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Proposed experimental probes of non-abelian
anyons
Ady Stern (Weizmann)
with: N.R. Cooper, D.E. Feldman, Eytan Grosfeld, Y. Gefen,
B.I. Halperin, Roni Ilan, A. Kitaev, K.T. Law, B. Rosenow, S.
Simon
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Outline:
1. Non-abelian anyons in quantum Hall states – what they
are, why they are interesting, how they may be useful
for topological quantum computation.
2. How do you identify a non-abelian quantum Hall state
when you see one ?
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Introductory
pedagogical
Comprehensive
More precise and relaxed presentations:
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The quantized Hall effect and unconventional
quantum statistics
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• zero longitudinal resistivity - no dissipation, bulk
energy gap current flows mostly along the edges of the
sample
• quantized Hall resistivity2
1
e
hxy
The quantum Hall effect
is an integer,
or q even
or a fraction with q odd,qp
IB
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Extending the notion of quantum statistics
),..,;............,.........( 411 RRrr NA ground state:
Adiabatically interchange the position of two excitations
Energy gap
ie
Laughlin quasi-
particlesElectrons
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In a non-abelian quantum Hall state, quasi-particles obey
non-abelian statistics, meaning that (for example)
with 2N quasi-particles at fixed positions, the ground state is
-degenerate.
Interchange of quasi-particles shifts between ground states.
N2
More interestingly, non-abelian statistics (Moore and Read, 91)
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ground statesN2
...,2..
...,2..
...,1..
21
21
21
RRsg
RRsg
RRsg
N
…..
..., 21 RR
position of quasi-particles
Permutations between quasi-particles positions unitary transformations in the ground state subspace
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Topological quantum computation (Kitaev 1997-2003)
• Subspace of dimension 2N, separated by an energy gap from the continuum of excited states.• Unitary transformations within this subspace are defined by the topology of braiding trajectories • All local operators do not couple between ground states
– immunity to errors
Up to a global phase, the unitary transformation depends only on the topology of the trajectory
1
1
3
2
2
3
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The goal:
experimentally identifying non-abelian quantum Hall
states
The way: the defining characteristics of the most
prominent candidate, the =5/2 Moore-Read state,
are
1. Energy gap.
2. Ground state degeneracy exponential in the
number N of quasi-particles, 2 N/2.
3. Edge structure – a charged mode and a Majorana
fermion mode
4. Unitary transformation applied within the ground
state subspace when quasi-particles are braided.
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In this talk:
1. Proposed experiments to probe ground state
degeneracy – thermodynamics
2. Proposed experiments to probe edge and bulk
braiding by electronic transport–
Interferometry, linear and non-linear Coulomb
blockade, Noise
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Probing the degeneracy of the ground state
(Cooper & Stern, 2008Yang & Halperin, 2008)
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Measuring the entropy of quasi-particles in the bulk
The density of quasi-particles is
Zero temperature entropy is then
02/5 2
544
Bnnn
2log2
54
0
Bn
To isolate the electronic contribution from other contributions:
T
m
B
s
Tn
s
;
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Leading to
2/5sgn2log5
2/5sgn2log2
0
T
m
T (~1.4)
(~12pA/mK)
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Probing quasi-particle braiding - interferometers
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When a vortex i encircles a vortex j, the ground state is multiplied by the operator ij
Nayak and WilczekIvanov
ii A localized Majorana operator .All ’s anti-commute, and 2=1.
