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

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

Introduction Experimental motivation Three QD’s: N-parallel 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

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

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 Carlo method (CPMC), Zhang, Carlson and Gubernatis, PRL 74 ,3652 (1995);PRB 59, 12788 (1999).

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)

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

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

Three coupled quantum dots Zitko, Bonca, Rejec, Ramsak, PRB 73, 153307 (2006)  MO AFM TSK Using NRG technique: Using GS – variational: NGS [1000,2000] Using CPMC: NCPMC [100,180]

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

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

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

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

Three coupled QDs Non-Fermi-Liquid MO AFM TSK

Quantum phase transitions in parallel QD’s

N - quantum dots Three different time-scales: 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:

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

Quantum phase transitions in parallel QD’s

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.

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