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: 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

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, (2003).
Constrained Path Monte Carlo method (CPMC), Zhang, Carlson and Gubernatis, PRL 74 ,3652 (1995);PRB 59, (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, (2003)

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

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

9. 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]

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) ln 2
TK(1)
AFM
SU(2)spin x SU(2)izospin MO
TK(2) MO
AFM
TSK
Zitko & Bonca
PRL
98, Kuzmenko et al.,Europhy.Lett
OBSERVATION
Potok et al.,
Cond-mat/
TK(1)
TK(2) TD
ZOOM
NFL

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

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

14. Three coupled QDs Non-Fermi-LiquidMO
AFM
TSK

15. Quantum phase transitions in parallel QD’s

16. 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:

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