Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity...
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![Page 1: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/1.jpg)
Avraham Schiller / Seattle 09
equilibrium: Real-time dynamics
Avraham Schiller
Quantum impurity systems out of
Racah Institute of Physics,
The Hebrew University
Collaboration: Frithjof B. Anders, Dortmund University
F.B. Anders and AS, Phys. Rev. Lett. 95, (2005)
F.B. Anders and AS, Phys. Rev. B 74, (2006)
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Avraham Schiller / Seattle 09
Outline
Confined nano-structures and dissipative systems:
Time-dependent Numerical Renormalization
Benchmarks for fermionic and bosonic baths
Spin and charge relaxation in ultra-small dots
Non-perturbative physics out of equilibrium
Group (TD-NRG)
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Avraham Schiller / Seattle 09
Coulomb blockade in ultra-small quantum dots
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Avraham Schiller / Seattle 09
Quantum dot
Coulomb blockade in ultra-small quantum dots
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Avraham Schiller / Seattle 09
Leads
Coulomb blockade in ultra-small quantum dots
![Page 6: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/6.jpg)
Avraham Schiller / Seattle 09
Lead Lead
Coulomb blockade in ultra-small quantum dots
![Page 7: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/7.jpg)
Avraham Schiller / Seattle 09
Lead Lead
Coulomb blockade in ultra-small quantum dots
![Page 8: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/8.jpg)
Avraham Schiller / Seattle 09
ULead Lead
Coulomb blockade in ultra-small quantum dots
![Page 9: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/9.jpg)
Avraham Schiller / Seattle 09
ULead Lead
Coulomb blockade in ultra-small quantum dots
![Page 10: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/10.jpg)
Avraham Schiller / Seattle 09
ULead Lead
Coulomb blockade in ultra-small quantum dots
i+U
U
![Page 11: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/11.jpg)
Avraham Schiller / Seattle 09
ULead Lead
Coulomb blockade in ultra-small quantum dots
![Page 12: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/12.jpg)
Avraham Schiller / Seattle 09
ULead Lead
Conductance vs gate voltage
Coulomb blockade in ultra-small quantum dots
![Page 13: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/13.jpg)
Avraham Schiller / Seattle 09
ULead Lead
Conductance vs gate voltage
Coulomb blockade in ultra-small quantum dots
![Page 14: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/14.jpg)
Avraham Schiller / Seattle 09
ULead Lead
Conductance vs gate voltage
dI/d
V (
e2 /h)
Coulomb blockade in ultra-small quantum dots
![Page 15: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/15.jpg)
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
![Page 16: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/16.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
![Page 17: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/17.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Tunneling to leads
![Page 18: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/18.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
![Page 19: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/19.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
![Page 20: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/20.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
![Page 21: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/21.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
Hybridization width )( 22RL tt
![Page 22: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/22.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
Hybridization width )( 22RL tt
![Page 23: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/23.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
dd U ,Condition for formation of local moment:
Hybridization width )( 22RL tt
![Page 24: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/24.jpg)
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
dd U ,Condition for formation of local moment:
Hybridization width )( 22RL tt
![Page 25: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/25.jpg)
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
RLdimp dtnUnnH
,
H.c.)0(
![Page 26: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/26.jpg)
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
EFd d+U
TK
RLdimp dtnUnnH
,
H.c.)0(
![Page 27: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/27.jpg)
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
EFd d+U
TK
A sharp resonance of width TK
develops at EF when T<TK
RLdimp dtnUnnH
,
H.c.)0(
![Page 28: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/28.jpg)
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
EFd d+U
Abrikosov-Suhl resonance
TK
A sharp resonance of width TK
develops at EF when T<TK
RLdimp dtnUnnH
,
H.c.)0(
![Page 29: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/29.jpg)
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
EFd d+U
TK
A sharp resonance of width TK
develops at EF when T<TK
Unitary scattering for T=0 and <n>=1
RLdimp dtnUnnH
,
H.c.)0(
![Page 30: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/30.jpg)
