- 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)

OUTLINE OF THIS TALK 1.NRG, Fermi Liquid description of the SIAM 2.Double quantum dot 3.Reservoir-mediated superexchange interaction 4.Conclusions

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)

Numerical Renormalization Group + H o H1H1 H2H2 HNHN H3H N... V          Energy resolution

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

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

SIAM renormalized parameters

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

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 (2000)  

Two-impurity Kondo problem R. Lopez R. Aguado and G. Platero, Phys. Rev. Lett (2002) Serial DQD, t C =0.5 J=25 x10 -4 J=25 x10 -4

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

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

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

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

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

Discrepancy for The Large U limit

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

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

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

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 (2000)

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?

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

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 ¡¡¡¡¡¡

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

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

Processes that generate J I

Initial state Final state J I S 1 S 2 J I Reservoir-mediated superexchange interaction

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

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

From the Symmetric U to the Infinite-U Case

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