1. http://ifisc.uib-csic.es - Mallorca - Spain Quantum Engineering of States and Devices: Theory and Experiments Obergurgl, Austria 2010 The two impurity Anderson Model revisited: Competition between Kondo effect and reservoir-mediated superexchange in double quantum dots Rosa López (Balearic Islands University,IFISC) Collaborators Minchul Lee (Kyung Hee University, Korea) Mahn-Soo Choi (Korea University, Korea) Rok Zitko (J. Stefan Institute, Slovenia) Ramón Aguado (ICMM, Spain) Jan Martinek (Institute of Molecular Physics, Poland)

2. http://ifisc.uib-csic.es OUTLINE OF THIS TALK 1.NRG, Fermi Liquid description of the SIAM 2.Double quantum dot 3.Reservoir-mediated superexchange interaction 4.Conclusions

3. http://ifisc.uib-csic.es Numerical Renormalization Group Spirit of NRG: Logarithmic discretization of the conduction band. The Anderson model is transformed into a Wilson chain Example: Single impurity Anderson Model (SIAM)

4. http://ifisc.uib-csic.es Numerical Renormalization Group + H o H1H1 H2H2 HNHN H3H3 0123 N... V          Energy resolution

5. http://ifisc.uib-csic.es Fermi liquid fixed point: SIAM renormalized parameters The low-temperature behavior of a impurity model can often be described using an effective Hamiltonian which takes exactly the same form as the original Hamiltonian but with renormalized parameters Example: SIAM, Linear conductance related with the phase shift and this related with the renormalized paremeters

6. http://ifisc.uib-csic.es Fermi liquid fixed point: SIAM renormalized parameters RENORMALIZED PARAMETERS E p(h) are the lowest particle and hole excitations from the ground state.They are calculated from the NRG output. g 00  is the Green function at the first site of the Wilson chain

7. http://ifisc.uib-csic.es SIAM renormalized parameters

8. http://ifisc.uib-csic.es TRANSPORT IN SERIAL DOUBLE QUANTUM DOTS LL RR tdtd We consider two Kondo dots connected serially This is the artificial realization of the “Two-impurity Kondo problem” 12 RL

9. http://ifisc.uib-csic.es Transport in double quantum dots in the Kondo regime For  G 0 ~ ( 2e 2 /h) t 2 For  G 0 ~ ( 2e 2 /h) t 2 For  G 0 =2e 2 /h, For  G 0 =2e 2 /h, For  G 0 decreases as  grows For  G 0 decreases as  grows Transport is governed by  =t/  R. Aguado and D.C Langreth, Phys. Rev. Lett. 85 1946 (2000)  

10. http://ifisc.uib-csic.es Two-impurity Kondo problem R. Lopez R. Aguado and G. Platero, Phys. Rev. Lett.89 136802 (2002) Serial DQD, t C =0.5 J=25 x10 -4 J=25 x10 -4

11. http://ifisc.uib-csic.es TRANSPORT IN SERIAL DOUBLE QUANTUM DOTS We consider two Kondo dots connected serially This is the artificial realization of the “Two-impurity Kondo problem” In the even-odd basis

12. http://ifisc.uib-csic.es TRANSPORT IN SERIAL DOUBLE QUANTUM DOTS We analyze three different cases: 1.Symmetric Case (  d =-U/2) 2.Infinity U Case 3.The transition from the finite U to the infinity U Case

13. http://ifisc.uib-csic.es Symmetric Case: Phase Shifts 1.When t d =0 both phase shifts are equal to  2 2.For large t d /  we have  e = ,  o =0 and the conductance vanishes 3. For certain value of t d /  the conductance is unitary ee oo ee oo 4. Particle-hole symmetry: Average occupation is one Friedel-Langreth sum rule fullfilled

14. http://ifisc.uib-csic.es Scaling function The position of the main peak, t d = t c1, is determined by the condition  =  /2, which coincides with the condition that the exchange coupling J is comparable to T K, or J = J c = 4t c1 2 /U ~ 2.2 T K The crossover from the Kondo state to the AF phase is described by a scaling function Scaling function

15. http://ifisc.uib-csic.es Crossover: Scaling Function 1.The appearence of the unitary-limit- value conductance is explained in terms of a crossover between the Kondo phase and the AF phase 2.When J<<T K each QD forms a Kondo state and then G 0 is very low (hopping between two Kondo resonances) 3.When J>>T K the dot spins are locked into a spin singlet state G 0 decreases