.... sgsg ji
Essential information on the Moore-Read state:
• Each quasi-particle carries a single Majorana
mode
• The application of the Majorana operators
takes one ground state to another within the
subspace of degenerate ground states
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The interference term depends on the number and quantum state of the quasi-particles in the loop.
even odd evenNumber of q.p.’s in the interference loop,
Interference term
Brattelli diagram
Interferometers:
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Odd number of localized vortices:vortex a around vortex 1 - a
1a left right
statescorestatescore arightleft 1
The interference term vanishes:
statescorestatescore arightleft 1*
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Even number of localized vortices:vortex a around vortex 1 and vortex 2 - aa
1a left right2
statescorestatescore rightleft 12
The interference term is multiplied by a phase:
statescorestatescorerightleft 12*
Two possible values, mutually shifted by
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Interference in the =5/2 non-abelian quantum Hall state: The Fabry-Perot interferometer
S1 D1
D2
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Gate
Volt
age, V
MG
(mV
)
Magnetic Field (or voltage on anti-dot)
cell area
The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field.
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Coulomb blockade vs interference
(Stern, Halperin 2006,Stern, Rosenow, Ilan, Halperin, 2009Bonderson, Shtengel, Nayak 2009)
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Interferometer (lowest order) Quantum dot
For non-interacting electrons – transition from one limit to
another via Bohr-Sommerfeld interference of multiply
reflected trajectories.
Can we think in a Bohr-Sommerfeld way on the transition
when anyons, abelian or not, are involved?Yes, we can (BO, 2008)
(One) difficulty – several types of quasi-particles may tunnel
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Thermodynamics is easier than transport. Calculate the
thermally averaged number of electrons on a closed dot.
Better still, look at NA
2ANNEE bgc
N
bgc
T
NNEZ
2)(exp
Poisson summation
The simplest case, =1.
Energy is determined by the number of electrons
Partition function
sbg
cs
bgc sAiNsE
TiNs
T
ANNEdNZ )(2)(exp~2
))((exp~ 2
2
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sbg
cNbg
c sAiNsE
TNN
T
E)(2)(exp~)(exp 22
• Sum over electron number. • Thermal suppression of high energy configurations
• Sum over windings. • Thermal suppression of high
winding number. • An Aharonov-Bohm phase
proportional to the winding
number.• At high T, only zero and one
windings remain
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The energy of the dot is made of • A charging energy
• An energy of the neutral mode. The spectrum is
determined by the number and state of the bulk
quasi-particles.
2ANNE bgc
)())((exp 2 TANNT
EZ N
Nbg
c
The neutral mode partition function χ depends on nqp and their state.
Poisson summation is modular invariance
And now for the Moore-Read state
Lv
TS
T
Lv
n
n
/
/
(Cappelli et al, 2009)
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The components of the vector correspond to
the different possible states of the bulk quasi-
particles, one state for an odd nqp (“”), and
two states for an even nqp (“1” and “ψ”).
A different thermal suppression factor for
each component.
The modular S matrix. Sab encodes the outcome of a quasi-particle of type a going around one of type b
Lv
TS
T
Lv
n
n
/
/
)(T
S
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Low T
High T
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Probing excited states at the edge –
non linear transport in the Coulomb blockade regime
(Ilan, Rosenow, Stern, 2010)
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A nu=5/2 quantum Hall system
=2
Goldman’s group, 80’s
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Non-linear transport in the Coulomb blockade
regime:
dI/dV at finite voltage – a resonance for each many-
body state that may be excited by the tunneling
event.dI/dV
Vsd
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Energy spectrum of the neutral mode on the edge
Single fermion:
For an odd number of q.p.’s En=0,1,2,3,….
For an even number of q.p.’s En= ½, 3/2, 5/2, …
Many fermions:
For an odd number of q.p.’s Integers only
For an even number of q.p.’s Both integers and half
integers (except 1!)
The number of peaks in the differential conductance
varies with the number of quasi-particles on the edge.
