Condensed Matter Search for (elusive?) Condensed Matter Search for (elusive?) MajoranaMajorana Modes Modes

• Sau, Lutchyn, Tewari, Stanescu, Lobos, Liu, Lin, Hui• Halperin, FlensbergHalperin, Flensberg• Alicea, Refael, von Oppen, Fisher, Stern, Brouwer, Loss,

Beenakker, Fu, Kane, … and many more (now a huge subject)• Kouwenhoven Marcus Yacoby and others (experiment)• Kouwenhoven, Marcus, Yacoby,… and others (experiment)

• 5/2 FQHE, SrRuO4 , TI, He-3, Cold Atoms,…..• Semiconductor/Superconductor Sandwich• Semiconductor/Superconductor Sandwich

Structures (2D, 1D, 0D)• s-SC+ proximity in SM + SO coupling + spin splittings SC+ proximity in SM + SO coupling + spin splitting• Strong SO-coupling + Low Disorder + Good Interface

MajoranaMajorana Returns!Returns!Plus a remarkably large number of theory proposals in cold atoms! ?

• Majorana/Fermi (1937) : “Real version” of Dirac Theory• Majorana disappears (1938)

cold atoms! ?

j pp ( )• Neutrino as Majorana (??): Double beta decay (lepton number!)• Neutralino of SUSY is a Majorana fermion• Read and Green (2000): 5/2 FQHE = Chiral p+ip SC• Kitaev (2001): Majorana 1D chain• Das Sarma, Freedman, Nayak (2005): 5/2 FQH ‘Majorana’ Qubit• Das Sarma, Nayak, Tewari (2006):SrRuO4 Majorana (half-vortex)• Fu Kane (2008): Topological Insulator + SC• Fu, Kane (2008): Topological Insulator + SC• Sau, Lutchyn, Tewari, Das Sarma (2010): SO/SM + SC + FMI• Alicea (2010): SO/SM + SCAlicea (2010): SO/SM SC• Lutchyn, Sau, Das Sarma (2010): Majorana Nanowire• Sau, Lin, Hui, Das Sarma (2012): QD + SC; ‘periodic struture’

2009-2011 (December)

Both the Majorana and the half-vortex have apparently already been seen

New Anomaly in the Transverse Acoustic Impedance of S fl id3H B ith W ll C t dSuperfluid3He-B with a Wall Coated by Several Layers of 4He

V i iV i i DD i li l S dS d( )Vortices in Vortices in 2D 2D spinlessspinless SuperconductorSuperconductor( )x yp ipCHIRAL, p-WAVE, SPINLESS; ZERO ENERGY MODE AT THE CORE

2 nOrder parameter phase rotates by around the core

Order parameter amplitude suppressed at the core2D Majorana is an anyon Bound states in vortex

Low energy normal bound states in the core

Not a regular fermion because it is zero-energy! Emergent Mode at low energy

cores are e-h symmetric

gyZERO-ENERGY CORE STATE IS MAJORANAPROTECTED BY INDEX THEOREM (e-h symmetry)

Qubit decoherence due to Majoranatunneling: The limit to topological protectiontunneling: The limit to topological protection

arXiv:1112 3662Fermion parity is the key here arXiv:1112.3662

As long as T is much less than the superconducting (proximity) gap, the topological protection applies even if the Majorana minigap is much smaller than T, i.e. even when the non-zero BdG subgap levels are occ pied in the ‘defect’ Not an ob io s res lt at all Visibilit ma beoccupied in the ‘defect’: Not an obvious result at all. Visibility may be affected, but not protection

Brief history of Majorana fermions Brief history of Majorana fermions

Simple clever modification of the Dirac

Question: Are equations for spin-1/2 particles necessarily complex ?

Simple clever modification of the Dirac equation that involves ONLY REAL

numbers

Majorana fermion - electrically neutral particle which is its own antiparticle

Experimental status:

p p

Relevance: Experimental status:

NOT observedparticle physics (neutrinos)

(neutralinos)EMERGENT MAJORANA?

NonNon‐‐abelianabelian statistics and topological statistics and topological qubitsqubitsincreasing simplicityincreasing simplicityincreasing simplicityincreasing simplicity

Fractional Quantum Hall 1D Kitaev chain

Chiral p‐wave superconductors

Majorana Fermions

Topological insulator/S‐wave superconductor

Majorana Fermions

S wave superconductor

Spin‐orbit coupled S i bit l dSpin‐orbit coupled Semiconductor/

S‐wave superconductor

Spin‐orbit coupled Nanowire/

S‐wave superconductor

Quantum Hall Effect: Which QH states might be non-Abelian?

The fragile 5/2 plateau is astrong candidate to be astrong candidate to be anon-Abelian quantum Hallstate (‘Pfaffian’)state ( Pfaffian ).

