Quantum criticality in a double-quantum-dot system

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Quantum criticality in a double-quantum-dot system
G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97,
166802 (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)
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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

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Kondo effect in quantum dot
edU
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)
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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
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Kondo effect in quantum dot
(J. von Delft)
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Kondo effect in quantum dot
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Kondo effect in quantum dot
Anderson Model

• New energy scale Tk Dexp(-pU/ G)
• For T lt 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
G 2pV 2?d
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Kondo Resonance of a single quantum dot
Spectral density at T0
Universal scaling of T/Tk
L. Kouwenhoven et al. science 2000
M. Sindel
particle-hole symmetry
phase shift
Fredel sum rule
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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.

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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

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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
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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

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Transport properties

• Current through the quantum dots

• Transmission coefficient

• Linear conductance

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NRG Flow of the lowest energy
Phase shift d
d
Kondo
KltKC
JC
Kondo
p/2
KgtKC
Spin-singlet
Spin-singlet
0
K
Kc
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
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Quantum phase transition of a double-quantum-dot
system
C.H. C and W. Hofstetter, cond-mat/0607772
JRKKYK
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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
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2-impurity Kondo problem

• Particle-hole symmetry V0

H ? H H under
Quantum phase transition as K is tuned
Kc 2.2 Tk
Affleck et al. PRB 52, 9528 (1995)
Jones and Varma, PRL 58, 843 (1989)
Jump of phase shift at Kc K lt Kc, d
p/2 K gtKC , d 0
Jones and Varma, PRB 40, 324 (1989)
Sakai et al. J. Phys. Soc. Japan 61, 7, 2333
(1992) ibdb. 61, 7, 2348 (1992)
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2-impurity Kondo problem
Zhu and Varma, cond-mat/0607426
Sharp phase transition
Smooth crossover
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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 KKc
Splitting between even and odd resonances
even
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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 t0
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Quantum criticality in a double-quantum dot
system
No direct hoping, t 0
Asymmetric limit
T1Tk1, T2 Tk2
2 channel Kondo System
QCP occurs when
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
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Optical conductivity
Sindel, Hofstetter, von Delft, Kindermann, PRL
94, 196602 (2005)
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Linear AC conductivity
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Transport of double-quantum-dot near QCP
NRG on DQDs without P-H and parity symmetry

At KKc
Affleck and Ludwig PRB 48 7279 (1993)
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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
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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
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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
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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