www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 J. Bonča Physics Department, FMF, University of Ljubljana, J. Stefan Institute, Ljubljana, SLOVENIA Conductance through coupled quantum dots

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 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

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007  Experimental motivation  Single QD: using three different methods: NRG, CPMC and GS – accurate results in a wide parameter regime  Three QD’s: Good agreement between CPMC and GS. Two 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, two-stage Kodo effect is found with an unstable non-Fermi liquid fixed point Introduction

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Nature, 391 (1998) Science, 281 (1998)

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Double- and multiple- dot structures Craig et el., Science 304, 565 (2004) Holleitner et el., Science 297, 70 (2002)

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot (Anderson single impurity problem) d

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded d=e d +U/2 U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d Meir-Wingreen, PRL 68, 2512 (1992) e d +U eded d=e d +U/2 U =1 G=dI/dV| V=0

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded d=e d +U/2 U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded d=e d +U/2 U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded d=e d +U/2 U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded d=e d +U/2 U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded d=e d +U/2 U =1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Quantum Dot d e d +U eded d=e d +U/2 ~ gate voltage U =1 D =U>> G

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Three alternative methods:  Constrained Path Monte Carlo method (CPMC), Zhang, Carlson and Gubernatis, PRL 74,3652 (1995);PRB 59, (1999).  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).  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).

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 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  ∞ G 0 =2e 2 /h Rejec, Ramšak, PRB 68, (2003) ~ ~

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Fisher – Lee relation … Conductance formalisms non-equilibrium transport: T ≠ 0, V ≠ 0 U = 0 Landauer – Büttiker formula linear response regime: T ≠ 0, V ~ 0zero-temperature linear response: T = 0, V ~ 0 U ≠ 0 Meir – Wingreen formula In Fermi liquid systems Kubo formula

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Comparison: CPMC,GS,NRG CPMC, GS-variational, Hartree-Fock: NRG: Meir-Wingreen, PRL 68, 2512 (1992) U<t; Wide-band Rejec, Ramšak, PRB 68, (2003)

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Comparison: CPMC,GS,NRG CPMC, GS-variational, Hartree-Fock: NRG: Meir-Wingreen, PRL 68, 2512 (1992) U>>t; Narrow-band

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Fermi-liquid E( f ) is a universal function of f Fermi-liquid E( f ) is a universal function of f Number of electrons odd Rejec, Ramšak, PRB 68, (2003)

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Zero-bias conductance Rejec, Ramšak, PRB 68, (2003)

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Rejec, Ramšak, PRB 68, (2003) Connection of G with charge stiffness

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 GS variational method Auxiliary Hamitonian Projection operators Variational wavefunctions: E H : the lowest eigenvalue gives the approximation to the GS of H

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 GS variational method – cont. Using Wick’s theorem

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Side-coupled Double Quantum Dot R.Z. & J.B. PRB 73, (2006)

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Large t d

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Large t d – Widths of conductance plateaus: Energies on isolated DQD: d1d1 d2d2

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Large t d – Kondo temperatures: Estimating T K using Schrieffer-Wolf:

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Large t d – Kondo temperatures: Estimating T K using Schrieffer-Wolf:

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Large t d – Adding FM coupling E S=1 E S=0 -J ad

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Small t d – Two-stage Kondo effect J eff <T K :Two Kondo temperatures: T K and T K 0 Vojta et al., PRB 65, (2002); Hofstetter, Schoeller, PRL 88, (2002), Cornaglia and Grempel, PRB 71, (2005), Wiel et al., PRL 88, (2002). Two energy scales: J eff =4t d 2 /U, T K

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Small t d – Two-stage Kondo effect J eff <T K :Two Kondo temperatures: T K and T K 0 TKTK TK0TK0 J eff <T K Vojta et al., PRB 65, (2002); Hofstetter, Schoeller, PRL 88, (2002), Cornaglia and Grempel, PRB 71, (2005), Wiel et al., PRL 88, (2002). Two energy scales: J eff =4t d 2 /U, T K

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Small t d – Two-stage Kondo effect J eff >T K J eff w

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Small t d – Two-stage Kondo effect J eff ~T K TKTK w TK0TK0

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Small t d – Two-stage Kondo effect J eff <T K TKTK w TK0TK0

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Small t d – Two-stage Kondo effect J eff <T K ~T TKTK w Experimental evidence Wiel et al., PRL 88, (2002).

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Large t d – Adding FM coupling Two-stage Kondo effect? Voja et al., PRB 65, (2002), Hofstetter, Schoeller, PRL 88, (2002),

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Three coupled quantum dots Zitko, Bonca, Rejec, Ramsak, PRB 73, (2006)  Using CPMC: N CPMC [100,180]  Using GS – variational: N GS [1000,2000]  Using NRG technique: MO AFM TSK

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Three coupled QDs 12 3 Oguri, Nisikawa,Hewson Half-filled case!

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Three coupled QDs 12 3 Oguri, Nisikawa,Hewson, cond-mat/ Half-filled case!

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Three QDs Non-Fermi-Liquid: Zitko & Bonca to appear in PRL T K (1) T K (2) TDTD T K (1) T K (2) MO AFM TSK NFL MO AFM ZOOM Cv~T lnT, c s ~lnT, S(T  0)=(1/2) ln 2 SU(2) spin x SU(2) izospin

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Three QDs Non-Fermi-Liquid: C v ~T lnT, c s ~ln T Zitko & Bonca to appear in PRL T K (1) T K (2) TDTD T K (1) T K (2) MO AFM TSK NFL MO AFM ZOOM

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Three coupled QDs Non-Fermi-Liquid MO AFM TSK Affletck et al. PRB 45, 7918 (1992)

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Three coupled QDs Non-Fermi-Liquid MO AFM TSK

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 N - quantum dots R.Z. & J.B. PRB 74, (2006) Schrieffer-Wolf Perturbation in V k 4 -th order

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 N - quantum dots  Three different time-scales:  Separation of time-scales:  Different temperature-regimes: S=N/2 N/8 N/4 S(S+1)/3 S=N/2-1

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Spin-charge Kondo effect, separation of spin and charge Nambu spinor: a) d)

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Spin-charge Kondo effect, separation of spin and charge

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Conclusions  Using three different methods: NRG, CPMC and GS – accurate results in a wide parameter regime  Three QD’s: Good agreement between CPMC, GS and NRG. Different phases exist:  t’’> G : three peaks in G( d ) due to 3 molecular levels  t’’< G : a single peak in G( d ) of width ~ U  Two-stage Kondo regime, when t’’<T K  NFL behavior is found in the crossover regime. A good candidate for the experimental observation.

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Conclusions  Using three different methods: NRG, CPMC and GS – accurate results in a wide parameter regime  DQD system: Large t d : Kondo regimes for odd DQD occupancy Small t d : Two-stage Kondo regime  Three QD’s: Good agreement between CPMC and GS. Different phases exist:  t’’> G : three peaks in G( d ) due to 3 molecular levels  t’’< G : a single peak in G( d ) of width ~ U  Two-stage Kondo regime, when t’’<T K  NFL behavior is found  N-dot system in parallel: RKKY interaction

www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007