First seen (1987) by Willett et al.

To determine if it is, we wouldhave to measure the effect ofquasiparticle braiding.There is sound theoretical

reason to believe that the

VERY SMALL GAP~ 10-100 mK!FRAGILE!

even-denominator 5/2 FQHE is a chiral p-wave SC with Majoranas as quasiparticles

arXiv:1008.1587; PRB 2011

Odd-even effect as in SC grains (in 5/2 but not 1/2 FQHE)

F(r) ~ SC gap parameter: finite!Presence of SC order(in 5/2, but not 1/2, FQHE)

Direct numerical demonstration of superconductivity in 5/2 FQHE

Presence of SC order

STORNI MORF DAS SARMA PRL 2010

5/2 FQHE: No Quantum Phase Transition as a function of x: Adiabatic continuity x=0 to 1STORNI, MORF, DAS SARMA PRL 2010

1/2 FQHE: Quantum Phase Transition as a function of x: No Adiabatic continuity x=0 to 1MORF 2008

Finite-Layer Thickness Stabilizes the Pfaffian State for the 5/2 Fractional Quantum Hall Effect: Wave Function Overlap and Topological Degeneracy

Mi h l R P t Th J li d S D SMichael. R. Peterson, Th. Jolicoeur, and S. Das Sarma

Phys. Rev. Lett. 101, 016807 (2008); See also, PRB 78, 155308 (2008)

Unity wavefunction overlap and the correct degeneracy show up for d/l~4

PROPOSED TOPOLOGICAL QC PROPOSED TOPOLOGICAL QC ARCHITECTURESARCHITECTURESARCHITECTURESARCHITECTURES

• OPTICAL LATTICES (analog simulation of Kitaevmodel, extended Hubbard model, etc.)

• COLD ATOMS (rotating BEC, p+ip fermionicsuperfluid, artificial B-field)

• ‘SUITABLE’ MAGNETIC LATTICESSUITABLE MAGNETIC LATTICES• SUPERCONDUCTOR sandwiches • p+ip SUPERCONDUCTORS (ruthenates)• TOPOLOGICAL INSULATORS + s-SC• NONABELIAN FQHE STATES

KEY IDEA KEY IDEA: Non-Abelian modes in p-SC (‘Majorana’)Sufficient for TQC up to some regular noisy gates (~ 20% error)

All involve Majorana modes as the basicAll involve Majorana modes as the basic qubit element: (SU2)2 conformal field theory

arXiv:1103.2770PRL2012

arXiv:1111.6600

Majorana bound states in SM/SC heterostructureMajorana bound states in SM/SC heterostructure

Topo condition does not involve SO coupling

MajoranaMajorana at spinat spin‐‐orbit coupled semiconductor (orbit coupled semiconductor (SmSm))‐‐SC interfacesSC interfaces

Need single non degenerate Fermi surfaceNeed single non-degenerate Fermi-surface HSm= k2 + Vzσz

E

Rashba induced chiralspin-texture allows singlet pairing

Fermion doubling avoided:

Band structure of spin-split spin-orbit coupled semiconductor

k

Majorana!

semiconductorRashba + Zeeman break inversion and timeRashba + Zeeman break inversion and time--reversal for chiral edge modereversal for chiral edge mode

Sau, Lutchyn, Tewari, Das Sarma PRL(2010)

Engineering Engineering spinlessspinless p+ipp+ip superconductor superconductor

Ordinary S-wave SCOrdinary S wave SC+

2D (or 1D) Semiconductorwith Strong SO Coupling

Sau et al. PRB (2010)Sau, Tewari, Das Sarma Ann Phys’10

Superconductors are natural hosts for MajoranaSuperconductors are natural hosts for Majorana

Topological protection of zeroTopological protection of zero--energy modeenergy mode

Semiconductor with spinSemiconductor with spin--orbit interactionorbit interaction

1D wires with spin-orbit: helical state

Majorana quantum wires

Lutchyn, Sau, Das Sarma PRL 2010

What are the roles of various parameters?

SC gap, SC-SM tunneling, SO coupling, Zeeman splitting, h i l t ti l SC di dchemical potential, SC disorder,

SM disorder, system configuration, environment……….

Large SC gapWeak SC-SM tunnelingWeak SC-SM tunnelingLarge SO couplingNot too large Zeeman splittingCl t ibl SM b lli tiCleanest possible SM ballisticLittle chemical pot. FluctuationsSC disorder (short-range) okay( g ) yMultiband betterBut not too many bands

Stanescu, Lutchyn, Das Sarma; arXiv 2011 PRB 2011

The goal is to have the largest The goal is to have the largest possible SC proximity induced possible SC proximity induced a i the SMa i the SM ith al o theith al o thegap in the SM gap in the SM with also the with also the 

largest possiblelargest possible minigapminigap oror BdGBdGlargest possible largest possible minigapminigap or or BdGBdGquasiparticlequasiparticle gap for the gap for the 

MajoranaMajorana mode mode staying within staying within the topolo i al re imethe topolo i al re imethe topological regimethe topological regime

This requires careful tuning ofThis requires careful tuning ofThis requires careful tuning of This requires careful tuning of all the system parameters!all the system parameters!