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Current-voltage characteristics (Ilan, Rosenow, AS 2010)
Source-drain voltage
Magnetic field
nE
nE
n
n
)2/1( Even number
Odd number
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Interference in the =5/2 non-abelian quantum Hall state:
Mach-Zehnder interferometer
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The Mach-Zehnder interferometer: (Feldman, Gefen, Kitaev, Law, Stern, PRB2007)
S D1D2
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Compare:
S D1D2
S1 D1
D2
M-Z
F-P
Main difference: the interior edge is/is not part of interference loop
For the M-Z geometry every tunnelling quasi-particle advances the system along the Brattelli diagram
(Feldman, Gefen, Law PRB2006)
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Number of q.p.’s in the interference loop
Interference term
• The system propagates along the diagram, with transition rates
assigned to each bond.
• The rates have an interference term that
• depends on the flux
• depends on the bond (with periodicity of four)
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The other extreme – some of the bonds are “broken”
Charge flows in “bursts” of many quasi-particles. The maximum expectation value is around 12 quasi-particles per burst – Fano factor of about three.
If all rates are equal, current flows in “bunches” of one quasi-particle each – Fano factor of 1/4.
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Summary:
Interference magnitude depends on the parity of the number of quasi-particlesPhase depends on the eigenvalue of
12
Fano factor changing between 1/4 and about three – a signature of non-abelian statistics in Mach-Zehnder interferometers
Mach-Zehnder:
Temperature dependence of the chemical potential and the magnetization reflect the ground state entropy
2/5sgn2log5
2/5sgn2log2
0
T
m
T
Coulomb blockade I-V characteristics may measure the spectrum of the edge Majorana mode
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even odd evenNumber of q.p.’s in the interference loop,
Interference term
S1 D1
D2For a Fabry-Perot interferometer, the state of the bulk determines the interference term.
Das-Sarma-Freedman-Nayak qubit
The interference phases are mutually shifted by
0
2/2)(m
immegg
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even odd evenNumber of q.p.’s in the interference loop,
Interference term
The sum of two interference phases, mutually shifted by
The area period goes down by a factor of two.
S1 D1
D2
0
4)(m
immegg
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Gate
Volt
age, V
MG
(mV
)
Magnetic Field (or voltage on anti-dot)
cell area
Ideally,
The magnetic field Quasi-particles
number
The gate voltage Area
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Are we getting there?
(Willett et al. 2008)
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From electrons at n=5/2 to non-abelian quasi-particles:
A half filled Landau level on top of two filled Landau levels
Step II:
the Chern-Simons transformation
from: electrons at a half filled Landau level
Step I:Read and Green (2000)
2
12
2
5
to: spin polarized composite fermions at zero (average) magnetic field
GM87R89ZHK89LF90HLR93KZ93
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Be-
B1/2 = 2ns0
B)c(
20
CF
)b(B
Electrons in a magnetic field B
Composite particles in a magnetic field )(2 0 rnB
Mean field (Hartree) approximation
02 0 nBB
H = E
ji ji rri
ii err)arg(2
})({})({
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Step IV: introducing quasi-particles into the super-conductor
- shifting the filling factor away from 5/2
The super-conductor is subject to a magnetic field and thus accommodates vortices. The vortices, which are charged, are the non-abelian quasi-particles.
Step III: fermions at zero magnetic field pair into Cooper pairs
Spin polarization requires pairing of odd angular momentum
a p-wave super-conductor of composite fermions
..)(0 chrdrHH
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-2 -1 0 1 2-1
0
1
2
3
4
5x 10
-29
Impinging current (nA)
Sho
t no
ise
(A2 /H
z) e/2
e/4
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Step IV: introducing quasi-particles into the super-conductor
- shifting the filling factor away from 5/2
The super-conductor is subject to a magnetic field and thus accommodates vortices. The vortices, which are charged, are the non-abelian quasi-particles.
Step III: fermions at zero magnetic field pair into Cooper pairs
Spin polarization requires pairing of odd angular momentum
a p-wave super-conductor of composite fermions
..)(0 chrdrHH
For a single vortex – there is a zero energy mode at the vortex’ core
Kopnin, Salomaa (1991), Volovik (1999)
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g(r) is a localized function in the vortex core
)()()()( * rRrgrRrgdr iii
ii
A zero energy solution is a spinor
A localized Majorana operator .
