When TuoPu meets PuTuo in 2011...May 22, 2011

The Search for Non-Abelian Anyons in The Search for Non-Abelian Anyons in Fractional Quantum Hall Systems Fractional Quantum Hall Systems

– The Past Five Years– The Past Five Years

Xin WanZhejiang Institute of Modern Physics, Hangzhou

Mini-Workshop on Topological Quantum ComputationMini-Workshop on Topological Quantum ComputationHangzhou, July 6-7, 2006Hangzhou, July 6-7, 2006

Theory oriented; FQHE + TQC (6 talks); 17 researchers and 10 studentsRecording available at http://zimp.zju.edu.cn/~xinwan/topo06/

CollaboratorsCollaborators

● Fractional quantum Hall effect in 2DEGs

Hua Chen (Zhejiang U), Zi-Xiang Hu (Princeton)

Ki Hoon Lee (Pohang), Ed Rezayi (Los Angeles)

Peter Schmitteckert (Karlsruhe), Kun Yang (Tallahassee)

● Universal topological quantum gate construction

Haitan Xu (U Maryland)

Michele Burrelle (Trieste), Giuseppe Mussardo (Trieste)

● Quantum Hall effect in rotating ultracold fermion systems with dipolar interaction

Ruizhi Qiu (ITP, Beijing), Su Yi (ITP, Beijing)

Zi-Xiang Hu (Princeton), Su-Peng Kou (Beijing Normal U)

OutlineOutline

● Motivation: Topological quantum computation

● A simple picture for the 5/2 FQH state

● Experimental progress on the 5/2 FQH state

– Shot noise, and

– Conductance, for charge tunneling across a narrow constriction

– Charge tunneling into localized states in the bulk

– Quasiparticle interference (and my understanding)

● Outlook

Motivation: Topological Quantum ComputationMotivation: Topological Quantum Computation

Fault-tolerant. Information is stored globally, while environmental noises are local. Thus decoherence due to noises is protected against. (Need non-Abelian anyons.)

Motivation: Topological Quantum ComputationMotivation: Topological Quantum Computation

Fault-tolerant. Information is stored globally, while environmental noises are local. Thus decoherence due to noises is protected against. (Need non-Abelian anyons.)

● In topological quantum computing, tensor decomposition is unnecessary and inconvenient – a leakage error occurs when a tensor decomposition is forced.

● Topological quantum gate construction, also known as topological quantum compiling:

– Given a set of fundamental gates (braids), finding a sequence approaching an arbitrary gate is a hard question.

● O(log(1/)) in time and O(log2(1/)) in length – beats currently the most efficient algorithms (i.e. the Solovay-Kitaev algorithm, c.f. the Nielsen & Chung book)

● Cited by Zhenghan Wang in Topological Quantum Computation (Published by American Mathematical Society, June 2010)

Haitan Xu and XW, Phys. Rev. A 78, 042325 (2008); Phys. Rev. A 80, 012306 (2009)

Burrello, Xu, Mussardo & XW, Phys. Rev. Lett. 104, 160502 (2010)Burrello, Mussardo & XW, arXiv:1009.5808, New J Phys (2011)

Topological Quantum Gate ConstructionTopological Quantum Gate Construction

Non-Abelian FQH StatesNon-Abelian FQH States

● Ising CFT: Moore & Read (1991); Morf (1998); Rezayi & Haldane (2000); Read & Green (2000)

● Parafermion CFTs: Read & Rezayi (1999)

● Das Sarma, Freedman & Nayak (2005)

● Experimental candidate: ν = 5/2 FQHE (Willett, 1987)

Moore-Read Pfaffian ~ p+ip superconductor? e/4 quasiparticles ~ flux hc/2e vortices

i=i

Majorana condition

Xia et al., PRL 93, 176809 (2004)

Microsoft Station Q @ UCSB

2DEG

FQH CondensatesFQH Condensates

● Condensate of composite bosons ( = 1/3)

● Condensate of composite fermions ( = 5/2 = 2 + 1/2)

e/3

e/4

e/4

qhe /4= ei/22

qhe /2=e i/2 , ei/2

Ground State DegeneracyGround State Degeneracy

A: (1, 2) (3, 4) B: (1, 3) (2, 4) C: (1, 4) (3, 2)

● Even though we fixed all the positions of the excitations, there are still internal degrees of freedom

● A, B, C are not linearly independent – though it is not obvious [Nayak & Wilczek, Nucl. Phys. B (1996)]