Xi 1111 2054arXiv:1111.2054PRB2012

>>0 !>>0 !

Gap as large as possible

SO coupling as large as possible

Disorder as small as possible

TS phase arises after the s-SC is killed by the Zeeman field

Density of states across phase transition

Experimental considerations

Nature 442, 667 (2006)

Tunneling experimentsTunneling experiments

arXiv:1106.3708

Calculatedspectrum wavefunctionand experimentalexperimentaltunneling results formultiband

inanowirein the presence of realistic disorder

Multi-band semiconductor nanowiresThe image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

Fractional ac Josephson effect

Josephson current through heterostructure

Experimental proposal: nanowire embedded into SQUID

The search for the Majorana ‘fermion’ may finally be coming to an endmay finally be coming to an endThe Majorana mode may soon be observed in table-top experiments as an emergent zero-energy mode in solid stateemergent zero energy mode in solid state semiconductor-superconductor sandwich t tstructures

‘Majorana’ may return after a 75-year j y f yhiatus – thanks to topological quantum computationcomputationActually, we get more than the original Majorana particle, we get the realization of the 2D Majorana operator and non-abelian statistics

Selected questions/comments/answers during the talk with appropriate slide numbers in the parenthesis of each item

Q. Marcus (slide 1): Why does topological phase (or Majorna) show up in so many different places theoretically?

A. Because the chiral spinless p-wave SC generically shows up in many places, d h ll h M j d i h i ‘d f ’and they all have Majorna modes in their ‘defects’

Q. Halperin (slide 3): Where is the Majorana claimed to be in the 3D TI system?A In the 2D topological surface layerA. In the 2D topological surface layer

Q. Millis (slide 3): Why is the Majorana hard to observe in cold atoms in spite of many theoretical proposals?many theoretical proposals?

A. Because the measurement techniques in cold atom systems are highly limited and there is little that could simulate transport or tunneling od solid state measurements

Q. Halperin (slide 10): What is the geometry for the numerical calculation?A. It is the standard spherical geometry.

Q. Kouwenhoven, Marcus, Stern, Halperin, Fisher (slide 15): What about Coulomb blockade? What about interaction effects in the QD/SC structure? Does the SC need to be phase coherent?

A. Coulomb blockade is incorporated. Interaction effects are not particularly important here. The SC must be phase coherent, i.e. fingers coming in from the same SC block.

Q. Kouwenhoven (slide 20): How is the Majorana protected?A. Majorana, being a zero energy mode which is neither electron nor hole, cannot

decay into the electron or hole statesdecay into the electron or hole states.

Q. Marcus/Kouwenhoven (slide 24): Can nuclear spins be important here?A. Yes, through its effect on the Overhauser field affecting the electrons.A. Yes, through its effect on the Overhauser field affecting the electrons.

Q. Freedman (slide 24): What is the intuitive reason for the magnetic field to suppress the topological gap?pp p g g p

A. If the electron spins are parallel, as would happen for a large magnetic field, the basic proximity effect would simply vanish since the SC is an s-wave SC

Q. Marcus (slide 27): How important is disorder and why?A. Disorder scale in the semiconductor must be smaller than the SC gap,

otherwise the p-wave gap will vanish since it is not immune to disorder

Q. Yacoby, Kouwenhoven, Stern (slide 31): What is the nature of the various subgap states? Do they move around with magnetic field and other parameters?

A. The subagap states arise from random impurities in the system (except for the zero-energy Majorana state in the topological phase above the critical magnetic field), and they all move around with system parameters such as the

i fi ld Th M j i i h l i lmagnetic field, etc. The Majorana remains at zero energy in the topological phase, and does not exist in the non-topological phase. It is either there as a robust state or it is not there.

Q. Halperin/Aleiner/Glazman/Millis: How is the interaction included in the proximity calculation?

A. The interaction is included in the mean field level in the proximity 2DA. The interaction is included in the mean field level in the proximity 2D calculation and in the RG treatment in the 1D calculation. For the 2D mean field calculation, it is just like the “mu*” effect in the Eliashberg theory.

Q. Falko: Can one see the Majorana in a SC metal wire if it also has a strong SO coupling and an externally applied magnetic field?

A. No, because far too many subbands (“channels”) will be occupied in metals

Q. Levitov: What are the experimental requirements?A. Necessary condition is zero-bias anomaly, sufficient is fractional ac Josephson