16. http://ifisc.uib-csic.es Discrepancy for The Large U limit

17. http://ifisc.uib-csic.es Infinite-U Case For t d = 0 we have Since U is very large, the dot occupation does not reach 1 up to t d /  ~ 1 the phase shifts show the same behavior as the symmetric case. Finally for large t d /  the phase shift difference saturates around  /2 The phase shift difference shows nonmonotonic behavior

18. http://ifisc.uib-csic.es Linear Conductance Why the unitary-limit-value depends on  ? The main peak is shifted toward larger t d  with increasing  and its width also increases with  Plateau of 2e 2 /h starting at  d : Spin Kondo in the even sector

19. http://ifisc.uib-csic.es Spin Kondo effect in the even sector Plateau in G 0 : As t d increases, the DD charge decreases to one 1.The one-e - even-orbital state |N=1, S=1/2> of isolated DD with energy  d -t d is lowered below the two-dots groundstate |N=2, S=0> and |N=2, S=1> with energy 2  d as soon as t d is increased beyond  d  2.The conductance plateau is then attributed to the formation of a single-impurity Kondo state in the even channel, leading to  e =  The odd channel becomes empty with  o ~0

20. http://ifisc.uib-csic.es Linear conductance For the infinity U case the exchange interaction vanishes. From Fermi Liquid theories (SBMFT, for example) we know that SBMFT marks the maximum for G 0 when t d * /2    t d /   This maximum is attributed to the formation of a  coherent superposition of Kondo states:  bonding -antibonding Kondo states R. Aguado and D.C Langreth,Phys. Rev. Lett. 85 1946 (2000)

21. http://ifisc.uib-csic.es Renormalized parameters 1.Fermi liquid theories, like SBMFT, predicts t d /2  t d * /2  * i.e., a universal behavior of G 0 independently on the  value 2.However, NRG results indicate that the peak position of G 0 depends strongly on  This surprising result suggests that t d /2  flows to larger values, so that t d /2  t d * /2  * Which is the origin of this discrepancy not noticed before?

22. http://ifisc.uib-csic.es Renormalize parameters: Symmetric U case The unitary value of G 0 coincides with =-1/4 denoting the formation of a spin singlet state between the dots spins due to the direct exchange interaction vv

23. http://ifisc.uib-csic.es Renormalize parameters: Infinity U Case Importantly: The unitary value of G 0 coincides with =-1/4 denoting the formation of a spin singlet state between the dots spins. However, for infinite U there is no direct exchange interaction ¡¡¡¡¡¡

24. http://ifisc.uib-csic.es Magnetic interactions 1.J U is the known direct coupling between the dots that vanishes for infinite U J U =4t d 2 /U 2.J I is a new exchange term that in general depends on U but does not vanish when this goes to infinity J I (U=0) does not vanish

25. http://ifisc.uib-csic.es Magnetic correlations 1.Indeed the essential features of the system state should not change whatever value of Coulomb interaction U is 2.The infinite U case is then also explained in terms of competition between an exchange coupling and the Kondo correlations. Therefore, there must exist two kinds of exchange couplings J=J U +J I

26. http://ifisc.uib-csic.es Processes that generate J I

27. http://ifisc.uib-csic.es Initial state Final state J I S 1 S 2 J I Reservoir-mediated superexchange interaction

28. http://ifisc.uib-csic.es Using the Rayleigh-Shr ö dinger perturbation theory for the infinite U case (to sixth order) yields For finite U case a more general expression can be obtained where the denominators in J I also depends on U It is expected then a universal behavior of the linear conductance as a function of a scaling function given by J I Reservoir-mediated superexchange interaction. Remarkably: This high order tunneling event is able to affect the transport properties

29. http://ifisc.uib-csic.es. SB theories should be in agreement with NRG calculations if ones introduces by hand this new term J I. This new term will renormalize t d in a different manner than it does for  and then t d /2  t d * /2  * This can explain the dependence on  of the peak position of the maximum in the linear conductance J 2 Reservoir-mediated superexchange interaction

30. http://ifisc.uib-csic.es From the Symmetric U to the Infinite-U Case

31. http://ifisc.uib-csic.es Conclusions Our NRG results support the importance of including magnetic interactions mediated by the conduction band in the theory in the Large-U limit. In this manner we have a showed an unified physical description for the DQD system when U  finite to U  Inf