1. Non-equilibrium transport of a quantum dot in the Kondo regime near quantum phase transitions Chung-Hou Chung 仲崇厚 Electrophysics Dept. National Chiao-Tung University Hsin-Chu, Taiwan Collaborators: Karyn Le Hur (Yale), Matthias Vojta (Koeln), Peter Woelfle (Karlsruhe), T.K. Ng (HKUST) *Chung, Le Hur, Woelfle, Vojta nonequilibrium transport near dissipative quantum phase transition, PRL 102, 216803 (2009)‏ *Chung, Le Hur, Woelfle, Vojta, Tunable Kondo-Luttinger system far from equilibrium, PRB 82, 115325 (2010)‏ *Chung, Latha, PRB, 82, 085120 (2010)‏ NTNU, Dec. 9, 2010

2. Introduction: Kondo effect in quantum dot Nonequilibrium transport of a dissipative quantum dot Nonequilibrium transport of a Kondo dot in Luttinger liquid: the 2-channel Kondo fixed point Kondo dot in 2D topological insulators Conclusions Outline

3. Kondo effect Kondo effect in quantum dot even odd conductance anomalies L.Kouwenhoven et al. science 289, 2105 (2000)‏ Glazman et al. Physics world 2001 Coulomb blockade  d +U dd VgVg V SD Single quantum dot Goldhaber-Gorden et al. nature 391 156 (1998)‏

4. Kondo effect in metals with magnetic impurities For T<Tk (Kondo Temperature), spin-flip scattering off impurities enhances Ground state is Resistance increases as T is lowered electron-impurity spin-flip scattering logT (Kondo, 1964)‏ (Glazman et al. Physics world 2001)‏

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

6. Kondo effect in quantum dot

7. Anderson Model local energy level : charging energy : level width : All tunable! Γ= 2πV 2 ρ d U  d ∝ V g New energy scale: T k ≈ Dexp  -  U   )‏ For T < T k : Impurity spin is screened (Kondo screening)‏ Spin-singlet ground state Local density of states developes Kondo resonance

8. Spectral density at T=0 Kondo Resonance of a single quantum dot phase shift Fredel sum rule particle-hole symmetry Universal scaling of T/T k L. Kouwenhoven et al. science 2000M. Sindel P-H symmetry  /2

9. Perturbative Renormalization Group (RG) approach: Anderson's poor man scaling and Tk H Anderson Reducing bandwidth by integrating out high energy modes Obtaining equivalent model with effective couplings Scaling equation  < Tk, J diverges, Kondo screening JJ J J  J Anderson 1964

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

11. II. Quantum phase transition in a dissipative quantum dot

12. Coulomb blockade  d +U dd VgVg V SD Quantum dot as charge qubit--quantum two-level system charge qubit-

13. Quantum dot as artificial spin S=1/2 system Quantum 2-level system

14. Dissipation driven quantum phase transition in a noisy quantum dot Noise ~ SHO of LC transmission line Noise = charge fluctuation of gate voltage Vg Caldeira-Leggett Model K. Le Hur et al, PRL 2004, 2005, PRB (2005), Impedence H = Hc + Ht + H HO N=1/2 Q=0 and Q=1 degenerate

15. H_ noisy-dot (bosonization + unitary transformation)  Spin Boson model

16. K. Le Hur et al, PRL 2004, Delocalized-Localized transition h ~ N -1/2 / delocalized localized ~ R

17. Charge Kondo effect in a quantum dot with Ohmic dissipation Kosterlitz-Thouless transition localized de-localized g=J H Ohmic spin-boson Anisotropic Kondo model K. Le Hur 05, Matveev 02 Unitary transformation refermionization

18. Equilibrium quantum transition in a dissipative quantum dot T Zarand et al, 05’ de-localized localized Dissipation strength TkTk V

19. Fresh Thoughts: nonequilibrium transport at transition What is the role of V at the transition compared to that of temperature T ? What is the scaling behavior of G(V, T) at the transition ? Important fundamental issues on nonequilibrium quantum criticality Will V smear out the transition the same way as T? Not exactly! Is there a V/T scaling in G(V,T) at transition? Yes! t t Steady-state current Spin Decoherence rate  K. Le Hur et al. Zarand et al New mapping: 2-lead anisotropic Kondo

20. Dissipative spinless 2-lead model New mapping: bosonization + unitary transformations + refermionization valid for small t, finite V, at KT transition and localized phase 2-lead anisotropic Kondo model  1 22 tt New idea! 2-lead setup Bias voltage V Nonequilibrium transport Effective leads: R,L Original leads: 1,2

21. f: pseudofermion Conduction electron spin: Impurity (quantum dot) spin:

22. Decoherence effect in Kondo dot Logarithmic divergence: signature of Kondo effect 1. Temperature broadening T 3 ways to cutoff the logarithmic divergence: 2. Magnetic field B 3. Finite bias voltage V Decoherence: spin-flips due to external energy (T or V), suppress the coherence AF Kondo resonance

23. Nonequilibrium perturbative functional RG approach to anisotropic Kondo model Decoherence  (spin-relaxation rate) from V Energy dependent Kondo couplings g in RG Keldysh nonequilibrium formulism Anderson’s poor man’s scaling P. Woelfle et. al. Ge Gf

24. Nonequilibrium decoherence rate of a Kondo dot Pseudofermion self-energy

25. Nonequilibrium perturbative functional RG approach to Kondo model G=dI / dV Noneq RG scaling equations for Kondo couplings Nonequilibrium current Nonequilibrium differential conductance

