1. Chung-Hou Chung Collaborators:
Quantum criticality in a double-quantum-dot system
G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, (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)

2. Outline
Introduction
Quantum criticality in a double-quantum-dot system:
particle-hole symmetry
Quantum criticality in a 2-impurity Kondo 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 (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
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
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

8. 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

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. Quantum phase transition and non-Fermi liquid state in Coupled quantum dots
G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, (2006)
C.H. C and W. Hofstetter, cond-mat/ 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

11. Coupled Quantum dots
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

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:
Linear conductance:

14. 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

15. Quantum phase transition of a double-quantum-dot system
C.H. C and W. Hofstetter, cond-mat/
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.
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. 2-impurity Kondo problem
Particle-hole symmetry V=0
H  H’ = H under
Non-fermi liquid Kc K T
Spin-singlet
Kondo
even
odd 1 2
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. 2-impurity Kondo problem
Particle-hole asymmetry
Zhu and Varma, cond-mat/
even
odd
Sharp phase transition
Smooth crossover

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

20. Quantum criticality in a double-quantum –dot system
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

21. 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

22. Optical conductivity Linear AC conductivity
Sindel, Hofstetter, von Delft, Kindermann, PRL 94, (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 and Ludwig PRB (1993)

24. 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

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 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

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