Download Quantum impurity systems out of equilibrium: Real

Quantum impurity systems out of
equilibrium: Real-time dynamics
Avraham Schiller
Racah Institute of Physics,
The Hebrew University
Collaboration: Frithjof B. Anders, Dortmund University
F.B. Anders and AS, Phys. Rev. Lett. 95, 196801 (2005)
F.B. Anders and AS, Phys. Rev. B 74, 245113 (2006)
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Outline
Confined nano-structures and dissipative systems:
Non-perturbative physics out of equilibrium
Time-dependent Numerical Renormalization
Group (TD-NRG)
Benchmarks for fermionic and bosonic baths
Spin and charge relaxation in ultra-small dots
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Coulomb blockade in ultra-small quantum dots
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Coulomb blockade in ultra-small quantum dots
Quantum dot
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Coulomb blockade in ultra-small quantum dots
Leads
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Coulomb blockade in ultra-small quantum dots
Lead
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Lead
Coulomb blockade in ultra-small quantum dots
Lead
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Lead
Coulomb blockade in ultra-small quantum dots
Lead
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U
Lead
Coulomb blockade in ultra-small quantum dots
Lead
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U
Lead
Coulomb blockade in ultra-small quantum dots
Dei+U
U
Lead
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U
Lead
Coulomb blockade in ultra-small quantum dots
Lead
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U
Lead
Coulomb blockade in ultra-small quantum dots
Conductance vs gate voltage
Lead
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U
Lead
Coulomb blockade in ultra-small quantum dots
Conductance vs gate voltage
Lead
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U
Lead
Coulomb blockade in ultra-small quantum dots
dI/dV (e2/h)
Conductance vs gate voltage
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Lead
U
Lead
The Kondo effect in ultra-small quantum dots
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The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 

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

 L,R


t  d  (0)  H.c.

The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



 L,R


t  d  (0)  H.c.

Tunneling to leads
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The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



Inter-configurational energies
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 L,R
ed


t  d  (0)  H.c.

and U+ed
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



Inter-configurational energies
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 L,R
ed


t  d  (0)  H.c.

and U+ed
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



Inter-configurational energies
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 L,R
ed


t  d  (0)  H.c.

and U+ed
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



Inter-configurational energies
Hybridization width
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 L,R
ed


t  d  (0)  H.c.

and U+ed
  (t L2  t R2 )
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



Inter-configurational energies
Hybridization width
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 L,R
ed


t  d  (0)  H.c.

and U+ed
  (t L2  t R2 )
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



Inter-configurational energies
Hybridization width
 L,R
ed


and U+ed
  (t L2  t R2 )
Condition for formation of local moment:
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
t  d  (0)  H.c.
  e d ,U  e d
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



Inter-configurational energies
Hybridization width
 L,R
ed


and U+ed
  (t L2  t R2 )
Condition for formation of local moment:
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
t  d  (0)  H.c.
  e d ,U  e d
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 

Avraham Schiller / Seattle 09


 L,R


t  d  (0)  H.c.

The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



 L,R


t  d  (0)  H.c.

TK
ed
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EF
ed+U
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



 L,R


t  d  (0)  H.c.

A sharp resonance of width TK
develops at EF when T<TK
TK
ed
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EF
ed+U
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



 L,R


t  d  (0)  H.c.

A sharp resonance of width TK
develops at EF when T<TK
Abrikosov-Suhl resonance
TK
ed
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EF
ed+U
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



 L,R


t  d  (0)  H.c.

A sharp resonance of width TK
develops at EF when T<TK
Unitary scattering for T=0 and <n>=1
TK
ed
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EF
ed+U
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



 L,R


t  d  (0)  H.c.

A sharp resonance of width TK
develops at EF when T<TK
Unitary scattering for T=0 and <n>=1
Nonperturbative scale:
  | e d | (U  e d ) 
TK  exp 

2U


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TK
ed
EF
ed+U
The Kondo effect in ultra-small quantum dots
H imp  e d  n  Un n 



 L,R


t  d  (0)  H.c.
A sharp resonance of width TK
develops at EF when T<TK

Perfect transmission for
symmetric structure
Unitary scattering for T=0 and <n>=1
Nonperturbative scale:
  | e d | (U  e d ) 
TK  exp 

