Quantum criticality in a double-quantum-dot system

Chung-Hou Chung Electrophysics Dept.

National Chiao-Tung University

Hsin-Chu, Taiwan

Collaborators:

Gergely Zarand (Budapest),

Matthias Vojta (TKM, Karlsruhe)

Pascal Simon (CNRS, Grenoble)

G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)

• 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

Outline

Kondo effect in quantum dot

even

odd

Coulomb blockade

Single quantum dot

conductance anomalies

Goldhaber-Gorden et al. nature 391 156 (1998)

Glazman et al. Physics world 2001

L.Kouwenhoven et al. science 289, 2105 (2000)

d+U

d Kondo effect

Vg

VSD

Kondo effect in metals with magnetic impurities

At low T, spin-flip scattering off impurities enhances

Ground state is spin-singlet

Resistance increases as T is lowered

electron-impurity scattering

via spin exchange coupling

logT

(Kondo, 1964)

(Glazman et al. Physics world 2001)

Kondo effect in quantum dot

(J. von Delft)

Kondo effect in quantum dot

Kondo effect in quantum dot

Anderson Model

local energy level :

charging energy :

level width :

All tunable!

Γ= 2πV 2ρd

U

d ∝ Vg

New energy scale: Tk ≈ Dexp-U )

For T < Tk :

Impurity spin is screened (Kondo screening)

Spin-singlet ground state

Local density of states developes Kondo resonance

Spectral density at T=0

Kondo Resonance of a single quantum dot

phase shift

Fredel sum rule

particle-hole symmetry

Universal scaling of T/Tk

L. Kouwenhoven et al. science 2000M. Sindel

P-H symmetry

/2

Recent experiments on coupled quantum dots

• 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.

(I). C.M. Macrus et al.

Science, 304, 565 (2004)

Quantum phase transition and non-Fermi liquid state in Coupled quantum dots

L1

L2 R2

R1

C.H. C and W. Hofstetter, cond-mat/0607772

G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)

Non-fermi liquid

KcK

T

Spin-singletKondo

• 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

Coupled Quantum dots

• 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

L1

L2

R1

R2

Izumida and Sakai PRL 87, 216803 (2001)

Vavilov and Glazman PRL 94, 086805 (2005)

Simon et al. cond-mat/0404540

triplet states

Hofstetter and Schoeller, PRL 88, 061803 (2002) singlet state

K

K

Numerical Renormalization Group (NRG)

Non-perturbative numerical method by Wilson to treat quantum impurity problem

Anderson impurity model is mapped onto a linear chain of fermions

Logarithmic discretization of the conduction band

Iteratively diagonalize the chain and keep low energy levels

K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)

W. Hofstetter, Advances in solid state physics 41, 27 (2001)

Transport properties

• Transmission coefficient:

• Current through the quantum dots:

• Linear conductance:

JC

NRG Flow of the lowest energy Phase shift

0

KKc

K<KC

K>KC

Two stable fixed points (Kondo and spin-singlet phases )

One unstable fixed point (critical fixed point) Kc, controlling the quantum phase transition

Jump of phase shift in both channels at Kc

Kondo

Spin-singlet

Kondo

Spin-singlet

Crossover energy scale T* k-kc

• 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.

Quantum phase transition of a double-quantum-dot system

J=RKKY=K

C.H. C and W. Hofstetter, cond-mat/0607772

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)

R/2-R/2

X

H0 =

Himp

Heavy fermions

2-impurity Kondo problem

Kc = 2.2 Tk

Non-fermi liquid

KcK

T

Spin-singletKondo1 2

Affleck et al. PRB 52, 9528 (1995)

Jones and Varma, PRL 58, 843 (1989)Jump of phase shift at Kc K < Kc, = /2 ; K >KC ,

Quantum phase transition as K is tuned

Jones and Varma, PRB 40, 324 (1989)

Sakai et al. J. Phys. Soc. Japan 61, 7, 2333 (1992); ibdb. 61, 7, 2348 (1992)

• Particle-hole symmetry V=0

H H’ = H under

even

odd

2-impurity Kondo problem

Smooth crossover

• Particle-hole asymmetry

even

odd

Zhu and Varma, cond-mat/0607426

Sharp phase transition

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 K=Kc

Splitting between even and odd resonances

even

odd

Quantum criticality in a double-quantum –dot system

V1 ,V2 break P-H sym and parity sym. QCP still survives as long as no direct hoping t=0

Non-fermi liquid

KcK

T

Spin-singletKondo

G. Zarand, C.H. C, P. Simon, M. Vojta, PRL, 97, 166802 (2006)

even 1 (L1+R1) even 2 (L2+R2)K

_

Quantum criticality in a double-quantum –dot system

K

_

No direct hoping, t = 0 Asymmetric limit: T1=Tk1, T2= Tk2

2 channel Kondo System

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

Goldhaber-Gordon et. al. PRL 90 136602 (2003)

QCP occurs when

Optical conductivity

Linear AC conductivity

Sindel, Hofstetter, von Delft, Kindermann, PRL 94, 196602 (2005)

1

Transport of double-quantum-dot near QCP

At K=Kc

Affleck and Ludwig PRB 48 7279 (1993)

NRG on DQDs without P-H and parity symmetry

The only relevant operator at QCP: direct hoping term t

charge transfer between two channels of the leads

dim[

(wr.t.QCP)

Relevant operator

Generate smooth crossover at energy scale

RG

most dangerous operators: off-diagonal J12

At scale Tk, typical quantum dot

may spoil the observation of QCP

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

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

Conclusions

• Coupled quantum dots in Kondo regime exhibit quantum phase transition

• The QCP is robust against particle-hole and parity asymmetries

•The QCP is destroyed by charge transfer between two channels

• The QCP of DQDs is identical to that of a 2-channel Kondo system

• The effect of charge transfer can be reduced by inserting additional even number of dots, making it possible to be observe QCP in experiments