Electronic Transport and Quantum Phase Transitions of Quantum Dots in Kondo Regime

1. Electronic Transport and Quantum Phase Transitions of Quantum Dots in Kondo Regime Chung-Hou Chung 1.Institut für Theorie der Kondensierten Materie Universität Karlsruhe, Karlsruhe, Germany 2. Electrophysics Dept. National Chiao-Tung University, HsinChu, Taiwan, R.O.C. Collaborators: Walter Hofstetter (Frankfurt), Gergely Zarand (Budapest), Peter Woelfle (TKM, Karlsruhe) Acknowledgement: Michael Sindel, Matthias Vojta

2. Outline • Introduction • Electronic transport and quantum phase transitions in coupled quantum dots: • Model (I): parallel coupled quantum dots, 2-channel Kondo, • non-trivial quantum critical point • Model (II): side-coupled quantum dots, 1-channel Kondo, • Kosterlitz-Thouless quantum transition • Conclusions and Outlook

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. V 1 2 V t V Interesting topics/questions • Non-equilibrium Kondo effect • Kondo effect in carbon nanotubes • Double quantum dots / Multi-level quantum dot: • Singlet-triplet Kondo effect and Quantum phase transitions

10. T g g c Quantum phase transitions Non-analyticity in ground state properties as a function of some control parameter g Avoided level crossing which becomes sharp in the infinite volume limit: Second-order transition True level crossing: Usually a first-order transition Sachdev, quantum phase transitions, Cambridge Univ. press, 1999 • 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. 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.

12. (II). Von der Zant et al. cond-mat/0508395, (PRL, 2005) • A quantum dot coupled to magnetic impurities in the leads • Antiferromagnetic spin coupling between impurity and dot suppresses Kondo effect • Kondo peak restored at finite temperatures and magnetic fields

13. Coupled quantum dots Model system (I): 2-channel parallel coupled quantum dots C.H. C and W. Hofstetter, cond-mat/0607772 L1 R1 L2 R2 G. Zarand, C.H. C, P. Simon, M. Vojta, cond-mat/0607255 Model system (II): 1-channel side-coupled quantum dots

14. 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

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

16. Model System (I) triplet states L1 R1 Izumida and Sakai PRL 87, 216803 (2001) Vavilov and Glazman PRL 94, 086805 (2005) 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 J, Magnetic field B • 2-channel Kondo physics, complete Kondo screening for B = J = 0

17. L1 R1 R2 L2 even 2 (L2+R2) even 1 (L1+R1) T Non-fermi liquid 1 2 Kondo Spin-singlet J Jc J-Jc Specific heat coefficient g -2 2-impurity Kondo problem Quantum phase transition as J is tuned For V1 = V2 and with p-h symmetry Jc = 2.2 Tk Affleck et al. PRB 52, 9528 (1995) Jump of phase shift at Jc J < Jc, d = p/2 ; J >JC , 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. n J-Jc Crossover energy scale T* NRG Flow of the lowest energy Phase shift d d Kondo J<JC JC Kondo p/2 J>JC Spin-singlet Spin-singlet 0 Jc J Two stable fixed points (Kondo and spin-singlet phases ) Jump of phase shift in both channels at Jc One unstable fixed point (critical fixed point) Jc, controlling the quantum phase transition

19. Quantum phase transition of Model System (I) • 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.

20. Restoring of Kondo resonance Singlet-triplet crossover at finite temperatures T NRG Result Experiment by von der Zant et al. T=0.003 T=0.004 • At T= 0, Kondo peak splits up due to large J. • Low energy spectral density increases as temperature increases • Kondo resonance reappears when T is of order of J • Kondo peak decreases again when T is increased further.