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
EFd d+U
TK
A sharp resonance of width TK
develops at EF when T<TK
Unitary scattering for T=0 and <n>=1
U
UT dd
K 2
)(||exp
Nonperturbative scale:
RLdimp dtnUnnH
,
H.c.)0(
![Page 31: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/31.jpg)
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
EFd d+U
TK
A sharp resonance of width TK
develops at EF when T<TK
Unitary scattering for T=0 and <n>=1
U
UT dd
K 2
)(||exp
Nonperturbative scale:
Perfect transmission for
symmetric structure
RLdimp dtnUnnH
,
H.c.)0(
![Page 32: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/32.jpg)
Avraham Schiller / Seattle 09
Electronic correlations out of equilibrium
![Page 33: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/33.jpg)
Avraham Schiller / Seattle 09
Electronic correlations out of equilibrium dI
/dV
(e2 /
h)
Differential conductance intwo-terminal devices
Steady state
van der Wiel et al.,Science 2000
![Page 34: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/34.jpg)
Avraham Schiller / Seattle 09
Electronic correlations out of equilibrium dI
/dV
(e2 /
h)
Differential conductance intwo-terminal devices
Steady state ac drive
Photon-assisted side peaks
Kogan et al.,Science 2004van der Wiel et al.,Science 2000
![Page 35: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/35.jpg)
Avraham Schiller / Seattle 09
Electronic correlations out of equilibrium dI
/dV
(e2 /
h)
Differential conductance intwo-terminal devices
Steady state ac drive
Photon-assisted side peaks
Kogan et al.,Science 2004van der Wiel et al.,Science 2000
![Page 36: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/36.jpg)
Avraham Schiller / Seattle 09
Nonequilibrium: A theoretical challenge
![Page 37: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/37.jpg)
Avraham Schiller / Seattle 09
Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
and/or nonzero driving fields
![Page 38: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/38.jpg)
Avraham Schiller / Seattle 09
Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
Required: Inherently nonperturbative treatment of nonequilibrium
and/or nonzero driving fields
![Page 39: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/39.jpg)
Avraham Schiller / Seattle 09
Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
Required: Inherently nonperturbative treatment of nonequilibrium
and/or nonzero driving fields
Problem: Unlike equilibrium conditions, density operator is not
known in the presence of interactions
![Page 40: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/40.jpg)
Avraham Schiller / Seattle 09
Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
Required: Inherently nonperturbative treatment of nonequilibrium
and/or nonzero driving fields
Problem: Unlike equilibrium conditions, density operator is not
Most nonperturbative approaches available in equilibrium
known in the presence of interactions
are simply inadequate
![Page 41: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/41.jpg)
Avraham Schiller / Seattle 09
Nonequilibrium: A theoretical challenge
Two possible strategies
Work directly atsteady state
e.g., construct the many-particle Scattering states
Evolve the system in time to reach steady
state
![Page 42: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/42.jpg)
Avraham Schiller / Seattle 09
Time-dependent numerical RG
![Page 43: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/43.jpg)
Avraham Schiller / Seattle 09
Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which asudden perturbation is applied at time t = 0
![Page 44: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/44.jpg)
Avraham Schiller / Seattle 09
Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which asudden perturbation is applied at time t = 0
LeadLead
Vg
t < 0
![Page 45: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/45.jpg)
Avraham Schiller / Seattle 09
Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which asudden perturbation is applied at time t = 0
LeadLead
Vg
t > 0
LeadLead
Vg
t < 0
![Page 46: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/46.jpg)
Avraham Schiller / Seattle 09
Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which asudden perturbation is applied at time t = 0
![Page 47: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/47.jpg)
Avraham Schiller / Seattle 09
Time-dependent numerical RG
Consider a quantum impurity (e.g., quantum dot) in equilibrium, to which asudden perturbation is applied at time t = 0
OeeOtO iHtiHt
t
ˆˆTraceˆ)(ˆTraceˆ0
0
Perturbed Hamiltonian
Initial density operator
![Page 48: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/48.jpg)
Avraham Schiller / Seattle 09
Wilson’s numerical RG
![Page 49: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/49.jpg)
Avraham Schiller / Seattle 09
Wilson’s numerical RG
-1 1--1 --2 --3 -1-2-3
/D
Logarithmic discretization of band: 1
![Page 50: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/50.jpg)
Avraham Schiller / Seattle 09
Wilson’s numerical RG
-1 1--1 --2 --3 -1-2-3
/D
Logarithmic discretization of band: 1
imp
After a unitary transformation the bath is represented by a semi-infinitechain
![Page 51: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/51.jpg)
Avraham Schiller / Seattle 09
Why logarithmic discretization?
