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

• 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

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

Vg

VSDSingle quantum dotGoldhaber-Gorden et al. nature 391 156 (1998)

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)

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

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

HAnderson

•Reducing bandwidth by integrating out high energy modes

•Obtaining equivalent model with effective couplings

•Scaling equation

< Tk, J diverges, Kondo screening

J J

J J

J

Anderson 1964

Quantum phase transitions

c

T

gg

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

II.

Quantum phase transition in a dissipative quantum dot

Coulomb blockade d+U

d

Vg

VSD

Quantum dot as charge qubit--quantum two-level system

charge qubit-

Quantum dot as artificial spin S=1/2 system

Quantum 2-level system

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 + HHO

N=1/2Q=0 and Q=1 degenerate

H_noisy-dot (bosonization + unitary transformation) Spin Boson model

K. Le Hur et al, PRL 2004,

Delocalized-Localized transition

h ~ N -1/2

/

delocalized localized

~ R

Charge Kondo effect in a quantum dot with Ohmic dissipation

Kosterlitz-Thouless transition

localized

de-localized

g=J

HOhmic spin-boson

Anisotropic Kondo modelK. Le Hur 05, Matveev 02

Unitary transformation refermionization

Equilibrium quantum transition in a dissipative quantum dot

T

Zarand et al, 05’

de-localized

localized

Dissipation strength

Tk

V

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 currentSpin Decoherence rate

K. Le Hur et al.

Zarand et al

New mapping: 2-lead anisotropic Kondo

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

t tNew idea!

2-lead setupBias voltage VNonequilibrium transport

Effective leads: R,L

Original leads: 1,2

f: pseudofermion

Conduction electron spin:

Impurity (quantum dot) spin:

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

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

Nonequilibrium decoherence rate of a Kondo dot

Pseudofermion self-energy

Nonequilibrium perturbative functional RG approach to Kondo model

G=dI / dV

Noneq RG scaling equations for Kondo couplings

Nonequilibrium currentNonequilibrium differential conductance

RG flows cut-off by Decoherence not by V

Nonequilibrium Decoherence rate: Highly nonlinear function in V !

Equilibrium decoherence rate: linear in T

Effective Kondo coupling I-V

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

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

V<<TEquilibrium scaling

V>>T Nonequilibrium profile

New!

V/T scaling in conductance G(V,T) at KT transition

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

III.

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

L

R

JLL

LL LL

JRR

JLR

HK = JLL SLL Sd + JRR SRR Sd + JLR SLR Sd + JRL SRL Sd

S = d

K <=1

Luttinger parameter

movers

Kondo dot coupled to Luttinger leads

Non-interacting limit: K=1

JLL

1-channel Kondo

Strongly interacting limit: K<1/2

d JLL(RR)/d ln D = - JLL(RR) - JLR2 2

d JLR/d ln D = - JLL JLR – JLR JRR

JLR, JRR(LL)

Tunneling DOS: ~1/K -1

Strongly suppresses JLR

d JLL(RR)/d ln D = - JLL(RR) - JLR2 2

d JLL(RR)/d ln D = - ½(1-1/K) JLR - JLL(RR) - JLR2 2

JLL

LL LL

JRR

2-channel Kondo

JRR(LL)

JLR

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

Equilibrium Linear Conductance G(T) = dI/dV|V->0 ~ JLR(T)2

1-channel Kondo, conducting 2-channel Kondo, insulating

0< K<1/21/2< K<1

[JLR]=(1+K)/2weak-coupling

strong-coupling

[JLR]=1/(2K)

E. Kim, cond-mat/0106575

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 currentSpin Decoherence rate

K. Le Hur et al.

Zarand et al

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

0.2 0.4 0.6 0.8 1

0.02

0.04

0.06

0.08

0.1

RG flows cut-off by Decoherence Vnot by V

V

D/D0

gLR

Dip-peak structures of frequency-dependent Kondo couplings

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 = gLR(T)~ 2

D0>>T>> Tk

Analytical approximated forms for G(V) at large bias

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

2-Channel Kondo physics in quantum dot coupled to 2D topological insulators

Hassn, Kane, arXiv:1002.3895

Helical edge states in 2D topological insulatorSpinful, nonchiral Luttinger liquid

g2 gterm: forward scattering,

Breaks SU(2) sym. of Kondo couplings under RG

Anisotropic 2-channel Kondo model

TK Ng et al. PRB (R) 2010

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-couplingstrong-coupling

TK Ng et al. PRB (R) 2010

2CK FP stable for K<1

Equilibrium and nonequilibrium differential conductance G(V T), G(V)

Kondo dot in 2D Topological Insulator

gLL/RR/LR0 = 0.001

J1 J1

J2

02CK fixed point Stablized 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

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 dot2. 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)

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