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 Wan Zhejiang Institute of Modern Physics, Hangzhou
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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)
Fault-tolerant. Information is stored globally, while environmental noises are local. Thus decoherence due to noises is protected against. (Need non-Abelian anyons.)
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)
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.
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)