Wilson’s numerical RG
![Page 52: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/52.jpg)
Avraham Schiller / Seattle 09
Why logarithmic discretization?
Wilson’s numerical RG
To properly account for the logarithmic infra-red divergences
![Page 53: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/53.jpg)
Avraham Schiller / Seattle 09
Why logarithmic discretization?
Wilson’s numerical RG
To properly account for the logarithmic infra-red divergences
imp
Hopping decays exponentially along the chain: 1,2/ nn
![Page 54: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/54.jpg)
Avraham Schiller / Seattle 09
Why logarithmic discretization?
Wilson’s numerical RG
imp
Hopping decays exponentially along the chain: 1,2/ nn
Separation of energy scales along the chain
To properly account for the logarithmic infra-red divergences
![Page 55: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/55.jpg)
Avraham Schiller / Seattle 09
Why logarithmic discretization?
Wilson’s numerical RG
imp
Hopping decays exponentially along the chain: 1,2/ nn
Exponentially small energy scales can be accessed, limited by T only
To properly account for the logarithmic infra-red divergences
Separation of energy scales along the chain
![Page 56: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/56.jpg)
Avraham Schiller / Seattle 09
Why logarithmic discretization?
Wilson’s numerical RG
imp
Hopping decays exponentially along the chain: 1,2/ nn
Iterative solution, starting from a core cluster and enlarging the chainby one site at a time. High-energy states are discarded at each step,refining the resolution as energy is decreased.
To properly account for the logarithmic infra-red divergences
Exponentially small energy scales can be accessed, limited by T only
Separation of energy scales along the chain
![Page 57: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/57.jpg)
Avraham Schiller / Seattle 09
Equilibrium NRG:
Geared towards fine energy resolution at low energies
Discards high-energy states
Wilson’s numerical RG
![Page 58: Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity systems out of Racah Institute of Physics, The Hebrew University.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d7f5503460f94a632f1/html5/thumbnails/58.jpg)
Avraham Schiller / Seattle 09
Equilibrium NRG:
Problem: Real-time dynamics involves all energy scales
Geared towards fine energy resolution at low energies
Discards high-energy states
Wilson’s numerical RG
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Avraham Schiller / Seattle 09
Equilibrium NRG:
Problem: Real-time dynamics involves all energy scales
Resolution: Combine information from all NRG iterations
Geared towards fine energy resolution at low energies
Discards high-energy states
Wilson’s numerical RG
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Avraham Schiller / Seattle 09
Time-dependent NRG
imp
r e
Basis set for the “environment” statesNRG eigenstate of relevant iteration
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Avraham Schiller / Seattle 09
Time-dependent NRG
imp
r e
Basis set for the “environment” statesNRG eigenstate of relevant iteration
For each NRG iteration, we trace over its “environment”
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Avraham Schiller / Seattle 09
Time-dependent NRG
N
m
trun
rs
tEEisr
mrs
mr
msemOtO
1 ,
)(red,, )()(
Sum over discarded NRG statesof chain of length m
Matrix element of Oon the m-site chain
Reduced density matrix for them-site chain
e
sr mesmerm ;,;,)( 0red,
(Hostetter, PRL 2000)
Sum over all chain lengths(all energy scales)
Trace over the environment, i.e., sitesnot included in chain of length m
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Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
k
kkdk
kkk cddcVddtEccH )()(
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Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
k
kkdk
kkk cddcVddtEccH )()(
0)0( tEd 0)0( 1 dd EtE
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Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
k
kkdk
kkk cddcVddtEccH )()(
0)0( tEd
We focus on )()( tddtnd and compare the TD-NRG to exact
analytic solution in the wide-band limit (for an infinite system)
Basic energy scale: 2V
0)0( 1 dd EtE
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Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
T = 0
Relaxed values(no runaway!)
