non-equilibrium transport of a quantum dot in the kondo regime near quantum phase transitions...
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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
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