1. Quantum transport at nano-scale National Chiao-Tung University
Zarand, Chung, Simon, Vojta, PRL (2006)
Chung, Hofstetter, PRB (2007), selected by Virtual Journal of Nanoscience and Technology Aug
Chung, Zarand, Woelfle, PRB 77, (2008), selected by Virtual Journal of Nanoscience and Technology Jan
Chung, Glossop, Fritz, Kircan, Ingersent,Vojta, PRB 76, (2007)
Chung, Le Hur, Vojta, Woelfle nonequilibrium transport near the quantum phase transition (arXiv: )
Chung-Hou Chung 仲崇厚
Electrophysics Dept.
National Chiao-Tung University
Hsin-Chu, Taiwan
Collaborators:
Matthias Vojta (Koeln), Gergely Zarand (Budapest), Walter Hofstetter (Frankfurt U.)
Pascal Simon (CNRS, Grenoble), Lars Fritz (Harvard),
Marijana Kircan (Max Planck, Stuttgart),
Matthew Glossop (Rice U.) , Kevin Ingersent (U. Florida)
Peter Woelfle (Karlsruhe), Karyn Le Hur (Yale U.)

2. Outline
Introduction
Quantum criticality in a double-quantum-dot system
Quantum phase transition in a dissipative quantum dot
Nonequilibrium transport in a noisy quantum dot
Conclusions

3. Quantum dot---A single-Electron-Transistor (SET)
Coulomb blockade
ed+U ed Vg
VSD
Single quantum dot
Goldhaber-Gorden et al. nature (1998)
Coulomb Blockade

4. Quantum dot---charge quantization
Goldhaber-Gorden et al. nature (1998)

5. Kondo effect in quantum dot
Coulomb blockade
ed+U ed Vg
VSD
Kondo effect
even
odd
conductance anomalies
Glazman et al. Physics world 2001
L.Kouwenhoven et al. science 289, 2105 (2000)

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

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

8. Kondo effect in quantum dot

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

10. 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
= p/2
phase shift
Fredel sum rule

11. 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

12. Perturbative Renormalization Group (RG) approach: Anderson's poor man scaling and Tk
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

13. 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 c T g
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

14. I.
Quantum phase transition in coupled double-quantum-dot system

15. 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.

16. Von der Zant et al. (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

17. Quantum phase transition in coupled double-quantum-dot system
G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, (2006)
C.H. C and W. Hofstetter, PRB (2007) R1 L1
Non-fermi liquid Kc K T
Spin-singlet
Kondo 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

18. 2-impurity Kondo problem
Affleck et al. PRB 52, 9528 (1995)
Jones and Varma, PRL 58, 843 (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 =

19. Quantum criticality of 2-impurity Kondo problem
Particle-hole symmetry V=0
H  H’ = H under
Non-fermi liquid Kc K T
Spin-singlet
Kondo 1 2
even
odd
Affleck et al. PRB 52, 9528 (1995)
Kc = 2.2 Tk
Jump of phase shift at Kc K < Kc, d = p/2 ; K >KC , d = 0
Quantum phase transition as K is tuned
Jones and Varma, PRL 58, 843 (1989)
Jones and Varma, PRB 40, 324 (1989)
Particle-hole asymmetry
Kc is smeared out, crossover
Misleading common belief ! We have corrected it!

20. Quantum Phase Transition in Double Quantum dots: P-H Symmetry
triplet states L1 R1
Izumida and Sakai PRL 87, (2001)
Vavilov and Glazman PRL 94, (2005) K
Simon et al. cond-mat/
Hofstetter and Schoeller, PRL 88, (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

21. Transport properties Current through the quantum dots:
Transmission coefficient:
Linear conductance:

22. NRG Flow of the lowest energy
Phase shift d d
Kondo
K<KC JC
Kondo
p/2
K>KC
Spin-singlet
Spin-singlet Kc K
Two stable fixed points (Kondo and spin-singlet phases )
Jump of phase shift in both channels at Kc
Crossover energy scale T*
k-kc n
One unstable fixed point (critical fixed point) Kc, controlling the quantum phase transition

23. Quantum phase transition of a double-quantum-dot system
J=RKKY=K
Chung, Hofstetter, PRB (2007)
J < Jc, transport properties reach unitary limit:
T( = 0) 2, G(T = 0) G0 where G0 = 2e2/h.
J > Jc spins of two dots form singlet ground state,
T( = 0) 0, G(T = 0) ; and Kondo peak splits up.
Quantum phase transition between Kondo (small J) and spin singlet (large J) phase.

24. Restoring of Kondo resonance in coupled quantum dots
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.

25. Singlet-triplet crossover at finite magnetic fields
J=0.007, Jc=0.005, Tk=0.0025, T= ,
in step of 400 B
J=-0.005, Tk=0.0025
B in Step of 0.001
NRG: P-h symmetry
EXP: P-h asymmetry
Ferromagnetic J<0
Antiferromagnetic J>0
J close to Jc, smooth crossover
J >> Jc, sharper crossover

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

27. Quantum criticality in a double-quantum –dot system
No direct hoping, t = 0
Asymmetric limit:
T1=Tk1, T2= Tk2 _ K G 1 2
QCP occurs when
2 channel Kondo System
Goldhaber-Gordon et. al. PRL (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

28. Transport of double-quantum-dot near QCP (only K, no t term)
NRG on DQDs without t, P-H and parity symmetry
At K=Kc
Affleck and Ludwig PRB (1993)

