov et al, PRL 102, 216404 (2009)

InAs+SC : Sau, Tewari & Das Sarma, PRB 84, 085109 (2011)

General theory in Majorana basis : Fleury, Li & Büttiker (writing in progress)






electron e = 1 + i2






electron e = 1 + i2

hole h = 1 – i 2

sign determined by the difference

of phases acquired by the two MFs :

= 1 - 2 = k.L + + n






electron e = 1 + i2

hole h = 1 – i 2



= 1 - 2 = k.L + + n Choosing proper

Majorana basis






electron e = 1 + i2

hole h = 1 – i 2



= 1 - 2 = k.L + + n Choosing proper

Majorana basis

Current in the drain : Interference between 2

chiral Majorana fermions

Zero frequency noise : Thermal noise only

(NO partition or exchange noise)


HANBURY BROWN - TWISS INTERFEROMETER

Strübi et al, PRL 107, 136403 (2011); Fleury, Li & Büttiker (writing in progress)




Choosing proper

Majorana basis




Choosing proper

Majorana basis

No average current in the two drains :

Phase-incoherent

Majorana pair

Phase-incoherent

Majorana pair




Choosing proper

Majorana basis

No average current in the two drains :

Phase-incoherent

Majorana pair

Phase-incoherent

Majorana pair

But non-zero cross-correlators (here exchange noise only):

!!!

: phase difference

CONCLUSION

Majorana fermions are predicted to appear as zero-energy midgap states in

…spinless p-wave topological superconductors

They should soon « materialize » … if they do exist

Experimental efforts in Delft, Harvard, UCSB, Saclay …

See also Sasaki et al, PRL 107, 217001 (nov. 2011)

Zero-bias conductance peak

They should give rise to fancy effects : anomalous Josephson effect, only exchange

…noise in interferometry experiments, non-local behaviour, non-abelian statistics …

Introduction to Majorana fermions in condensed matter physics

Documents

Transcript of Introduction to Majorana fermions in condensed matter physics