Theoretical Study of Quantum Dot Phase Evolution with Kondo Correlations

1. 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)

2. Determining the phase of a QD by using a two path Aharonov-Bohm interferometer F QD 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

4. 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)

5. 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)

6. 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

7. Scattering theory in 1D In the case when there is no magnetic moment in the dot (for instance in the Kondo regime at T=0), spin-flip scattering cannot occur outgoing incoming Asymptotic solutions

8. Scattering theory in 1D For the symmetric QD, following Ng and Lee PRL ’88 ^ ^ ^ Scattering theory Using exact results on Fermi liquid at T=0, one can show that Denoting the phase of by , one gets (Friedel sum rule see Langreth Phys.Rev.’66)

9. Using trigonometric arguments Using again exact results on Fermi liquid at T=0, one can show Putting altogether, one gets

10. Scattering off a composite system Generalized Levinson’s theorem where is the number of bound states is the number of states excluded by the Pauli principle Levinson’49 Swan ’55 Rosenberg and Spruch PRA’96 Example: scattering of an electron by an atom of hydrogen is the ground state of a hydrogen atom “Triplet” scattering : Sztot=1 “Singlet” scattering: Sztot=0 + e H 1s Phase shift 1s2 0 1s"+1s# 1s"+1s" 1s"1s" 0

11. Scattering theory in 1D Quantum dot = Artificial atom Generalized Levinson’s 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 anadditional 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

13. Scattering theory in 1D Landauer formula Aharonov-Bohm interferometry Consequences (at T=0, H=0) • Phase shift measured • Conductance measured

14. Scattering theory in 1D Experimental check of the prediction P. Vitushinky, A.Jerez, M.Lavagna Quantum Information and Decoherence in Nanosystems, p.309 (2004)

15. Bethe-Ansatz solution at T=0 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

16. Bethe-Ansatz solution at T=0 In the asymmetric regime, , n0 shows a universal behavior as a function of the renormalized energy Universal behavior occurs when Asymptotic behavior in the limitn0 0 The existence of both those universal and asymptotic behavior is of valuable help in fitting the experimental data

17. Bethe-Ansatz solution at T=0 Fit in the unitary limit and Kondo regimes All the experimental curves are shifted in order to get = at the symmetric limit

18. Bethe-Ansatz solution at T=0 (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, PRL’05

19. Conclusions • 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. • 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. • 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. • We have also checked the prediction with experimental data on G(VG) and d(VG)and also found a very good agreement.