A subspace of degenerate ground states, with the ’s operating in that subspace.
In particular, when a vortex i encircles a vortex j, the ground state is multiplied by the operator ij
Nayak and Wilczek (1996)Ivanov (2001)
All ’s anti-commute, and 2=1.
.... sgsg ji
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Effective charge span the range from 1/4 to about three. The
dependence of the effective charge on flux is a consequence of
unconventional statistics.
Charge larger than one is due to the Brattelli diagram having
more than one “floor”, which is due to the non-abelian statistics
In summary, flux dependence of the effective charge in a Mach-Zehnder
interferometer may demonstrate non-abelian statistics at =5/2
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Closing the island into a quantum dot:
Interference involving multiple scatterings, Coulomb blockade
0 50 100 150 2000
5
10
15
-9.0 -7.5 -6.0 -4.5 -3.0
cell area
Curr
ent
(a.u
.)
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12 1ais very different from
But, 121
212 a
so, interference of even number of windings always survives.
Equal spacing between peaks for odd number of localized vorticesAlternate spacing between peaks for even number of localized vortices
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nis – a crucial quantity. How do we know it’s time independent?
What is ? )0()( isis ntn
isis ne
Q4
By the fluctuation-dissipation theorem,
02)0()( tt
isis CTeQtQ
C – capacitance
t0 – relaxation time = C/G
G – longitudinal conductance
Best route – make sure charging energy >> Temperature
A subtle question – the charging energy of what ??
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And what if nis is time dependent?
A simple way to probe exotic statistics:
tItnGtn isis )()(
A new source of current noise.
For Abelian states ():
q
nGnG is
is
2cos1)( 0
For the state:
Chamon et al. (1997)
G = G0 (nis odd) G0[1 ± cos( + nis/4)] (nis even)
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time
G
G
0)0()( tt
isis entn
022
0
2 tVGI
compared to shot noise GVe*
bigger when t0 is long enough
close in spirit to 1/f noise, but unique to FQHE states.
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When multiple reflections are taken into account, the average conductance and the noise, satisfy
4cos102 I
and
311 , BBB
III
I
A signature of the =5/2 state
(For abelian Laughlin states – the power is ) 11
A “cousin” of a similar scaling law for the Mach-Zehndercase (Law, Feldman and Gefen, 2005)
4cos1III BB
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For a lattice, expect a tight-binding Hamiltonian
..,
chtHji
jiij
Analogy to the Hofstadter problem.
j
iijdlAii
ij etett
The phases of the tij’s determine the flux in each plaquette
Finally, a lattice of vortices
When vortices get close to one another, degeneracy is lifted by tunneling.
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Since the tunneling matrix elements must be imaginary. ii
ji
jiijitH,
The question – the distribution of + and -
For a square lattice:
ii i
i
Corresponds to half a flux quantum per
plaquette.
A unique case in the Hofstadter problem –
no breaking of time reversal symmetry.
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Spectrum – Dirac: kvkE 0)( tav 0
E
k
A mechanism for dissipation, without a motion of the charged vortices
0v is varied by varyingdensity
22~0,0 qqe
Exponential dependence on density
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Protection from decoherence:
• The ground state subspace is separated from the rest
of the spectrum by an energy gap
• Operations within this subspace are topological
But:
• In present schemes, the read-out involves interference
of two quasi-particle trajectories (subject to
decoherence).
• In real life, disorder introduces unintentional quasi-
particles. The ground state subspace is then not fully
accounted for.
A theoretical challenge!
(Kitaev, 1997-2003)
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Summary
1. A proposed interference experiment to address the non-abelian
nature of the quasi-particles, insensitive to localized quasi-particles.
2. A proposed “thermodynamic” experiment to address the
non-abelian nature of the quasi-particles, insensitive to localized
quasi-particles.