26. RG flows cut-off by Decoherence  not by V

27. Nonequilibrium Decoherence rate: Highly nonlinear function in V ! Equilibrium decoherence rate: linear in T

28. Effective Kondo coupling I-V

29. Nonequilibrium Conductance at KT transition Large V, G(V) shows different profile V and T play a very different role at the transition at large V Small V, nonequilibrium scaling G(V, T=0) ~ G(V=0,T) equilibrium scaling New! eq log

30. Scaling of nonequilibrium conductance G(V,T=0) in localized phase near KT transition (Equilibrium V->0)‏ New! (Non-Equilibrium V large)‏ Black--Equilibrium Color-- Nonequilibrium

31. V<<T Equilibrium scaling V>>T Nonequilibrium profile New! V/T scaling in conductance G(V,T) at KT transition

32. Charge Decoherence rate  Spinful Kondo model: Spin relaxzation rate due to spin flips Spin Decoherence rate  Dissipative quantum dot: charge flip rate between Q=0 and Q=1 Nonequilibrium :Decoherence rate  cuts off the RG flow Nonlinear function in V ! Equilibrium :Temperature cuts off the RG flow Conclusions At KT transition

33. III. Quantum phase transition of a quantum dot coupled to interacting Luttinger liquid leads

34.  L RR J LL LL J RR J LR H K = J LL S LL S d + J RR S RR S d + J LR S LR S d + J RL S RL S d S = d K <=1 Luttinger parameter movers Kondo dot coupled to Luttinger leads

35. Non-interacting limit: K=1 J LL 1-channel Kondo Strongly interacting limit: K<1/2 d J LL(RR) /d ln D = - J LL(RR) - J LR 22 d J LR /d ln D = - J LL J LR – J LR J RR  J LR, J RR(LL) Tunneling DOS:~ 1/K -1 Strongly suppresses J LR d J LL(RR) /d ln D = - J LL(RR) - J LR 22 d J LL(RR) /d ln D = - ½(1-1/K) J LR - J LL(RR) - J LR 22 J LL LL J RR 2-channel Kondo  J RR(LL)  J LR

36. At the 2CK fixed point, Conductance g(V ds ) 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 The single quantum dot can get Kondo screened via 2 different channels: At low temperatures, blue channel finite conductance; red channel zero conductance

37. Equilibrium Linear Conductance G(T) = dI/dV| V - >0 ~ J LR (T)‏ 2 1-channel Kondo, conducting 2-channel Kondo, insulating 0< K<1/21/2< K<1 [J LR ]=(1+K)/2 weak-coupling strong-coupling [J LR ]=1/(2K)‏ E. Kim, cond-mat/0106575

38. Quantum phase transition out of equilibrium What is the role of V at the transition compared to that of temperature T ? What is the scaling behavior of G(V, T) at the transition ? Important fundamental issues on nonequilibrium quantum criticality Will V smear out the transition the same way as T? Not exactly! Is there a V/T scaling in G(V,T) at transition? Yes! t t Steady-state current Spin Decoherence rate  K. Le Hur et al. Zarand et al

39. Nonequilibrium perturbative RG approach to Kondo model Decoherence  (spin-relaxation rate) from V Energy dependent Kondo couplings g in RG P. Woelfle et. al. G=dI / dV

40. RG flows cut-off by Decoherence  V  not by V V D/D 0  g LR

41. Dip-peak structures of frequency-dependent Kondo couplings

42. Nonequilibrium Conductance of Kondo dot coupled to Luttinger leads Large V, G(V) shows different profile Small V, nonequilibrium scaling G(V, T=0) ~ G(V=0,T) equilibrium scaling New! G(T) eq = g LR (T)~ 2 D 0 >>T>> Tk

43. Analytical approximated forms for G(V) at large bias

44. Non-universal crossover for G(V) at V>> Tk

45. 2-Channel Kondo physics in quantum dot coupled to 2D topological insulators Hassn, Kane, arXiv:1002.3895 Helical edge states in 2D topological insulator Spinful, nonchiral Luttinger liquid

46. g 2 gterm: forward scattering, Breaks SU(2) sym. of Kondo couplings under RG Anisotropic 2-channel Kondo model

47. TK Ng et al. PRB (R) 2010

48. Near weak-coupling fixed point J1, J2 -->0: Scaling dimensions: [ ]=1,[ ]=K, most relevant term [ ]= [ ] = 1/2(K+1/K) >1 Relevant: Irrelevant: Near strong-coupling fixed point: cuts Luttinger wire into 2 parts at x=0 [ ] =1/K, irrelevant for K<1 weak-coupling strong-coupling TK Ng et al. PRB (R) 2010 2CK FP stable for K<1

49. Equilibrium and nonequilibrium differential conductance G(V  T), G(V)‏ Kondo dot in 2D Topological Insulator g LL/RR/LR0 = 0.001

50. J1J1 J1J1 J2J2 0 2CK fixed pointStablized for K<1 No spin gap, finite spin current Insulator, charge gap S.C. Zhang et al PRL 2006 Kondo screening cloud=> spin current vortex

51. V and T play different role in transport— Equilibrium :Temperature cuts off the RG flow Nonequilibrium :Decoherence rate G cuts off the RG flow G(V,T=0) different from G(T, V=0) at large bias voltages Conclusions Interactions in Luttinger liquid leads-- 1.suppress charge transport through quantum dot 2.Favor insulating 2-channel Kondo fixed point Spinful non-chiral Luttinger liquid leads-- 2CK is stable for K<1/2 Helical edge state in 2D topological insulators– 2CK is more stable, K<1 2CK in 2D Topological Insulators-- --charge gap (insulator)‏ -- no spin gap (finite spin current)‏

52. Single Kondo dot in nonequilibrium, large bias V and magnetic field B Paaske Woelfle et al, J. Phys. Soc.,Japan (2005)‏ Paaske, Rosch, Woelfle et al, PRL (2003)‏ Exp: Metallic point contact