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Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
T = 0
T > 0
Relaxed values(no runaway!)
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Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
T = 0
T > 0
Relaxed values(no runaway!)
The deviation of the relaxed T=0 value from the new thermodynamic valueis a measure for the accuracy of the TD-NRG on all time scales
For T > 0, the TD-NRG works well up to Tt /1
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Avraham Schiller / Seattle 09
T = 0 Ed (t < 0) = -10Ed (t > 0) = = 2
Source of inaccuracies
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Avraham Schiller / Seattle 09
T = 0 Ed (t < 0) = -10Ed (t > 0) = = 2
Source of inaccuracies
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Avraham Schiller / Seattle 09
T = 0 Ed (t < 0) = -10Ed (t > 0) = = 2
Source of inaccuracies
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Avraham Schiller / Seattle 09
T = 0 Ed (t < 0) = -10Ed (t > 0) = = 2
TD-NRG is essentially exact on the Wilson chain
Source of inaccuracies
Main source of inaccuracies is due to discretization
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Avraham Schiller / Seattle 09
Analysis of discretization effects
Ed (t < 0) = -10Ed (t > 0) =
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Avraham Schiller / Seattle 09
Analysis of discretization effects
Ed (t < 0) = -10Ed (t > 0) =
Ed (t < 0) = Ed (t > 0) = -10
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Avraham Schiller / Seattle 09
Bosonic benchmark: Spin-boson model
iiii
zx
iiii bbbbH )(
22
ssc
iiicJ 12 2)()(
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Avraham Schiller / Seattle 09
Bosonic benchmark: Spin-boson model
iiii
zx
iiii bbbbH )(
22
ssc
iiicJ 12 2)()(
Setting =0, we start from the pure spin state
BathThermalxxt ˆ11)0(ˆ
and compute 1)(ˆ1)(01 zBathz tTrt
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Avraham Schiller / Seattle 09
Bosonic benchmark: Spin-boson model
)(01 t
Excellent agreement between TD-NRG (full lines) and theexact analytic solution (dashed lines) up to Tt /1
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Avraham Schiller / Seattle 09
Bosonic benchmark: Spin-boson model
For nonzero and s = 1 (Ohmic bath), we prepare the system suchthat the spin is initially fully polarized (Sz = 1/2)
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Avraham Schiller / Seattle 09
Bosonic benchmark: Spin-boson model
For nonzero and s = 1 (Ohmic bath), we prepare the system suchthat the spin is initially fully polarized (Sz = 1/2)
Damped oscillations
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Avraham Schiller / Seattle 09
Bosonic benchmark: Spin-boson model
For nonzero and s = 1 (Ohmic bath), we prepare the system suchthat the spin is initially fully polarized (Sz = 1/2)
Monotonic decay
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Avraham Schiller / Seattle 09
Bosonic benchmark: Spin-boson model
For nonzero and s = 1 (Ohmic bath), we prepare the system suchthat the spin is initially fully polarized (Sz = 1/2)
Localized phase
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Avraham Schiller / Seattle 09
Anderson impurity model
ddtHtEccH dk
kkk )(2
)(,
dddUdcddctVk
kk
,
)()(
t < 0
02 V
t > 0
12 V
2/UEd
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Avraham Schiller / Seattle 09
Anderson impurity model: Charge relaxation
Charge relaxation is governed by tch=1/1
TD-NRG works better for interacting case!
Exact newEquilibrium
values
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Anderson impurity model: Spin relaxation
1t
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Anderson impurity model: Spin relaxation
KTt 1t
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Avraham Schiller / Seattle 09
Anderson impurity model: Spin relaxation
Spin relaxes on a much longer time scale
Spin relaxation is sensitive to initial conditions!
chsp tt
Starting from a decoupled impurity, spin relaxation approaches a
universal function of t/tsp with tsp=1/TK
KTt 1t
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Avraham Schiller / Seattle 09
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