3. Current noise probes unconventional quantum statistics.
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Closing the island into a quantum dot:
Coulomb blockade !
Transport thermodynamics
For a conventional super-conductor, spacing alternates between
charging energy Ec (add an even electron)
charging energy Ec + superconductor gap
(add an odd electron)
The spacing between conductance peaks translates to the energy cost of adding an electron.
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Reason: consider a compact geometry (sphere). By Dirac’s
quantization, the number of flux quanta (h/e) is quantized to an integer,
the number of vortices (h/2e) is quantized to an even
integer
In a non-compact closed geomtry, the edge “completes” the pairing
But this super-conductor is anything but conventional…
For the p-wave super-conductor at hand, crucial dependence on the number of bulk localized quasi-particles, nisa gapless (E=0) edge mode if nis is odd corresponds
to=0
a gapfull (E≠0) edge mode if nis is even corresponds
to ≠0
The gap diminishes with the size of the dot ∝ 1/L
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So what about peak spacings?
When nis is odd, peak spacing is “unaware” of
peaks are equally spaced
When nis is even, peak spacing is “aware” of
periodicity is doubled
B
cell area
odd
even
even
No interference
No interference
B
cell area
odd
even
even
Interference patternCoulomb peaks
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From electrons at =5/2 to a lattice of non-abelian quasi-particlesin four steps:
A half filled Landau levelon top of Two filled Landau levels
Step II:
From a half filled Landau level of electrons to composite fermions
at zero magnetic field - the Chern-Simons transformation
Step I:
Read and Green (2000)
2
12
2
5
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• The original Hamiltonian:
• Schroedinger eq. H • Define a new wave function:
Hm
P A re
r rii i ji j
1
2
1
2
22
( )| |,
ji ji rri
ii err)arg(2
})({})({
)()()( rarAPrAP ii
The Chern-Simons transformation
})({ ir
})({ ir
describes electrons (fermions)
describes composite fermions
The effect on the Hamiltonian:
r
r
arg
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|tleft + tright|2 for an even number of localized quasi-particles
|tright|2 + |tleft|2 for an odd number of localized quasi-particles
The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field.
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The new magnetic field:)(
~)( 0 rra
B
ns
e-)a(
B1/2 = 2ns0
B)c(
20
CF
)b(B
Electrons in a magneticfield
BA
Composite particles ina magnetic field
)(2 0 rnBaA
Mean field (Hartree) approximation
02 0
nBaA
B
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Spin polarized composite fermions at zero (average) magnetic field
Step III: fermions at zero magnetic field pair into Cooper pairs
Spin polarization requires pairing of odd angular momentum
a p-wave super-conductorRead and Green (2000)
Step IV: introducing quasi-particles into the super-conductor
- shifting the filling factor away from 5/2
022 10 nBaAB
The super-conductor is subject to a magnetic field
an Abrikosov lattice of vortices in a p-wave super-conductor
Look for a ground state degeneracy in this lattice
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..)2
()2
(),('0 chr
Rr
RrRdrdrHH
)(),( rfRrR
Dealing with Abrikosov lattice of vortices in a p-wave super-conductor
First, a single vortex – focus on the mode at the vortex’ core
A quadratic Hamiltonian – may be diagonalized
(Bogolubov transformation)
E
EEEEH 0
)()()()( rrvrrudrE BCS-quasi-particle annihilation operator
Ground state degeneracy requires zero energy modes
Kopnin, Salomaa (1991), Volovik (1999)
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The functions are solutions of the Bogolubov de-Gennes eqs.)(),( rvru
Ground state should be annihilated by all ‘sE
For uniform super-conductorsrkiervru )()(
kkkk vacccgsg 1..
.const
)()()()( rrvrrudrE
For a single vortex – there is a zero energy mode at the vortex’ core
Kopnin, Salomaa (1991), Volovik (1999)
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g(r) is a localized function in the vortex core
)()()()( * rRrgrRrgdr iii
ii
A zero energy solution is a spinor
A localized Majorana operator .
