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J. BončaJ. BončaPhysics Department, FMF, University of Ljubljana, Physics Department, FMF, University of Ljubljana,
J. Stefan Institute, Ljubljana, SLOVENIAJ. Stefan Institute, Ljubljana, SLOVENIA
Conductance through coupled Conductance through coupled
quantum dotsquantum dots
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Collaborators:Collaborators:
R. R. ŽitkoŽitko, , J. Stefan Inst., Ljubljana, J. Stefan Inst., Ljubljana, SloveniaSlovenia
A.Ramšak and T. RejecA.Ramšak and T. Rejec,, FMF, Physics dept., FMF, Physics dept., University of Ljubljana and J.University of Ljubljana and J. Stefan Inst., Stefan Inst., Ljubljana, Ljubljana, SloveniaSlovenia
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Experimental motivationExperimental motivation Three QD’s:Three QD’s:
Good agreement between Good agreement between CPMCCPMC and and GSGS and and NRG NRG approachesapproaches
Many different regimesMany different regimes• t’’>t’’>: three peaks in : three peaks in G(G()) due to 3 molecular levels due to 3 molecular levels• t’’<t’’<: a single peak in : a single peak in G(G()) of width ~ of width ~ UU
At t”<<D, in the At t”<<D, in the crossover regimecrossover regime an unstable non-Fermi an unstable non-Fermi liquid (NFL) fixed point exists liquid (NFL) fixed point exists
Two-stage Kodo effect is also followed by the NFL Two-stage Kodo effect is also followed by the NFL N-parallel QD’s: N-parallel QD’s:
~0: ~0: S=N/S=N/2 Kondo effect 2 Kondo effect ~~U/U/2: Quantum phase transitions2: Quantum phase transitions
IntroductionIntroduction
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Double- and multiple- dot structuresDouble- and multiple- dot structures
Craig et el., Science Craig et el., Science 304 ,304 , 565 (2004) 565 (2004)
Holleitner et el., Science Holleitner et el., Science 297,297, 70 (2002) 70 (2002)
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Three alternative methods:Three alternative methods:
Numerical Renormalization Group using Numerical Renormalization Group using Reduced Density Matrix (NRG), Reduced Density Matrix (NRG), Krishna-murthy, Krishna-murthy, Wilkins and Wilson, PRB Wilkins and Wilson, PRB 2121, 1003 (1980); Costi, Hewson and , 1003 (1980); Costi, Hewson and ZlatiZlatićć, J. Phys.: Condens. Matter , J. Phys.: Condens. Matter 66, 2519, (1994); Hofstetter, , 2519, (1994); Hofstetter, PRLPRL 85, 1508 (2000).85, 1508 (2000).
Projection – variational metod (GS)Projection – variational metod (GS), , Schonhammer, Z. Phys. B Schonhammer, Z. Phys. B 2121, 389 (1975); PRB , 389 (1975); PRB 1313, 4336 (1976), , 4336 (1976), Gunnarson and Shonhammer, PRB Gunnarson and Shonhammer, PRB 3131, 4185 (1985), Rejec and , 4185 (1985), Rejec and RamRamššak, PRB 68, 035342 (2003).ak, PRB 68, 035342 (2003).
Constrained Path Monte CarloConstrained Path Monte Carlo methodmethod (CPMC),(CPMC), Zhang, Carlson and Gubernatis, PRL 74 ,3652 Zhang, Carlson and Gubernatis, PRL 74 ,3652
(1995);PRB (1995);PRB 5959, 12788 (1999)., 12788 (1999).
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How to obtain G from GS properties:How to obtain G from GS properties:
CPMCCPMC and and GSGS are zero-temperature are zero-temperature methods methods Ground state energy Ground state energy
Conditions: System is a Conditions: System is a Fermi Fermi liquidliquid
N-(noninteracting) N-(noninteracting) sites, sites, N N ∞∞
GG00=2e=2e22/h/h
Rejec, RamRejec, Ramššak, PRB 68, 035342 (2003)ak, PRB 68, 035342 (2003)
~
~
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Comparison: Comparison: CPMC,GS,NRGCPMC,GS,NRG
• CPMC,CPMC, • GS-variational,GS-variational,• Hartree-Fock:Hartree-Fock:
• NRG:NRG:
Meir-Wingreen, PRL Meir-Wingreen, PRL 6868,, 2512 (1992)2512 (1992)
U<t;Wide-band
Rejec, RamRejec, Ramššak, PRB 68, 035342 ak, PRB 68, 035342 (2003)(2003)
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Comparison: Comparison: CPMC,GS,NRGCPMC,GS,NRG
• CPMC, • GS-variational,• Hartree-Fock:
• NRG:
Meir-Wingreen, PRL 68, 2512 (1992)
U>>t;Narrow-band
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Three coupled quantum dotsThree coupled quantum dotsŽŽitko, Bonitko, Bončča, Rejec, Rama, Rejec, Ramššak, PRB ak, PRB 7373, 153307 (2006), 153307 (2006)
Using NRG technique:Using NRG technique: Using GS – variational: NUsing GS – variational: NGSGS [1000,2000] [1000,2000]
Using CPMC: NUsing CPMC: NCPMCCPMC [100,180] [100,180]
MOMO
AFMAFM
TSKTSK
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Three coupled quantum dotsThree coupled quantum dots
Using NRG technique:Using NRG technique: Using GS – variational: NUsing GS – variational: NGSGS [1000,2000] [1000,2000]
Using CPMC: NUsing CPMC: NCPMCCPMC [100,180] [100,180]
MOMO
AFMAFM
TSKTSK
Half-filled case!Half-filled case!
