Proposed experimental probes of non-abelian anyons

1
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

2

• Outline
• Non-abelian anyons in quantum Hall states what
  they are, why they are interesting, how they may
  be useful for topological quantum computation.
• How do you identify a non-abelian quantum Hall
  state when you see one ?

3
More precise and relaxed presentations
Introductory
pedagogical
Comprehensive
4
The quantized Hall effect and unconventional
quantum statistics
5
The quantum Hall effect

• zero longitudinal resistivity - no dissipation,
  bulk energy gap current flows mostly along the
  edges of the sample
• quantized Hall resistivity

B
I
n is an integer,
or q even
6
Extending the notion of quantum statistics
Laughlin quasi-particles
Electrons
A ground state
Energy gap
Adiabatically interchange the position of two
excitations
7
More interestingly, non-abelian statistics
(Moore and Read, 91)
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.
8
ground states
position of quasi-particles
..
Permutations between quasi-particles positions
unitary transformations in the
ground state subspace
9
Up to a global phase, the unitary transformation
depends only on the topology of the trajectory
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

10

• The goal
• experimentally identifying non-abelian quantum
  Hall states
• The way the defining characteristics of the most
  prominent candidate, the n5/2 Moore-Read state,
  are
• Energy gap.
• Ground state degeneracy exponential in the number
  N of quasi-particles, 2 N/2.
• Edge structure a charged mode and a Majorana
  fermion mode
• Unitary transformation applied within the ground
  state subspace when quasi-particles are braided.

11

• In this talk
• Proposed experiments to probe ground state
  degeneracy thermodynamics
• 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, 2008 Yang Halperin, 2008)
13
Measuring the entropy of quasi-particles in the
bulk The density of quasi-particles is Zero
temperature entropy is then
To isolate the electronic contribution from
other contributions
14
Leading to
(1.4)
(12pA/mK)
15
Probing quasi-particle braiding - interferometers
16

• 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

When a vortex i encircles a vortex j, the ground
state is multiplied by the operator gigj
Nayak and Wilczek Ivanov
17
Interferometers
The interference term depends on the number and
quantum state of the quasi-particles in the loop.
18
Odd number of localized vortices vortex a around
vortex 1 - g1ga
The interference term vanishes
19
Even number of localized vortices vortex a
around vortex 1 and vortex 2 -
g1gag2ga g2g1
2
1
a
The interference term is multiplied by a phase
Two possible values, mutually shifted by p
20
Interference in the n5/2 non-abelian quantum
Hall state The Fabry-Perot interferometer
D2
D1
S1
21
The number of quasi-particles on the island may
be tuned by charging an anti-dot, or more simply,
by varying the magnetic field.
cell area
22
Coulomb blockade vs interference
(Stern, Halperin 2006, Stern, Rosenow, Ilan,
Halperin, 2009 Bonderson, 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
24
Thermodynamics is easier than transport.
Calculate the thermally averaged number of
electrons on a closed dot. Better still, look at
The simplest case, n1. Energy is determined by
the number of electrons Partition function
Poisson summation
25

• 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

• Sum over electron number.
• Thermal suppression of high energy
  configurations

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And now for the Moore-Read state

• 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.

The neutral mode partition function ? depends on
nqp and their state. Poisson summation is modular
invariance
(Cappelli et al, 2009)
27
The components of the vector correspond to the
different possible states of the bulk
quasi-particles, one state for an odd nqp (s),
and two states for an even nqp (1 and ?). 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
28
Low T
High T
29
Probing excited states at the edge non linear
transport in the Coulomb blockade regime
(Ilan, Rosenow, Stern, 2010)
30
A nu5/2 quantum Hall system
31
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
En0,1,2,3,. 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.
33
Current-voltage characteristics
(Ilan, Rosenow, AS 2010)
Source-drain voltage
Magnetic field
34
Interference in the n5/2 non-abelian quantum
Hall state Mach-Zehnder interferometer
35
The Mach-Zehnder interferometer
(Feldman, Gefen, Kitaev, Law, Stern,
PRB2007)
S
D1
D2
36
Compare
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)
37
Interference term
Number of q.p.s in the interference loop

• 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)