A subspace of degenerate ground states, with the ’s operating in that subspace.
In particular, when a vortex i encircles a vortex j, the ground state is multiplied by the operator ij
Nayak and WilczekIvanov
All ’s anti-commute, and 2=1.
.... sgsg ji
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backscattering = |tleft+tright|2
Interference experiment: Stern and Halperin (2005)Following Das Sarma et al (2005)
interference pattern is observed by varying the cell’s area
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vortex a around vortex 1 - a
vortex a around vortex 1 and vortex 2 - aa
The effect of the core states on the interference of backscattering amplitudes depends crucially on the parity of the number of localizedstates.
statescorerightleft Before encircling
1a left right2
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After encircling
statescorestatescore arightleft 1
for an even number of localized vorticesonly the localized vortices are affected(a limited subspace)
for an odd number of localized vorticesevery passing vortex acts on a different subspaceinterference is dephased
statescorestatescore rightleft 12
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|tleft + tright|2 for an even number of localized quasi-particles
|tright|2 + |tleft|2 for an odd number of localized quasi-particles
• the number of quasi-particles on the dot may be tuned by a gate• insensitive to localized pinned charges
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cell area
even
odd
even
occupation ofanti-dot
interference
no interference
interference
Localized quasi-particles shift the red lines up/down
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Be-
B1/2 = 2ns0
B)c(
20
CF
)b(B
Electrons in a magnetic field B
Composite particles in a magnetic field )(2 0 rnB
Mean field (Hartree) approximation
02 0 nBB
H = E
ji ji rri
ii err)arg(2
})({})({
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A yet simpler version:
B
cell area
equi-phase lines
odd
even
even
No interference
No interference
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For a lattice, expect a tight-binding Hamiltonian
..,
chtH jji
iij
Analogy to the Hofstadter problem.
j
iijdlAii
ij etett
The phases of the tij’s determine the flux in each plaquette
And now to a lattice of quasi-particles.
When vortices get close to one another, degeneracy is lifted by tunneling.
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Since the tunneling matrix elements must be imaginary. ii
ji
jiijitH,
The question – the distribution of + and -
For a square lattice:
ii i
i
Corresponds to half a flux quantum per
plaquette.
A unique case in the Hofstadter problem –
no breaking of time reversal symmetry.
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Spectrum – Dirac: kvkE 0)( tav 0
E
k
What happens when an electric field E(q,) is applied?
0v is varied by varyingdensity
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E
k
Given a perturbation
tAJ cos
the rate of energy absorption is
f
iffAJi
22
Distinguish between two different problems –
1. Hofstadter problem – electrons on a lattice
2. Present problem – Majorana modes on a lattice
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f
iffAJi
22
E
k
For both problems the rate of energy absorption
is
20
~)(
~ field electric
vdos
AiE
2Re E
The difference between the two problems is in the matrix elements
tev
ev
fJi
if0
0 for the electrons
for the Majorana modes
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So the real part of the conductivity is
h
e
8
2
for the electrons
22
8
th
e for the Majorana modes
The reason – due the particle-hole symmetry of the Majorana mode,it does not carry any current at q=0.
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From the conductivity of the Majorana modes to the electronic
response
The conductivity of the p-wave super-conductor of composite fermions, in the presence of the lattice of vortices
2
022
2
0
22
18
0
0
18
,
qvqae
m
ni
qv
qae
m
ni
qCF
From composite fermions to electrons
02
202
11
e
hCFe
2~,0 qqe 0, 0 qvqe
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Summary
1. A proposed interference experiment to address the non-abelian
nature of the quasi-particles.