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Three QDs Three QDs NNon-on-FFermi-ermi-LLiquid: iquid:
ŽŽitko & Bonitko & BonččaaPRLPRL
98, 04720398, 047203 Kuzmenko Kuzmenko et al.,Europhy.Lett. et al.,Europhy.Lett. 6464
218 2003218 2003OBSERVATIONOBSERVATION
Potok et al.,Potok et al.,Cond-mat/0610721Cond-mat/0610721
TTKK(1)(1)
TTKK(2)(2)
T
TTKK(1)(1)
TTKK(2)(2)
MOMO
AFMAFM
TSKTSK
NFLNFL
MOMO
AFMAFM
ZOOMZOOM
Cv~T lnT , s~lnT, S(T0)=(1/2) ln 2
SU(2)SU(2)spinspin x SU(2) x SU(2)izospinizospin
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Three QDs Three QDs NNon-on-FFermi-ermi-LLiquid: iquid:
CCvv~~TT ln lnTT , , ss~ln ~ln TT
TTKK(1)(1)
TTKK(2)(2)
T
TTKK(1)(1)
TTKK(2)(2)
MOMO
AFMAFM
TSKTSK
NFL
MOMO
AFMAFM
ZOOM
Žitko & BončaPRL
98, 047203
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Three coupled QDs Three coupled QDs NNon-on-FFermi-ermi-LLiquidiquid
MOMO
AFMAFM
TSKTSK
Affletck et al. PRB 45, 7918 (1992)
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Three coupled QDs Three coupled QDs NNon-on-FFermi-ermi-LLiquidiquid
MOMO
AFMAFM
TSKTSK
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Quantum phase transitions in parallel QD’sQuantum phase transitions in parallel QD’s
R.R.ŽitkoŽitko. & J.B. & J.Bončaonča PRB PRB 7474, 045312 (2006), 045312 (2006)
Schrieffer-Wolf
Perturbation in Vk4-th order
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N - quantum dotsN - quantum dots
Three different time-scales:Three different time-scales:
Separation of time-scales:Separation of time-scales:
Different temperature-regimes:Different temperature-regimes:
S=N/2
N/8
N/4
S(S+1)/3
S=N/2-1
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Quantum phase transitions in parallel Quantum phase transitions in parallel QD’sQD’s
~0: ~0: SS==NN/2 Kondo effect/2 Kondo effect UU/2/2Discontinuities in Discontinuities in
GG Discontinuities in Discontinuities in G G
Quantum phase transitionsQuantum phase transitions
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Quantum phase transitions in parallel Quantum phase transitions in parallel QD’sQD’s
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ConclusionsConclusions Three QD’s in series:Three QD’s in series:
Good agreement between Good agreement between NRG,NRG, GSGS, and , and CPMCCPMC.. Different phases exist:Different phases exist:
• t’’>t’’>: three peaks in : three peaks in G(G()) due to 3 molecular levels ( due to 3 molecular levels (MOMO), t’’<), t’’<: a : a single peak in single peak in G(G()) of width ~ of width ~ U U in the in the AFMAFM regime regime
• Two-stage Kondo Two-stage Kondo (TSK)(TSK) regime, when t’’<T regime, when t’’<TKK
• NFLNFL behavior is found in the behavior is found in the crossover regimecrossover regime. . A good A good candidate for the experimental observation. candidate for the experimental observation.
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ConclusionsConclusions Three QD’s in series:Three QD’s in series:
Good agreement between Good agreement between NRG,NRG, GSGS, and , and CPMCCPMC..
Different phases exist:Different phases exist:• t’’>t’’>: three peaks in : three peaks in G(G()) due to 3 molecular levels ( due to 3 molecular levels (MOMO), ),
t’’<t’’<: a single peak in : a single peak in G(G()) of width ~ of width ~ U U in the in the AFMAFM regime regime
• Two-stage Kondo Two-stage Kondo (TSK)(TSK) regime, when t’’<T regime, when t’’<TKK
• NFLNFL behavior is found in the behavior is found in the crossover regimecrossover regime. . A good A good candidate for the experimental observation. candidate for the experimental observation.
N-parallel QD’s: N-parallel QD’s: ~0: ~0: S=N/S=N/2 Kondo effect 2 Kondo effect ~~U/U/2: Quantum phase transitions2: Quantum phase transitions