21. At T = B = 0, Kondo peak splits up due to large J. • T = 0 singlet-triplet crossover at finite magnetic fields. • Splitting of Kondo peaks gets smaller as B increases. • B J, Kondo resonance restored, T( = 0) 1 reaches • unitary limit of a single-channel S = ½ Kondo effect. • B > J, Kondo peak splits again. • B J, T() shows 4 peaks in pairs around = (B J). Singlet-triplet crossover at finite magnetic fields Jc=0.00042 Tk=0.0002 Effective S=1/2 Kondo effect Glazman et al. PRB 64, 045328 (2001) Hofstetter and ZarandPRB 69, 235301 (2002)

22. J=-0.005, Tk=0.0025 B in Step of 0.001 Singlet-triplet crossover at finite field and temperature J=0.007, Jc=0.005, Tk=0.0025, T=0.00001, in step of 400 B NRG: P-h symmetry EXP: P-h asymmetry Ferromagnetic J<0 Antiferromagnetic J>0 J close to Jc, smooth crossover splitting of Kondo peak due to Zeemann splitting of up and down spins J >> Jc, sharper crossover splitting is linearly proportional to B

23. Model System (II) 1 2 V J even Vojta, Bulla, and Hofstetter, PRB 65, 140405, (2002) Cornaglia and Grempel, PRB 71, 075305, (2005) • Two coupled quantum dots, only dot 1 couples to single-channel leads • Antiferrimagnetic exchange interaction J • 1-channel Kondo physics, dot 2 is Kondo screened for any J > 0. • Kosterlitz-Thouless transition, Jc = 0

24. Anderson's poor man scaling and Tk HAnderson • Reducing bandwidth by integrating out high energy modes Anderson 1964 J J J J • Obtaining equivalent model with effective couplings • Scaling equation w < Tk, J diverges, Kondo screening J 0

25. Jk 4V2/U J: AF coupling btw dot 1 and 2 Tk D rc 1/G 2 stage Kondo effect 1st stage Kondo screening Jk: Kondo coupling 2nd stage Kondo screening dip in DOS of dot 1

26. Log (T*) 1/J J 0 8 Kondo spin-singlet NRG:Spectral density of Model (II) U=1 ed=-0.5 G=0.1 Tk=0.006 L=2 Kosterlitz-Thouless quantum transition No 3rd unstable fixed point corresponding to the critical point Crossover energy scale T* exponentially depends on |J-Jc|

27. Dip in DOS of dot 1: Perturbation theory 1 2 when w Dip in DOS of dot 1 J = 0 d1 wn< Tk J > 0 but weak self-energy vertex sum over leading logarithmic corrections

28. Dip in DOS: perturbation theory U=1, ed=-0.5, G= 0.1, L=2,J=0.0005, Tk=0.006, T*=8.2x10-10 • Excellence agreement between Perturbation theory (PT) and NRG for T* << w << Tk • PT breaks down for w T* • Deviation at larger w > O(Tk)due to interaction U

29. 1 1 2 2 More general model of 1-channel 2-stage Kondo effect Vojta, Bulla, and Hofstetter, PRB 65, 140405, (2002) Jk1 Two-impurity, S=1, underscreened Kondo I Jk2 Jk1 J ( Jk2 = 0 ) I < Ic: Tcimp = 1/4 residual spin-1/2 I > Ic: Tcimp = 0 spin-singlet Ic ~ Jk1 Jk2 D

30. Optical conductivity J 1 U=1 ed=-0.5 G=0.1 Tk=0.006 L=2 Dot 2 • Sindel, Hofstetter, von Delft, Kindermann, PRL 94, 196602 (2005) ‘ ‘ Linear AC conductivity

31. L1 R1 1 1 Jk1 R2 even 2 (L2+R2) L2 even 1 (L1+R1) J Jk2 2 2 Jk J n J x J-Jc T* 8 J 8 0 Jc 0 Kondo spin-singlet Kondo spin-singlet Comparison between two models Model (I) Model (II) 2 impurity, S=1, Two-channel Kondo 2 impurity, S=1, One-channel Kondo complete Kondo screening quantum critical point underscreened Kondo K-T transition

32. L1 R1 R2 L2 Conclusions • Coupled quantum dots in Kondo regime exhibit quantum phase transition Model system (I): 2-channel Kondo physics Quantum phase transition between Kondo and spin-singlet phases Singlet-triplet crossover at finite field and temperatures, qualitatively agree with experiments Model system (II): 1-channel Kondo physics, two-stage Kondo effect Kosterlitz-Thouless quantum transition, Provide analytical and numerical understanding of the transition • Our results have applications in spintronics and quantum information

33. V T g g c Outlook Quantum critical and crossover in transport properties near QCP Non-equilibrium transport in various coupled quantum dots Quantum phase transition out of equilibrium Quantum phase transition in quantum dots with dissipation