2. Transport properties of an array of non-abelian quasi-particles.
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i
i
e
e localized function in the direction perpendicular to the =0 line
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When vortex i encircles vortex i+1, the unitary transformation operating on the ground state is
)'()'()'()'(
)()()()('
11 *11
*1
rerwrerw
rerwrerwdrdr
ii
ii
ii
ii
ii
iiii
• How does the zero energy state at the i’s vortex “know” that it is encircled by another vortex?
• No tunneling takes place?
Unitary transformations:
A more physical picture?
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• 2N localized intra-vortex states, each may be filled (“1”) or empty (“0”)
• Notation: means 1st, 3rd, 5th vortices filled, 2nd, 4th vortices empty....11001
1111100101010110
1010110000110000
1111100101010110
1010110000110000
iiii
iiii
eeee
eeee
and all possible combinations with odd numbers of filled states
1110110110110111
1000010000100001
1110110110110111
1000010000100001
iiii
iiii
eeee
eeee
Product states are not ground states:
0110*0110 2222
4321 iiii
ieeiee
The emerging picture – two essential ingredients:
•Full entanglement: Ground states are fully entangled super-positions of
all possible combinations with even numbers of filled states
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• Phase accumulation depend on occupation
When a vortex traverses a closed
trajectory, the system’s wave-function
accumulates a phase that is N2
N – the number of fluid particles encircled by the trajectory
(|0000 + |1100 ) (|0000 - |1100 )
HalperinArovas, Schrieffer, Wilczek
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Vortex 2 encircling vortex 3
1111100101010110
1010110000110000
1111100101010110
1010110000110000
iiii
iiii
eeee
eeee
A “+” changing into a “”
Vortex 2 and vortex 3 interchanging positions
1111100101010110
1010110000110000
1111100101010110
1010110000110000
iiii
iiii
eeee
eeee
Permutations of vortices change relative phases in the superposition
Four vortices:
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A vortex going around a loop generates a unitary transformation in the ground state subspace
1
2
34
A vortex going around the same loop twice does not generate any transformation
1
2
34
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The Landau filling range of 2<<4
Unconventional fractional quantum Hall states:
1. Even denominator states are observed
2. Observed series does not follow the rule.
3. In transitions between different plateaus, is non-monotonous12
p
p8
19,
2
7,
2
5
xy
as opposed to
(Pan et al., PRL, 2004)
Focus on =5/2
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The effect of the zero energy states on interference
Dephasing, even at zero temperature
No dephasing (phase changes of )4
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More systematically: what are the ground states? N2
The goal: ground statesN2
...,2..
...,2..
...,1..
21
21
21
RRsg
RRsg
RRsg
N
…..
that, as the vortices move, evolve without being mixed.
..., 21 RR
position of vortices
The condition: nkfornsgR
ksgi
0....
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To answer that, we need to • define a (partial) single particle basis, near each vortex• find the wave function describing the occupation of these states
*
0
0
w
wis a purely zero energy state
*
0
0
w
wis a purely non-zero energy state =
EEEEE CC *
1*
0*
10
iww
iwwCY
EEE
defines a localized function correlates its occupation with that of
1w
0w
There is an operator Y for each vortex.
How does the wave function near each vortex look?
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We may continue the process
*
1
1
w
w
*
1
1
w
w E
EE chC ..)2(
E
EE chC ..)2(
2
*1
*21)2()2(
iww
iwwCY
EEE
defines a localized function correlates its occupation with that of
2w
10, ww
This generates a set of orthogonal vortex states near each vortex (the process must end when states from different vortices start overlapping).
lwwww ,...,, 210
The requirements for determine the occupations of the states near each vortex.
0.. sgY j
kww ..0
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The functions are solutions of the Bogolubov de-Gennes eqs.)(),( rvru
Ground state should be annihilated by all ‘sE
For uniform super-conductorsrkiervru )()(
kkkk vacccgsg 1..
