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Quantum criticality in a double-quantum-dot system - 交通大學電子

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									Quantum criticality in a double-quantum-dot system
         G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)




               Chung-Hou Chung
                       Electrophysics Dept.
              National Chiao-Tung University
                        Hsin-Chu, Taiwan

                     Collaborators:
                 Gergely Zarand (Budapest),
             Matthias Vojta (TKM, Karlsruhe)
             Pascal Simon (CNRS, Grenoble)
                         Outline
• Introduction
• Quantum criticality in a double-quantum-dot system:
            particle-hole symmetry

• Quantum criticality in a 2-impurity Kondo system
• Quantum criticality in a double-quantum-dot system:
       more general case: no P-H or parity symmetry
• Realization of QCP in a proposed experimental setup

• Conclusions
                            Kondo effect in quantum dot

                                   Coulomb blockade    ed+U



                                                        ed
                                                                                Kondo effect
                                                          Vg
   Single quantum dot
Goldhaber-Gorden et al. nature 391 156 (1998)          VSD




                                                                                   odd



                                                                              even


                              conductance anomalies                     Glazman et al. Physics world 2001

                        L.Kouwenhoven et al. science 289, 2105 (2000)
        Kondo effect in metals with magnetic impurities
                                                                         (Kondo, 1964)




                     logT




                                                      electron-impurity scattering
                                                      via spin exchange coupling
(Glazman et al. Physics world 2001)


               At low T, spin-flip scattering off impurities enhances
               Ground state is spin-singlet
               Resistance increases as T is lowered
Kondo effect in quantum dot




                              (J. von Delft)
Kondo effect in quantum dot
                       Kondo effect in quantum dot




                                 Anderson Model


                                     New energy scale: Tk ≈ Dexp(-pU/ G)
local energy level :   ed ∝ Vg
                                                For T < Tk :
charging energy : U
                                     Impurity spin is screened (Kondo screening)
level width :   Γ= 2πV    2ρ
                            d
                                     Spin-singlet ground state
All tunable!
                                     Local density of states developes Kondo resonance
      Kondo Resonance of a single quantum dot

Spectral density at T=0             Universal scaling of T/Tk
                   M. Sindel       L. Kouwenhoven et al. science 2000




 particle-hole
 symmetry                        P-H symmetry

 phase shift

 Fredel sum rule                         = p/2
      Recent experiments on coupled quantum dots

    (I). C.M. Macrus et al.
    Science, 304, 565 (2004)




Two quantum dots coupled through an open conducting region
•
which mediates an antiferromagnetic spin-spin coupling
• For odd number of electrons on both dots, splitting of zero bias
Kondo resonance is observed for strong spin exchange coupling.
        Quantum phase transition and non-Fermi liquid state
                   in Coupled quantum dots
   G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)

   C.H. C and W. Hofstetter, cond-mat/0607772

      L1                                   R1
                                                       T                   Non-fermi liquid

                             K

                                                                   Kondo                  Spin-singlet
      L2                                  R2                                                             K
                                                                                  Kc




• Critical   point is a novel state of matter
• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures
• Quantum critical region exhibits universal power-law behaviors
                                                   Coupled Quantum dots

                                                                               triplet states
Izumida and Sakai PRL 87, 216803 (2001)           L1                 R1
Vavilov and Glazman PRL 94, 086805 (2005)

Simon et al. cond-mat/0404540                                K
Hofstetter and Schoeller, PRL 88, 061803 (2002)
                                                  L2                  R2         singlet state



                • Two quantum dots (1 and 2) couple to two-channel leads
                • Antiferrimagnetic exchange interaction K, Magnetic field B
                • 2-channel Kondo physics, complete Kondo screening for B = K = 0




                                                   K
              Numerical Renormalization Group (NRG)
                                             K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)
                                            W. Hofstetter, Advances in solid state physics 41, 27 (2001)
 Non-perturbative numerical method by Wilson to treat quantum impurity problem

 Logarithmic discretization of the conduction band



 Anderson impurity model is mapped onto a linear chain of fermions



 Iteratively diagonalize the chain and keep low energy levels
                      Transport properties
• Current through the quantum dots:




• Transmission coefficient:




 • Linear conductance:
NRG Flow of the lowest energy                                               Phase shift d




                                         K<KC     Kondo       d

                                                  JC
                                                                    Kondo
                                                             p/2
                                          K>KC
                                             Spin-singlet
                                                                                     Spin-singlet

                                                               0                Kc                   K




Two stable fixed points (Kondo and spin-singlet phases )    Jump of phase shift in both channels at Kc


