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Conductance through coupled quantum dots

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Conductance through coupled quantum dots

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  1. Conductance through coupled quantum dots J. Bonča Physics Department, FMF, University of Ljubljana, J. Stefan Institute, Ljubljana, SLOVENIA

  2. Collaborators: • R. Žitko, J. Stefan Inst., Ljubljana, Slovenia • A.Ramšak and T. Rejec,FMF, Physics dept., University of Ljubljana and J. Stefan Inst., Ljubljana, Slovenia

  3. Introduction • Experimental motivation • Three QD’s: • Good agreement between CPMC and GS and NRG approaches • Many different regimes • t’’>G: three peaks in G(d) due to 3 molecular levels • t’’<G: a single peak in G(d) of width ~ U • At t”<<D, in the crossover regime an unstable non-Fermi liquid (NFL) fixed point exists • Two-stage Kodo effect is also followed by the NFL • N-parallel QD’s: • d~0: S=N/2 Kondo effect • d~U/2: Quantum phase transitions

  4. Double- and multiple- dot structures Holleitner et el., Science 297, 70 (2002) Craig et el., Science 304 , 565 (2004)

  5. Three alternative methods: • Numerical Renormalization Group using Reduced Density Matrix (NRG), Krishna-murthy, Wilkins and Wilson, PRB 21, 1003 (1980); Costi, Hewson and Zlatić, J. Phys.: Condens. Matter 6, 2519, (1994); Hofstetter, PRL 85, 1508 (2000). • Projection – variational metod (GS), Schonhammer, Z. Phys. B 21, 389 (1975); PRB 13, 4336 (1976), Gunnarson and Shonhammer, PRB 31, 4185 (1985), Rejec and Ramšak, PRB 68, 035342 (2003). • Constrained Path Monte Carlomethod(CPMC),Zhang, Carlson and Gubernatis, PRL 74 ,3652 (1995);PRB 59, 12788 (1999).

  6. How to obtain G from GS properties: • CPMC and GS are zero-temperature methods  Ground state energy • Conditions: System is a Fermi liquid ~ N-(noninteracting) sites, N ∞ ~ G0=2e2/h Rejec, Ramšak, PRB 68, 035342 (2003)

  7. Comparison: CPMC,GS,NRG • CPMC, • GS-variational, • Hartree-Fock: Rejec, Ramšak, PRB 68, 035342 (2003) U<t; Wide-band • NRG: Meir-Wingreen, PRL 68,2512 (1992)

  8. Comparison: CPMC,GS,NRG • CPMC, • GS-variational, • Hartree-Fock: • NRG: U>>t; Narrow-band Meir-Wingreen, PRL 68,2512 (1992)

  9. Three coupled quantum dotsŽitko, Bonča, Rejec, Ramšak, PRB 73, 153307 (2006) MO AFM TSK • Using NRG technique: • Using GS – variational: NGS [1000,2000] • Using CPMC: NCPMC [100,180]

  10. Three coupled quantum dots Half-filled case! MO AFM TSK • Using NRG technique: • Using GS – variational: NGS [1000,2000] • Using CPMC: NCPMC [100,180]

  11. Three QDs Non-Fermi-Liquid: Cv~T lnT , cs~lnT, S(T0)=(1/2)ln2 TK(1) AFM SU(2)spin x SU(2)izospin MO TK(2) MO AFM TSK Žitko & Bonča PRL 98, 047203Kuzmenko et al.,Europhy.Lett. 64 218 2003 OBSERVATION Potok et al., Cond-mat/0610721 TK(1) TK(2) TD ZOOM NFL

  12. Three QDs Non-Fermi-Liquid: Cv~T lnT , cs~ln T Žitko & Bonča PRL 98, 047203 TK(1) MO AFM MO AFM ZOOM TSK TK(2) TK(1) TK(2) TD NFL

  13. Three coupled QDs Non-Fermi-Liquid MO AFM TSK Affletck et al. PRB 45, 7918 (1992)

  14. Three coupled QDs Non-Fermi-Liquid MO AFM TSK

  15. Quantum phase transitions in parallel QD’sR.Žitko. & J.Bonča PRB 74, 045312 (2006) Schrieffer-Wolf Perturbation in Vk4-th order

  16. N - quantum dots S=N/2-1 S=N/2 • Three different time-scales: S(S+1)/3 N/4 N/8 • Separation of time-scales: • Different temperature-regimes:

  17. Quantum phase transitions in parallel QD’s • d~0: S=N/2 Kondo effect • d~U/2 Discontinuities in G • Discontinuities in G  Quantum phase transitions

  18. Quantum phase transitions in parallel QD’s

  19. Conclusions • Three QD’s in series: • Good agreement between NRG,GS, and CPMC. • Different phases exist: • t’’>G: three peaks in G(d) due to 3 molecular levels (MO), t’’<G: a single peak in G(d) of width ~ U in the AFM regime • Two-stage Kondo (TSK) regime, when t’’<TK • NFL behavior is found in the crossover regime. A good candidate for the experimental observation.

  20. Conclusions • Three QD’s in series: • Good agreement between NRG,GS, and CPMC. • Different phases exist: • t’’>G: three peaks in G(d) due to 3 molecular levels (MO), t’’<G: a single peak in G(d) of width ~ U in the AFM regime • Two-stage Kondo (TSK) regime, when t’’<TK • NFL behavior is found in the crossover regime. A good candidate for the experimental observation. • N-parallel QD’s: • d~0: S=N/2 Kondo effect • d~U/2: Quantum phase transitions