38
If all rates are equal, current flows in
bunches of one quasi-particle each Fano
factor of 1/4.
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.
39
Summary
Temperature dependence of the chemical potential
and the magnetization reflect the ground state
entropy
Coulomb blockade I-V characteristics may measure
the spectrum of the edge Majorana mode
Fano factor changing between 1/4 and about three
a signature of non-abelian statistics in
Mach-Zehnder interferometers
Mach-Zehnder
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For a Fabry-Perot interferometer, the state of
the bulk determines the interference term.
D2
D1
S1
Interference term
Number of q.p.s in the interference loop,
The interference phases are mutually shifted by
p.
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Interference term
Number of q.p.s in the interference loop,
The sum of two interference phases, mutually
shifted by p.
The area period goes down by a factor of two.
42
Ideally, The magnetic field
Quasi-particles number The gate voltage
Area
cell area
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Are we getting there?
(Willett et al. 2008)
44
From electrons at n5/2 to non-abelian
quasi-particles
Read and Green (2000)
Step I
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
to spin polarized composite fermions at zero
(average) magnetic field
GM87 R89 ZHK89 LF90 HLR93 KZ93
45
B
Electrons in a magnetic field B
e-
H y E y
Composite particles in a magnetic field
Mean field (Hartree) approximation
46
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.
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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.
For a single vortex there is a zero energy mode
at the vortex core
Kopnin, Salomaa (1991), Volovik (1999)
49
A zero energy solution is a spinor
g(r) is a localized function in the vortex core
A localized Majorana operator .
All gs anti-commute, and g21.
A subspace of degenerate ground states, with the
gs operating in that subspace. In particular,
when a vortex i encircles a vortex j, the ground
state is multiplied by the operator gigj
Nayak and Wilczek (1996) Ivanov (2001)
50
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 n5/2
51
Closing the island into a quantum dot
n5/2
Interference involving multiple scatterings,
Coulomb blockade
52
n5/2
is very different from
But,
so, interference of even number of windings
always survives.
Equal spacing between peaks for odd number of
localized vortices
Alternate spacing between peaks for even number
of localized vortices
53
nis a crucial quantity. How do we know its
time independent?
What is ?
By the fluctuation-dissipation theorem,
C capacitance t0 relaxation time C/G G
longitudinal conductance
Best route make sure charging energy gtgt
Temperature
A subtle question the charging energy of what ??
54
And what if nis is time dependent?
A simple way to probe exotic statistics
A new source of current noise. For Abelian
states (n1/3)
Chamon et al. (1997)
For the n5/2 state
G G0 (nis
odd) G01 b cos(f nis/4) (nis
even)
55
G
dG
time
compared to shot noise
bigger when t0 is long enough
close in spirit to 1/f noise, but unique to FQHE
states.
56
When multiple reflections are taken into account,
the average conductance and the noise, satisfy
and
A signature of the n5/2 state
(For abelian Laughlin states the power is
)
A cousin of a similar scaling law for the
Mach-Zehnder case (Law, Feldman and Gefen, 2005)
57
Finally, a lattice of vortices
When vortices get close to one another,
degeneracy is lifted by tunneling.
For a lattice, expect a tight-binding Hamiltonian
Analogy to the Hofstadter problem.
The phases of the tijs determine the flux in
each plaquette
58
For a square lattice
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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Exponential dependence on density
60

• 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)
61

• Summary
• A proposed interference experiment to address the
  non-abelian
• nature of the quasi-particles, insensitive
  to localized quasi-particles.
• 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 !
n5/2
Transport thermodynamics
The spacing between conductance peaks translates
to the energy cost of adding an electron.
For a conventional super-conductor, spacing
alternates between charging energy Ec
(add an even electron) charging
energy Ec superconductor gap D
(add an odd electron)
63
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, nis
Reason consider a compact geometry (sphere). By
Diracs 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
64

• So what about peak spacings?
• When nis is odd, peak spacing is unaware of D
• peaks are equally spaced
• When nis is even, peak spacing is aware of D
• periodicity is doubled

Interference pattern
Coulomb peaks
65
From electrons at n5/2 to a lattice of
non-abelian quasi-particles in four steps
Read and Green (2000)
Step I
A half filled Landau level on 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
66
The Chern-Simons transformation