● We use two of their linear combinations as a basis set

Braiding = Quantum Evolution Braiding = Quantum Evolution = Quantum Computation = Quantum Computation

Non-Abelian StatisticsNon-Abelian Statistics

Ground state degeneracy robust against local perturbation!

ground state manifold

Excited states

Gap

Key Issues to ProveKey Issues to Prove

● Spin fully polarized

● Quasiparticles carry charge e/4 (not necessarily non-Abelian)

● Non-Abelian statistics (interferometry)

Tim

e

Noise, not cleanNoise, not cleanDolev et al., Nature 452, 829 (2008)

Tunneling fits, but not without an effortTunneling fits, but not without an effortRadu et al., Science 320, 899 (2008)

Local IncompressibilityLocal IncompressibilityVenkatachalam et al., Nature 469, 285 (2011)

With comparable gap, the disorder potential not altered. In the limit of an isolated compressible puddle surrounded by an incompressible fluid, incompressibility scales with local charge.

FQH Quasiparticle InterferometryFQH Quasiparticle Interferometry

edge of the ½ dropletpath via point contact 1

path via point contact 2

Gates controlling the strength of tunneling

Side gate controlling the number of quasiparticles on the central antidot

Edge of the filled Landau level is not included.

Das Sarma, Freedman & Nayak (05)Stern & Halperin (06); Bonderson, Kitaev & Shtengel (06)

G∝∣t1 U 1t 2 U 2∣ ⟩∣2=∣t1∣

2∣t2∣

22ℜ {t1

∗ t2 ei ⟨∣M n∣ ⟩ }

Observing Non-Abelian StatisticsObserving Non-Abelian Statistics

i− j , ji

Ivanov, Phys. Rev. Lett. 86, 268 (2001) In general, Us do not commute. In general, Us do not commute.

U ij

U ij=

1

21 ji

U 1a 2=a1

a1

1 2 a U 2a 2U 1a

2=a2a1=12

Dependent of the circling anyon!odd:

even:

no interference pattern

e.g. [a1 , b1] ≠ 0

To be or not to beTo be or not to be

B or Vbg

I

B or Vbg

I

Even number of non-Abelian quasiparticles inside the interference loop

Odd number of non-Abelian quasiparticles inside the interference loop

Model Experimental SystemsModel Experimental Systems

Pfeiffer et al., Appl. Phys. Lett. 55, 18 (1989)

Background charge (+Ne)

dElectron

layer (-Ne)

Φ = ΝΦ0 / ν

mmm

mmlnnlmmnl

lmn ccUccccVH +

+++

+ ∑∑ +=2

1

Coulomb interaction Confining potential

Advantage of disk geometry: interplay of edge modes and bulk quasiholes.

Pfaffian Stable in VPfaffian Stable in VCoulombCoulomb

+ U + UConfiningConfining

XW, Hu, Rezayi & Yang, PRB 77, 165316 (2008)

overlap ~ 0.5

Coulomb Interaction ( = 0)

repulsive 3-bodypure CoulombH=1−HCH 3B

Simple Pictures for FQH Edge ExcitationsSimple Pictures for FQH Edge Excitations

Integer QH edge: chiral Fermi liquid

ν = 1/3 FQH edge: chiral Luttinger liquid

Edge Spectrum AnalysisEdge Spectrum Analysis

M=∑l b

nb lb l b∑l f

n f l f l f

E=∑l b

nb l bb l b∑l f

n f l f f l f

H=12

H C12

H 3B

Right panels:

Bulk, bosonic, and fermionic edge excitations clearly distinguishable.

Left panel: Coulomb only.Bulk and edge excitations mixed up!

XW, Yang & Rezayi, Phys. Rev. Lett. (2006)

XW, Hu, Rezayi & Yang, PRB (2008)

Conclusion: Bose-Fermi separation Fermionic edge-mode velocity is much lower than the bosonic edge-mode velocity.

N=12 ; M gs=126

Non-Abelian Signatures at the edgeNon-Abelian Signatures at the edge

no quasihole one e/4 quasihole two e/4 quasiholes = one e/2 quasihole

Confirmation: The edge Majorana fermion mode has anti-periodic boundary condition (due to the 2 spinor rotation) in the presence of even number of charge e/4 quasiholes in the bulk, but periodic boundary condition in the presence of odd number of charge e/4 quasiholes.

