www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 J. Bona 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. itko, J. Stefan Inst., Ljubljana, Slovenia A.Ramak 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 QDs: 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 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 Ramak, PRB 68, (2003). Numerical Renormalization Group using Reduced Density Matrix (NRG), Krishna-murthy, Wilkins and Wilson, PRB 21, 1003 (1980); Costi, Hewson and Zlati, 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, Ramak, 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 Bttiker 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; 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, Ramak, PRB 68, (2003) www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Zero-bias conductance Rejec, Ramak, PRB 68, (2003) www-f1.ijs.si/~bonca/work.html New 3 SC-6, Sydney, 2007 Rejec, Ramak, 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 Wicks 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