• The original Hamiltonian

• Schroedinger eq. H y E y
• Define a new wave function

describes electrons (fermions)
describes composite fermions
The effect on the Hamiltonian
67
n5/2
tleft tright2 for an even
number of localized quasi-particles
tright2 tleft2 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.
68
The new magnetic field
(a)
Electrons in a magnetic field
e-
B
ns
Composite particles in a magnetic field
Mean field (Hartree) approximation
69
Spin polarized composite fermions at zero
(average) magnetic field
70
Dealing with Abrikosov lattice of vortices in a
p-wave super-conductor
First, a single vortex focus on the mode at the
vortex core
Kopnin, Salomaa (1991), Volovik (1999)
71
The functions are solutions of the
Bogolubov de-Gennes eqs.
Ground state should be annihilated by all s
For uniform super-conductors
For a single vortex there is a zero energy mode
at the vortex core
Kopnin, Salomaa (1991), Volovik (1999)
72
A zero energy solution is a spinor
g(r) is a localized function in the vortex core
A localized Majorana operator .
All gs anti-commute, and g21.
A subspace of degenerate ground states, with the
gs operating in that subspace. In particular,
when a vortex i encircles a vortex j, the ground
state is multiplied by the operator gigj
Nayak and Wilczek Ivanov
73
Interference experiment
Stern and Halperin (2005) Following Das Sarma et
al (2005)
n5/2
backscattering tlefttright2
interference pattern is observed by varying the
cells area
74
vortex a around vortex 1 - g1ga vortex a
around vortex 1 and vortex 2 -
g1gag2ga g2g1
1
a
75
After encircling
for an even number of localized vortices only the
localized vortices are affected (a limited
subspace)
for an odd number of localized vortices every
passing vortex acts on a different
subspace interference is dephased
76
n5/2
tleft tright2 for an even
number of localized quasi-particles
tright2 tleft2 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

77
occupation of anti-dot
interference
no interference
interference
cell area
Localized quasi-particles shift the red lines
up/down
78
B
Electrons in a magnetic field B
e-
H y E y
Composite particles in a magnetic field
Mean field (Hartree) approximation
79
equi-phase lines
B
cell area
80
And now to a lattice of quasi-particles.
When vortices get close to one another,
degeneracy is lifted by tunneling.
81
For a square lattice
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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Given a perturbation
the rate of energy absorption is

• Distinguish between two different problems
• Hofstadter problem electrons on a lattice
• Present problem Majorana modes on a lattice

84
For both problems the rate of energy absorption
is
The difference between the two problems is in the
matrix elements
for the electrons
for the Majorana modes
85
The reason due the particle-hole symmetry of
the Majorana mode, it does not carry any current
at q0.
So the real part of the conductivity is
for the electrons
for the Majorana modes
86
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
From composite fermions to electrons
87

• Summary
• 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.

88
localized function in the direction
perpendicular to the m0 line
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Unitary transformations
When vortex i encircles vortex i1, the unitary
transformation operating on the ground state is

• No tunneling takes place?

• How does the zero energy state at the is vortex
  know that it is
• encircled by another vortex?

A more physical picture?
91
The emerging picture two essential ingredients

• 2N localized intra-vortex states, each may be
  filled (1) or empty (0)
• Notation means 1st, 3rd, 5th
  vortices filled, 2nd, 4th vortices empty.

• Full entanglement Ground states are fully
  entangled super-positions of all possible
  combinations with even numbers of filled states

and all possible combinations with odd numbers
of filled states
Product states are not ground states
92

• Phase accumulation depend on occupation

When a vortex traverses a closed trajectory, the
systems wave-function accumulates a phase that
is
Halperin Arovas, Schrieffer, Wilczek
N the number of fluid particles encircled by
the trajectory
93
Permutations of vortices change relative phases
in the superposition
Four vortices
Vortex 2 encircling vortex 3
Vortex 2 and vortex 3 interchanging positions
A changing into a -
94
A vortex going around a loop generates a unitary
transformation in the ground state subspace
A vortex going around the same loop twice does
not generate any transformation
2
4
3
1
95

• The Landau filling range of 2ltnlt4
• Unconventional fractional quantum Hall states
• Even denominator states are observed
• Observed series does not follow the
  rule.
• In transitions between different plateaus,
  is non-monotonous

as opposed to
(Pan et al., PRL, 2004)
Focus on n5/2
96
The effect of the zero energy states on
interference
Dephasing, even at zero temperature
No dephasing (phase changes of )
97
More systematically what are the ground
states?
The goal ground states
position of vortices
..
that, as the vortices move, evolve without being
mixed.
The condition
98
How does the wave function near each vortex look?