H W=W c0 c0

XW, Yang & Rezayi, Phys. Rev. Lett. (2006)

Edge-mode VelocitiesEdge-mode Velocities

vc=5×106 cm/ s

vn=4×105 cm/ s

vc~e2

ℏ=

c

Neutral velocity is significantly smaller!

H =1−H CH 3B

Experimentally,

vc=8~15×106 cm/ s = 1Marcus grouparXiv:0903.5097

vc=4×106 cm/ s = 1/3Goldman groupPRB (2006)

Depends on interaction & comfinement!

vc=5×106 cm/ s

= 1/2)

XW, Hu, Rezayi & Yang, PRB (2008)

Interferometry AnalysisInterferometry Analysis

n bulk non-Abelian e/4 quasiparticles

odd-even effect: se /4 = { ±1 /2 n even0 n odd

Aharonov-Bohm effect

I 12∝∑qsq∣t1∣∣t2∣e

−∣x1−x2∣/ L cos 2 qe0

qarg t1∗ t2

tunneling amplitude

coherence length due to thermal smearing

favors e/4 qps favors e/2 qpsL=1

2 k B T g c

vc

gn

vn−1

se /2 = 1

qhe /4= ei/2 2

qhe /2= ei/2 ,e i/2

The Realistic Expectation (Low T)The Realistic Expectation (Low T)

B or Vbg

I

B or Vbg

I

Even number of non-Abelian quasiparticles inside the interference loop

Odd number of non-Abelian quasiparticles inside the interference loop

XW, Hu, Rezayi & Yang, PRB (2008)

At Higher TemperaturesAt Higher Temperatures

B or Vbg

I

B or Vbg

I

Even number of non-Abelian quasiparticles inside the interference loop

Odd number of non-Abelian quasiparticles inside the interference loop

XW, Hu, Rezayi & Yang, PRB (2008)

Predictions ObservedPredictions Observed

● At 10 mK, e/4 pattern observable only when device size < 4 µm

● Both e/4 and e/2 interference patterns observable

● At higher temperatures, e/4 pattern suppressed

● 25 mK, size ~ 1 µm

Wan, Hu, Rezayi & Yang, PRB (2008)

Willett, Pfeiffer & West, PNAS (2009)

Temperature DependenceTemperature Dependence

Parameters:B = 6 T = 13.1|x

1 – x

2| = 1 m

e/4: sensitive on interaction and confining potential

e/2: less sensitive

incoherent Hu, Rezayi, XW & Yang, PRB (2009)

only e/2 oscillations

e/4 & e/2Opposite trend for anti-Pfaffian

Willett Willett et alet al., PRB (2010)., PRB (2010)

Bishara & Nayak, Phys. Rev. B 77, 165302 (2008)Wan, Hu, Rezayi & Yang, Phys. Rev. B 77, 165316 (2008)

Bishara, Bonderson, Nayak, Shtengel & Slingerland, Phys. Rev. B 80, 155303 (2009)Chen, Hu, Yang, Rezayi & Wan, Phys. Rev. B 80, 235305 (2009)

See Viewpoint by K. Shtengel, Physics 3, 93 (2010)

OutlookOutlook

● The field is more exciting than ever

● Fractional quantum Hall effect in 2DEGs

– Independent experimental checks

– Samples & devices: innovating design

● Other systems

– Cold atomic systems (with Kou, Qiu, Yi)

– Graphene (with Bhatt, Hu, Yang)

– p-wave superconductors

– Noncentrosymmetric superconductors

Supported by NSFC, PCSIRT, MOST, Max-Planck Society, Korea MEST

FQHE ReferencesFQHE References

● Non-Abelian Fractional Quantum Hall States

– Model non-Abelian FQH states with Coulomb interaction; non-Abelian signatures in edge excitations; predictions on quasiparticle interferometry

– Estimate edge-mode velocities → coherence length/temperature

– Estimate quasiparticle tunneling amplitudes

– Scaling in quasiparticle tunneling amplitude → conformal dimensions

– Explore edge physics using Jack polynomials

Chen, Hu, Yang, Rezayi & XW, Phys. Rev. B. 80, 235305 (2009)

Hu, Rezayi, XW & Yang, Phys. Rev. B 80, 235330 (2009)

Hu, Lee, Rezayi, XW & Yang, New J. Phys. 13, 035020 (2011)

XW, Hu, Rezayi & Yang, PRB 77, 165316 (2008); Phys. Rev. Lett. 97, 256804 (2006)

Lee, Hu & XW, in preparation