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- 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)

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

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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)

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Numerical Renormalization Group + H o H1H1 H2H2 HNHN H3H N... V Energy resolution

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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

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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

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SIAM renormalized parameters

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TRANSPORT IN SERIAL DOUBLE QUANTUM DOTS LL RR tdtd We consider two Kondo dots connected serially This is the artificial realization of the “Two-impurity Kondo problem” 12 RL

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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)

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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

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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

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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

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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 ee oo ee oo 4. Particle-hole symmetry: Average occupation is one Friedel-Langreth sum rule fullfilled

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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

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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<

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Discrepancy for The Large U limit

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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

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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

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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

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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)

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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?

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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

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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 ¡¡¡¡¡¡

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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

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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

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Processes that generate J I

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

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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

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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

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From the Symmetric U to the Infinite-U Case

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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

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