2U


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TK
ed
EF
ed+U
Electronic correlations out of equilibrium
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Electronic correlations out of equilibrium
dI/dV (e2/h)
Steady state
Differential conductance in
two-terminal devices
van der Wiel et al.,Science 2000
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Electronic correlations out of equilibrium
ac drive
dI/dV (e2/h)
Steady state
Differential conductance in
two-terminal devices
Photon-assisted side peaks
van der Wiel et al.,Science 2000
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Kogan et al.,Science 2004
Electronic correlations out of equilibrium
ac drive
dI/dV (e2/h)
Steady state
w
w
Differential conductance in
two-terminal devices
Photon-assisted side peaks
van der Wiel et al.,Science 2000
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Kogan et al.,Science 2004
Nonequilibrium: A theoretical challenge
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Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
and/or nonzero driving fields
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Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
and/or nonzero driving fields
Required: Inherently nonperturbative treatment of nonequilibrium
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Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
and/or nonzero driving fields
Required: Inherently nonperturbative treatment of nonequilibrium
Problem: Unlike equilibrium conditions, density operator is not
known in the presence of interactions
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Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
and/or nonzero driving fields
Required: Inherently nonperturbative treatment of nonequilibrium
Problem: Unlike equilibrium conditions, density operator is not
known in the presence of interactions
Most nonperturbative approaches available in equilibrium
are simply inadequate
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Nonequilibrium: A theoretical challenge
Two possible strategies
Work directly at
steady state
e.g., construct the manyparticle Scattering states
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Evolve the system in
time to reach steady
state
Time-dependent numerical RG
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Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a
sudden perturbation is applied at time t = 0
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Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a
sudden perturbation is applied at time t = 0
t <0
Lead
Lead
Vg
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Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a
sudden perturbation is applied at time t = 0
t <0
Lead
t >0
Lead
Vg
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Lead
Vg
Lead
Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a
sudden perturbation is applied at time t = 0
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Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which a
sudden perturbation is applied at time t = 0
Initial density operator
Oˆ
t 0




 Trace ˆ (t )Oˆ  Trace e iHt ˆ 0eiHtOˆ
Perturbed Hamiltonian
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Wilson’s numerical RG
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Wilson’s numerical RG
L 1
Logarithmic discretization of band:
e/D
-1
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-L-1
-L-2 -L-3
L-3 L-2
L-1
1
Wilson’s numerical RG
L 1
Logarithmic discretization of band:
e/D
-1
-L-1
-L-2 -L-3
L-3 L-2
L-1
1
After a unitary transformation the bath is represented by a semi-infinite
chain
imp
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x0
x1
x2
x3
Wilson’s numerical RG
Why logarithmic discretization?
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Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
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Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Hopping decays exponentially along the chain:
imp
x0
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x1
x2
x n  Ln / 2 , L  1
x3
Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Hopping decays exponentially along the chain:
imp
x0
x1
x2
Separation of energy scales along the chain
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x n  Ln / 2 , L  1
x3
Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Hopping decays exponentially along the chain:
imp
x0
x1
x2
x n  Ln / 2 , L  1
x3
Separation of energy scales along the chain
Exponentially small energy scales can be accessed, limited by T only
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Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Hopping decays exponentially along the chain:
imp
x0
x1
x2
x n  Ln / 2 , L  1
x3
Separation of energy scales along the chain
Exponentially small energy scales can be accessed, limited by T only
Iterative solution, starting from a core cluster and enlarging the chain
by one site at a time. High-energy states are discarded at each step,
refining the resolution as energy is decreased.
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Wilson’s numerical RG
Equilibrium NRG:
Geared towards fine energy resolution at low energies
Discards high-energy states
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Wilson’s numerical RG
Equilibrium NRG:
Geared towards fine energy resolution at low energies
Discards high-energy states
Problem:
Real-time dynamics involves all energy scales
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Wilson’s numerical RG
Equilibrium NRG:
Geared towards fine energy resolution at low energies
Discards high-energy states
Problem:
Real-time dynamics involves all energy scales
Resolution:
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Combine information from all NRG iterations
Time-dependent NRG
imp
x0
x1
r
NRG eigenstate of relevant iteration
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
e
Basis set for the “environment” states
Time-dependent NRG
imp
x0
x1
r
NRG eigenstate of relevant iteration

