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Theoretical study of the phase evolution in a
quantum dot in the presence of Kondo correlations
Mireille LAVAGNA
Work done in collaboration with A. JEREZ (ILL
Grenoble and Rutgers University) and P.
VITUSHINSKY (CEA-Grenoble)
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Determining the phase of a QD by using a two
path Aharonov-Bohm interferometer
Experimental context quantum dots studied by
Aharonov-Bohm interferometry
Aharonov-Bohm oscillations of the conductance as
a function of the magnetic flux F
ref
source
drain
the phase d introduced by the QD is deduced from
the shift of the oscillations with magnetic field
3
Quantum interferometry
allows to determine the phase and visibility of
the QD
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Evolution of the phase when reducing coupling
strength
Uncomplete phase lapse
Unitary limit
Kondo regime
Coulomb blockade
plateau
Ji, Heiblum et Shtrikman PRL 88, 076601 (2002)
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Theoretical context
for the Kondo effect in bulk metals
Langreth PR 150, 516 (66) and Nozières JLTP 17,
31 (74)
for the Kondo effect in QD NRG and Bethe-Ansatz
calculations

Gerland, von Delft, Costi, Oreg PRL 84, 3710
(2000)
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Theoretical interpretation
2-reservoir Anderson model
Glazman and Raikh JETP Lett. 47, 452 (88) Ng and
Lee PRL 61, 1768 (88)
1-reservoir Anderson model
where
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Scattering theory in 1D
In the case when there is no magnetic moment in
the dot (for instance in the Kondo regime at
T0), spin-flip scattering cannot occur
outgoing
incoming
Asymptotic solutions
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Scattering theory in 1D
For the symmetric QD, following Ng and Lee PRL
88



Scattering theory
Using exact results on Fermi liquid at T0, one
can show that
Denoting the phase of
by , one gets
(Friedel sum rule see Langreth Phys.Rev.66)
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Using trigonometric arguments
Using again exact results on Fermi liquid at T0,
one can show
Putting altogether, one gets
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Scattering off a composite system
Generalized Levinsons theorem
where
is the number of bound states
is the number of states excluded by the Pauli
principle
Levinson49 Swan 55 Rosenberg and Spruch PRA96
Example scattering of an electron by an atom of
hydrogen
is the ground state of a hydrogen atom
Singlet scattering Sztot0
Triplet scattering Sztot1
e
H
1s
Phase shift
1s2
0
1s"1s 1s"1s"
0
1s"1s"
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Scattering theory in 1D
Quantum dot Artificial atom
Generalized Levinsons theorem
The single level Anderson model (SLAM) is not
sufficient to capture the whole physics contained
in the experimental device which can be viewed as
an artificial atom. One may try to start with a
many level Anderson model (MLAM) description of
the system. We have chosen another route and
introduced the missing ingredients through an
additional multiplicative factor in front
of the S-matrix of the SLAM.
12
is chosen in order that satisfies
the generalized Levinson theorem. It is easy to
show that
with
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Scattering theory in 1D
Landauer formula
Aharonov-Bohm interferometry
Consequences (at T0, H0)

• Phase shift measured
• Conductance measured

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Scattering theory in 1D
Experimental check of the prediction
P. Vitushinky, A.Jerez, M.Lavagna Quantum
Information and Decoherence in Nanosystems, p.309
(2004)
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Bethe-Ansatz solution at T0
We have numerically solved the Bethe ansatz
equations to derive n0 and hence d/p as a
function of the parameters of the model
(Wiegmann et al. JETP Lett. 82 and Kawakami and
Okiji, JPSJ 82)
A.Jerez, P.Vitushinsky, M.Lavagna PRL 95, 127203
(2005)
Particle-hole symmetry
symmetric limit
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Bethe-Ansatz solution at T0
In the asymmetric regime,
, n0 shows a universal behavior as
a function of the renormalized energy
Universal behavior occurs when
Asymptotic behavior in the limit n0 0
The existence of both those universal and
asymptotic behavior is of valuable help in
fitting the experimental data
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Bethe-Ansatz solution at T0
Fit in the unitary limit and Kondo regimes
All the experimental curves are shifted in order
to get ?? at the symmetric limit
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Bethe-Ansatz solution at T0
(a) Unitary limit
Fit in the unitary limit and Kondo regimes
Very good agreement in presence of a single
fitting parameter ?/U (we consider linear
correspondence between ?0 and VG )
(b) Kondo regime
A.Jerez, P.Vitushinsky, M.Lavagna, PRL05
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Conclusions

1. We have shown that there is a factor of 2
   difference between the phase of the S-matrix
   responsible for the shift in the AB oscillations
   and the phase controlling the conductance.
2. This result is beyond the simple single-level
   Anderson model (SLAM) description and supposes to
   consider the generalisation to the multi-level
   Anderson model (MLAM). Done here in a minimal way
   by introducing a multiplicative factor in
   front of the S-matrix in order to guarantee the
   generalized Levinson theorem.
3. Then the phase measured by A.B. experiments is
   related to the total occupation n0 of the dot
   which is exactly determined by Bethe-Ansatz
   calculations. We have obtained a quantitative
   agreement with the experimental data for the
   phase in two regimes.
4. We have also checked the prediction
   with experimental data on G(VG) and
   d(VG) and also found a very good agreement.