One unstable fixed point (critical fixed point) Kc,                                              n
                                                             Crossover energy scale T*       k-kc
controlling the quantum phase transition
           Quantum phase transition of a double-quantum-dot system




                                C.H. C and W. Hofstetter, cond-mat/0607772

• J < Jc, transport properties reach unitary limit:
                                                                             J=RKKY=K
T( = 0)     2, G(T = 0)   2G0 where G0 =    2e2/h.
• J > Jc spins of two dots form singlet ground state,
T( = 0)     0, G(T = 0)   0; and Kondo peak splits up.
• Quantum phase transition between Kondo (small J) and spin singlet (large J) phase.
                                        2-impurity Kondo problem
Affleck et al. PRB 52, 9528 (1995)             Jones and Varma, PRL 58, 843 (1989)   Jones and Varma, PRB 40, 324 (1989)

 Sakai et al. J. Phys. Soc. Japan 61, 7, 2333 (1992); ibdb. 61, 7, 2348 (1992)



                                                  1          K         2
                                                             X                       Heavy fermions
                                                      -R/2       R/2


                                              H = H0 + Himp


                                                 H0 =
                                2-impurity Kondo problem

  • Particle-hole symmetry V=0
                                              H  H’ = H        under




               even         T                    Non-fermi liquid
               odd

                                                                        1      2
                                     Kondo                            Spin-singlet
                                                                                          K
                                                          Kc
Quantum phase transition as K is tuned

 Kc = 2.2 Tk                                                                Affleck et al. PRB 52, 9528 (1995)


Jump of phase shift at Kc       K < Kc,   d   = p/2 ; K >KC , d = 0         Jones and Varma, PRL 58, 843 (1989)
                                                                            Jones and Varma, PRB 40, 324 (1989)

                                                                            Sakai et al. J. Phys. Soc. Japan 61, 7, 2333 (1992);
                                                                            ibdb. 61, 7, 2348 (1992)
                         2-impurity Kondo problem


                  • Particle-hole   asymmetry   Zhu and Varma, cond-mat/0607426




                                       even

                                       odd




Sharp phase transition                          Smooth crossover
                             2-impurity Kondo problem


P-H asymmetry             plus                          QCP destroyed  crossover

                                                                  Zhu and Varma, cond-mat/0607426

 V12 : Effective potential scattering terms generated
      Relevant operator at K=Kc




   Splitting between even and odd resonances



                   odd




                   even
          Quantum criticality in a double-quantum –dot system

                                    G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)

                                                            _                        _
                                                            G1                       G2

                                                                         K
                                                                                       even 2 (L2+R2)
                                                even 1 (L1+R1)




                      T                Non-fermi liquid




                               Kondo                       Spin-singlet
                                                                             K
                                               Kc
V1 ,V2 break P-H sym and parity sym.  QCP still survives as long as no direct hoping t=0
     Quantum criticality in a double-quantum –dot system

 No direct hoping, t = 0               Asymmetric limit:                               T1=Tk1, T2= Tk2


      _
      G1                     _2
                             G

                 K




QCP occurs when                                             2 channel Kondo System

                                                            Goldhaber-Gordon et. al. PRL 90 136602 (2003)


                     QC state in DQDs identical to 2CKondo state

                     Particle-hole and parity symmetry are not required


                           Critical point is destroyed by
                           charge transfer btw channel 1 and 2
           Optical conductivity


                   Sindel, Hofstetter, von Delft, Kindermann, PRL 94, 196602 (2005)



1


    Linear AC conductivity
            Transport of double-quantum-dot near QCP




NRG on DQDs without P-H and parity symmetry




                                              At K=Kc


                                                        Affleck and Ludwig PRB 48 7279 (1993)
       The only relevant operator at QCP: direct hoping term t




                          charge transfer between two channels of the leads


dim[     ] = 1/2            Relevant operator
                   Generate smooth crossover at energy scale
 (wr.t.QCP)


           RG

           most dangerous operators: off-diagonal J12

 At scale Tk,                            typical quantum dot




                                       may spoil the observation of QCP
How to suppress hoping effect and observe QCP in double-QDs



                          assume


                          effective spin coupling between 1 and 2




                            off-diagonal Kondo coupling




                <<             more likely to observe QCP
                               of DQDs in experiments
                             Conclusions

• Coupled quantum dots in Kondo regime exhibit quantum phase transition


 • The QCP of DQDs is identical to that of a 2-channel Kondo system


• The QCP is robust against particle-hole and parity asymmetries


•The QCP is destroyed by charge transfer between two channels



• The effect of charge transfer can be reduced by inserting additional
even number of dots, making it possible to be observe QCP in experiments

								
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