e
Basis set for the “environment” states
For each NRG iteration, we trace over its “environment”
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Time-dependent NRG
Sum over all chain lengths
(all energy scales)
N trun
O(t )   O  (m)e
m 1 s , r
Sum over discarded NRG states
of chain of length m
m
s ,r
red
r ,s
i ( Esm  Erm ) t
Reduced density
matrix for the
m-site chain
Matrix element of O
on the m-site chain
 (m)   r , e; m 0 s, e; m
red
r ,s
e
Trace over the environment, i.e., sites
not included in chain of length m
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(Hostetter, PRL 2000)
Fermionic benchmark: Resonant-level model
H   e k ck ck  Ed (t )d  d  V  (ck d  d  ck )
k
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k
Fermionic benchmark: Resonant-level model
H   e k ck ck  Ed (t )d  d  V  (ck d  d  ck )
k
Ed (t  0)  0
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k
Ed (t  0)  Ed1  0
Fermionic benchmark: Resonant-level model
H   e k ck ck  Ed (t )d  d  V  (ck d  d  ck )
k
k
Ed (t  0)  0
We focus on
Ed (t  0)  Ed1  0
nd (t )  d  d (t ) and compare the TD-NRG to exact
analytic solution in the wide-band limit (for an infinite system)
Basic energy scale:
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  V 2
Fermionic benchmark: Resonant-level model
T=0
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Relaxed values
(no runaway!)
Fermionic benchmark: Resonant-level model
T=0
T>0
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Relaxed values
(no runaway!)
Fermionic benchmark: Resonant-level model
T=0
Relaxed values
(no runaway!)
T>0
For T > 0, the TD-NRG works well up to t  1 / T
The deviation of the relaxed T=0 value from the new thermodynamic value
is a measure for the accuracy of the TD-NRG on all time scales
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Source of inaccuracies
T=0
Ed (t < 0) = -10
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Ed (t > 0) = 
L= 2
Source of inaccuracies
T=0
Ed (t < 0) = -10
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Ed (t > 0) = 
L= 2
Source of inaccuracies
T=0
Ed (t < 0) = -10
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Ed (t > 0) = 
L= 2
Source of inaccuracies
T=0
Ed (t < 0) = -10
Ed (t > 0) = 
TD-NRG is essentially exact on the Wilson chain
Main source of inaccuracies is due to discretization
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L= 2
Analysis of discretization effects
Ed (t < 0) = -10
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Ed (t > 0) = 
Analysis of discretization effects
Ed (t < 0) = -10
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Ed (t > 0) = 
Ed (t < 0) = 
Ed (t > 0) = -10
Bosonic benchmark: Spin-boson model
H   wi bibi 
i
D

x  z
2
2


(
b
 i i  bi )
i
J (w  wc )    i2  (w  wi )  2wcs 1w s
i
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Bosonic benchmark: Spin-boson model
H   wi bibi 
i
D

x  z
2
2


(
b
 i i  bi )
i
J (w  wc )    i2  (w  wi )  2wcs 1w s
i
Setting D=0, we start from the pure spin state
ˆ (t  0)   x  1  x  1  ˆThermalBath
and compute
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01(t )   z  1 TrBathˆ (t ) z  1
Bosonic benchmark: Spin-boson model
01 (t )
Excellent agreement between TD-NRG (full lines) and the
exact analytic solution (dashed lines) up to t  1 / T
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Bosonic benchmark: Spin-boson model
For nonzero D and s = 1 (Ohmic bath), we prepare the system such
that the spin is initially fully polarized (Sz = 1/2)
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Bosonic benchmark: Spin-boson model
For nonzero D and s = 1 (Ohmic bath), we prepare the system such
that the spin is initially fully polarized (Sz = 1/2)
Damped oscillations
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Bosonic benchmark: Spin-boson model
For nonzero D and s = 1 (Ohmic bath), we prepare the system such
that the spin is initially fully polarized (Sz = 1/2)
Monotonic decay
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Bosonic benchmark: Spin-boson model
For nonzero D and s = 1 (Ohmic bath), we prepare the system such
that the spin is initially fully polarized (Sz = 1/2)
Localized phase
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Anderson impurity model



H   e c c    Ed (t )  H (t ) d d
2

k ,
 

k k k
 V (t ) (ck d  d ck )  Ud d  d  d 
k ,
t<0
t>0
V 2  0
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Ed  U / 2
V 2  1
Anderson impurity model: Charge relaxation
Exact new
Equilibrium
values
Charge relaxation is governed by tch=1/1
TD-NRG works better for interacting case!
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Anderson impurity model: Spin relaxation
t  1
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Anderson impurity model: Spin relaxation
t  1
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t  TK
Anderson impurity model: Spin relaxation
t  1
t  TK
Spin relaxes on a much longer time scale tsp  t ch
Spin relaxation is sensitive to initial conditions!
Starting from a decoupled impurity, spin relaxation approaches a
universal function of t/tsp with tsp=1/TK
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Conclusions
A numerical RG approach was devised to track the real-time
dynamics of quantum impurities following a sudden perturbation
Works well for arbitrarily long times up to 1/T
Applicable to fermionic as well as bosonic baths
For ultra-small dots, spin and charge typically relax on
different time scales
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