Quantum criticality in a double-quantum-dot system

1. Quantum criticality in a double-quantum-dot system G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006) Chung-Hou Chung 仲崇厚 Electrophysics Dept. National Chiao-Tung University Hsin-Chu, Taiwan Collaborators: Gergely Zarand (Budapest), Matthias Vojta (TKM, Karlsruhe) Pascal Simon (CNRS, Grenoble)

2. 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

3. Kondo effect in quantum dot ed+U Coulomb blockade ed Kondo effect Single quantum dot Vg 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)

4. 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

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

6. Kondo effect in quantum dot

7. Kondo effect in quantum dot AndersonModel New energy scale: Tk ≈ Dexp(-pU/G) For T < Tk : Impurity spin is screened (Kondo screening) Spin-singlet ground state Local density of states developesKondoresonance d ∝ Vg local energy level : charging energy : level width : All tunable! U Γ=2πV 2ρd

8. P-H symmetry = p/2 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 phase shift Fredel sum rule

9. 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.

10. T Non-fermi liquid Kondo Spin-singlet K Kc 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 R1 L1 K L2 R2 • 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

11. Coupled Quantum dots triplet states L1 R1 Izumida and Sakai PRL 87, 216803 (2001) Vavilov and Glazman PRL 94, 086805 (2005) K Simon et al. cond-mat/0404540 Hofstetter and Schoeller, PRL 88, 061803 (2002) L2 singlet state R2 • 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

12. 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

13. Transport properties • Current through the quantum dots: • Transmission coefficient: • Linearconductance:

14. n k-kc Crossover energy scale T* NRG Flow of the lowest energy Phase shift d d Kondo K<KC 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, controlling the quantum phase transition

15. J < Jc, transport properties reach unitary limit: • 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. Quantum phase transition of a double-quantum-dot system C.H. C and W. Hofstetter, cond-mat/0607772 J=RKKY=K

16. 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 2 K X Heavy fermions -R/2 R/2 H = H0+ Himp H0 =

17. T Non-fermi liquid 1 2 even Kondo Spin-singlet K odd Kc 2-impurity Kondo problem • Particle-hole symmetry V=0 H  H’ = H under 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)

18. Particle-hole asymmetry even odd 2-impurity Kondo problem Zhu and Varma, cond-mat/0607426 Sharp phase transition Smooth crossover

19. odd 2-impurity Kondo problem QCP destroyed  crossover P-H asymmetry plus Zhu and Varma, cond-mat/0607426 V12 : Effective potential scattering terms generated Relevant operator at K=Kc Splitting between even and odd resonances even

20. _ _ G G 1 2 K even 2 (L2+R2) even 1 (L1+R1) T Non-fermi liquid Kondo Spin-singlet K Kc Quantum criticality in a double-quantum –dot system G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006) V1 ,V2 break P-H sym and parity sym.  QCP still survives as long as no direct hoping t=0

21. _ _ G G 1 2 K Quantum criticality in a double-quantum –dot system No direct hoping, t = 0 Asymmetric limit: T1=Tk1, T2= Tk2 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

22. Optical conductivity • Sindel, Hofstetter, von Delft, Kindermann, PRL 94, 196602 (2005) 1 Linear AC conductivity

23. Transport of double-quantum-dot near QCP NRG on DQDs without P-H and parity symmetry At K=Kc Affleck andLudwig PRB 48 7279 (1993)

24. charge transfer between two channels of the leads Relevant operator Generate smooth crossover at energy scale The only relevant operator at QCP: direct hoping term t dim[ ] = 1/2 (wr.t.QCP) RG most dangerous operators: off-diagonal J12 typical quantum dot At scale Tk, may spoil the observation of QCP

25. << 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

26. The single quantum dot can get Kondo screened via 2 different channels: At low temperatures, blue channel finite conductance; red channel zero conductance At the 2CK fixed point, Conductance g(Vds) scales as The single quantum dot can get Kondo screened via 2 different channels: At low temperatures, blue channel finite conductance; red channel zero conductance The 2CK fixed point observed in recent Exp. by Goldhaber-Gorden et al. Goldhaber-Gorden et al, Nature 446, 167 ( 2007) At the 2CK fixed point, Conductance g(Vds) scales as

27. 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