.const
)()()()( rrvrrudrE
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The simplest model – take a free Hamiltonian with a potential part
only )()( rrh
To get a localized mode of zero energy,
we need a localized region of . A vortex is a closed curve of =0with a phase winding of 2in the order parameter
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The phase winding is turned into a boundary condition
A change of sign
Spinor is
i
i
e
e localized function in the direction perpendicular to the =0 line
The phase depends on the direction of the =0 line. It changes by around the square. A vortex is associated with a localized Majorana operator.
i
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For a lattice, expect a tight-binding Hamiltonian
..,
chtH jji
iij
Questions –
1. What are the tij?
2. How do we calculate electronic response functions from the
spinors’ Hamiltonian?
Analogy to the Hofstadter problem.
j
iijdlAii
ij etett
The phases of the tij’s determine the flux in each plaquette
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Two close vortices:Solve along this line toget the tunneling matrixelement
We find tij=i
A different case – the line going through the tunneling region changes the sign of the tunneling matrix element.
ii i
i
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These requirements are satisfied for a given vortex by either one of two wave functions:
vacuumcccccp .....210 11...111
or:
vacuumcccccm .....210 11...111
The occupation of all vortex states is particle-hole symmetric.
Still, two states per vortex, altogether and not N2N22
We took care of the operators
jj
jjj
iww
iwwY
*1
*
1)(
For the last state, we should take care of the operatorwhich creates and annihilates quasi-particles
k
k
w
w
*
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Doing that, we get ground states that entangle states of different vortices (example for two vortices):
12
21
2..
1..
envpmmpenvmmppsg
envpmmpenvmmppsg
For 2N vortices, ground states are super-positions of states of the form
i
Ni
ii
ii vacuumccc )2()2()1( 1...11
1st vortex2nd vortex 2N’th vortex
The operator creates a particle at the state near the vortex. When the vortex is encircled by another
ic )(rwi
ii cc
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Open questions:
1. Experimental tests of non-abelian states
2. The expected QH series in the second Landau level
3. The nature of the transition between QH states in the second
Landau level
4. Linear response functions in the second Landau level
5. Physical picture of the clustered parafermionic states
6. Exotic directions – quantum computing, BEC’s
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One fermion mode, two possible states, two pi-shifted interference patterns
Several comments on the Das Sarma, Freedman and Nayak proposed experiment
n=0n=1
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Comment number 1: The measurement of the interference patterninitializes the state of the fermion mode
initial core state ii 2121
after measurement current has been flown through the system
21 2121 currenticurrenti
A measurement of the interference pattern implies
021 currentcurrent
The system is now either at the =i or at the =-i state.
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NA
At low temperature, as we saw
odd nqp (“”) even nqp (“1” and “ψ”).
At high temperature, the charge part will thermally
suppress all but zero and one windings, and the neutral
part will thermally suppress all but the “1” channel
(uninteresting) and the “” channel. The latter is what
one sees in lowest order interference (Stern et al, Bonderson
et al, 2006).
High temperature Coulomb blockade gives the same
information as lowest order interference.
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1 2
For both cases the interference pattern is shifted by by the transition of one quasi-particle through the gates.
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Summary
The geometric phase accumulated by a moving vortex
Entanglement between the occupation of states near
different vortices
Non-abelian statistics
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2/5sgn2log5
2/5sgn2log2
0
T
m
T
The positional entropy of the quasi-particles
If all positions are equivalent, other than hard core constraint –
positional entropy is ∝n log(n), but -
Interaction and disorder lead to the localization of the quasi-
particles – essential for the observation of the QHE – and to
the suppression of their entropy.
(~1.4)
(~12pA/mK)
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The positional entropy of the quasi-particles depends on their
spectrum:
qhe gap
ground state
excitations
non-abelianstemperature
localized states
Phonons of a quasi-particles Wigner crystal
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Shot noise as a way to measure charge:
p
1-p
I2Binomial distribution
For p<<1, current noise is
S=2eI2
coin tossingS
D1
D2