• 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

is a purely zero energy state
is a purely non-zero energy state
defines a localized function correlates its
occupation with that of
There is an operator Y for each vortex.
99
We may continue the process
defines a localized function correlates its
occupation with that of
This generates a set of orthogonal vortex states
near each vortex (the process must end when
states from different vortices start
overlapping).
The requirements for j1..k
determine the occupations of the states
near each vortex.
100
The functions are solutions of the
Bogolubov de-Gennes eqs.
Ground state should be annihilated by all s
For uniform super-conductors
101
The simplest model take a free Hamiltonian with
a potential part only
To get a localized mode of zero energy,
we need a localized region of m0. A vortex is a
closed curve of m0 with a phase winding of 2p in
the order parameter D.
102
The phase winding is turned into a boundary
condition
A change of sign
localized function in the direction
perpendicular to the m0 line
Spinor is

• The phase a depends on the direction of the m0
  line. It changes by
• around the square.
• A vortex is associated with a localized Majorana
  operator.

103
For a lattice, expect a tight-binding Hamiltonian
Analogy to the Hofstadter problem.
The phases of the tijs determine the flux in
each plaquette

• Questions
• What are the tij?
• How do we calculate electronic response functions
  from the
• spinors Hamiltonian?

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These requirements are satisfied for a given
vortex by either one of two wave functions
or
The occupation of all vortex states is
particle-hole symmetric.
Still, two states per vortex, altogether
and not
We took care of the operators
For the last state, we should take care of the
operator which creates and annihilates E?0
quasi-particles.
106
Doing that, we get ground states that entangle
states of different vortices (example for two
vortices)
For 2N vortices, ground states are
super-positions of states of the form
1st vortex
2nd vortex
2Nth vortex
The operator creates a particle at the
state near the vortex. When the
vortex is encircled by another
107

• Open questions
• Experimental tests of non-abelian states
• The expected QH series in the second Landau level
• The nature of the transition between QH states in
  the second
• Landau level
• 4. Linear response functions in the second
  Landau level
• Physical picture of the clustered parafermionic
  states
• Exotic directions quantum computing, BECs

108
Several comments on the Das Sarma, Freedman and
Nayak proposed experiment
One fermion mode, two possible states, two
pi-shifted interference patterns
n0
n1
109
Comment number 1 The measurement of the
interference pattern initializes the state of the
fermion mode
initial core state
after measurement current has been flown through
the system
A measurement of the interference pattern implies
The system is now either at the g1g2i or at the
g1g2-i state.
110
At low temperature, as we saw
odd nqp (s)
even nqp (1 and ?).
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 s
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.
111
For both cases the interference pattern is
shifted by p by the transition of one
quasi-particle through the gates.
112
Summary
Entanglement between the occupation of states
near different vortices
The geometric phase accumulated by a moving
vortex
Non-abelian statistics
113
(1.4)
(12pA/mK)
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.
114
The positional entropy of the quasi-particles
depends on their spectrum
excitations
localized states
qhe gap
Phonons of a quasi-particles Wigner crystal
temperature
non-abelians
ground state
115
Immunity to perturbations

• Diagonalized Hamiltonian
• Energy of all ground states is set to zero
• E is a general name for a positive energy of the
  excites states
• Lowest value of E is the energy gap

Perturbation No matrix element that connects
ground states Shaded part is protected. Virtual
transitions introduce exponentially small
splitting
116
Physical picture of the effect on magnetization

• The magnetization is determined by the current
  on the edge, which is proportional to the number
  of particles on the edge.
• The edge and the bulk are two systems at thermal
  equilibrium.
• As the temperature is varied, the free energy
  per particle in the two sub-systems varies, and
  the two subsystems exchange particles, hence
  changing the magnetization of the sample.

117
Interferometers
(Stern Halperin) (Bonderson, Shtengel, Kitaev)
(FeldmanKitaev) (Law, Feldman, Gefen, Kitaev
Stern)
118
Shot noise as a way to measure charge
D1
1-p
S
coin tossing
p
I2
D2
Binomial distribution For pltlt1, current noise is
S2eI2