29. The only relevant operator at QCP: direct hoping term t
charge transfer between two channels of the leads
Relevant operator
Generate smooth crossover at energy scale
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

30. 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

31. 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
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

32. Side-coupled double quantum dots1 2 V J
even
Chung, Zarand, Woelfle, PRB 77, (2008),
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

33. 2 stage Kondo effect 1st stage Kondo screening
Jk: Kondo coupling Jk
4V2/U
2nd stage Kondo screening
J: AF coupling btw dot 1 and 2 D Tk
dip in DOS of dot 1 rc
1/G

34. NRG:Spectral density of Model (II)
Log (T*)
1/J
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 8 J
Kondo
spin-singlet
Crossover energy scale T* exponentially depends on |J-Jc|

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

36. 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

37. Summary I
Coupled quantum dots in Kondo regime exhibit quantum phase transition L2 L1 R1 R2
quantum critical point T*
J-Jc n x Jc J
Kondo
spin-singlet 8
The QCP of DQDs is identical to that of a 2-channel Kondo system
correct common misleading belief: 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
K-T transition 8 J
Kondo
spin-singlet

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

39. Quantum dot as charge qubit--quantum two-level system
Coulomb blockade
ed+U ed Vg
VSD
charge qubit-

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

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

42. Spin Boson model

43. Delocalized-Localized transition
h ~ N -1/2
K. Le Hur et al, PRL 2004, /

44. Charge Kondo effect in a quantum dot with Ohmic dissipation
Hdissipative dot
non-interacting lead
N=1/2
Q=0 and Q=1degenerate
Anisotropic Kondo model
de-localized
Jz = -1/2 R
localized
g=J
Kosterlitz-Thouless transition

45. Generalized dissipative boson bath (sub-ohmic noise){
Sub-Ohmic

46. Generalized fermionic leads: Power-law DOS
Anderson model
Quantum phase transition in the pseudogap Anderson/Kondo model
d-wave superconductors and graphene: r =1
Fradkin et al. PRL 1990
Local moment (LM) J X
Kondo Jc

47. Delocalized-Localized transition in Pseudogap Fermi-Bose Anderson model
C.H.Chung et al., PRB 76, (2007)
Pseudogap Fermionic bath
Sub-ohmic bosonic bath

48. Phase diagram
Field-theoretical RG

49. Critical properties via perturbative RG
exact

50. Critical properties via NRG

51. Critical properties via NRG
Spectral function

52. Spectral function

53. Summary II
Delocalized-localized quantum phase transition exists in the new paradimic
pseudogap Bose-Fermi Anderson (BFA) model: relevant for describing
a noisy quantum dot
Kosterlitz-Thouless quantum transition between localized and delocalized phases in a noisy quantum dot with Ohmic dissipation
For metallic leads, our model maps onto Spin-boson model
Excellent agreement between perturbative RG and numerical RG on the critical properties of the BFA model

54. III. Nonequilibrium transport near quantum phase transition

55. Nonequilibrium transport in Kondo dot
Steady state nonequilibrium current at finite bias V generates decoherence
spin-flip scattering at finite V, similar to the effect of temperatures
Decoherence (spin-relaxation rate)
cuts-off logrithmic divergence of Kondo couplings
suppresses coherence Kondo conductance
Energy dependent Kondo couplings g in RG
Keldysh formulism for nonequilibrium transport

56. Nonequilibrium transport near quantum phase transition
in a dissipative quantum dot
dissipative quantum dot
Effective Kondo model
: Dissipation strength

57. t m1 m2 New idea! New! 2-lead anisotropic Kondo model
2-lead setup
Bias voltage V
Nonequilibrium transport
New idea!
Dissipative spinless 2-lead model
New!
New mapping:
valid for small t, finite V, at KT transition and localized phase
2-lead anisotropic Kondo model

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

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

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

61. Paaske, Rosch,Woelfle, et al, Nature physics, 2, 460 (2006)

62. Delocalized (Kondo) phase
P. Woelfle et. al. 2003
At KT transition?
In localized phase?

64. Scaling of nonequilibrium conductance G(V,T=0)
New!
Black--Equilibrium
Color--Nonequilibrium
(Equilibrium V=0)
(Non-Equilibrium V>0)
G noneq =dI noneq / dV
At KT transition
Localized phase:

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

66. Nonequilibrium Conductance at critical point
log eq
At KT transition:
Small V, nonequilibrium scaling G(V, T=0) ~ G(V=0,T) equilibrium scaling
Large V, G(V) gets a logrithmic correction
V and T play the “similar” role but with a correction
New!

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

68. Nonequilibrium transport at localized-delocalized transition
Chung, Le Hur, Woelfle, Vojta (unpublished, work-in progress)
Important fundamental issues of nonequilibrium quantum criticality
What is the role of V at the transition compared to that of temperature T ?
Will V smear out the transition the same way as T?
What is the scaling behavior of G(V, T) at the transition ?
Is there a V/T scaling in G(V,T) at transition?

69. Nonequilibrium RG scaling equations of effective Kondo model

70. New! V>>T Nonequilibrium scaling V<<T Equilibrium scaling
V and T play the similar role but with a logrithmic correction

71. Outlook
Quantum critical and crossover in transport properties near QCP c T g
Non-equilibrium transport in various coupled quantum dots V
Quantum phase transition out of equilibrium
Kondo effect in carbon nanotubes

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

73. Spin-boson model: NRG results
N.-H. Tong et al, PRL 2003