View
218
Download
3
Tags:
Embed Size (px)
Citation preview
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)
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)
Avraham Schiller / Seattle 09
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
Quantum dot
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
Leads
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
Lead Lead
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
Lead Lead
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
ULead Lead
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
ULead Lead
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
ULead Lead
Coulomb blockade in ultra-small quantum dots
i+U
U
Avraham Schiller / Seattle 09
ULead Lead
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
ULead Lead
Conductance vs gate voltage
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
ULead Lead
Conductance vs gate voltage
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
ULead Lead
Conductance vs gate voltage
dI/d
V (
e2 /h)
Coulomb blockade in ultra-small quantum dots
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Tunneling to leads
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
Avraham Schiller / Seattle 09
RLdimp dtnUnnH
,
H.c.)0(
The Kondo effect in ultra-small quantum dots
Inter-configurational energies d and U+d
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
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
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
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
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
RLdimp dtnUnnH
,
H.c.)0(
Avraham Schiller / Seattle 09
The Kondo effect in ultra-small quantum dots
EFd d+U
TK
RLdimp dtnUnnH
,
H.c.)0(
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(
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(
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(
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(
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(
Avraham Schiller / Seattle 09
Electronic correlations out of equilibrium
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
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
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
Avraham Schiller / Seattle 09
Nonequilibrium: A theoretical challenge
Avraham Schiller / Seattle 09
Nonequilibrium: A theoretical challenge
The Goal: The description of nano-structures at nonzero bias
and/or nonzero driving fields
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
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
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
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
Avraham Schiller / Seattle 09
Time-dependent numerical RG
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
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
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
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
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
Avraham Schiller / Seattle 09
Wilson’s numerical RG
Avraham Schiller / Seattle 09
Wilson’s numerical RG
-1 1--1 --2 --3 -1-2-3
/D
Logarithmic discretization of band: 1
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
Avraham Schiller / Seattle 09
Why logarithmic discretization?
Wilson’s numerical RG
Avraham Schiller / Seattle 09
Why logarithmic discretization?
Wilson’s numerical RG
To properly account for the logarithmic infra-red divergences
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
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
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
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
Avraham Schiller / Seattle 09
Equilibrium NRG:
Geared towards fine energy resolution at low energies
Discards high-energy states
Wilson’s numerical RG
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
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
Avraham Schiller / Seattle 09
Time-dependent NRG
imp
r e
Basis set for the “environment” statesNRG eigenstate of relevant iteration
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”
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
Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
k
kkdk
kkk cddcVddtEccH )()(
Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
k
kkdk
kkk cddcVddtEccH )()(
0)0( tEd 0)0( 1 dd EtE
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
Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
T = 0
Relaxed values(no runaway!)
Avraham Schiller / Seattle 09
Fermionic benchmark: Resonant-level model
T = 0
T > 0
Relaxed values(no runaway!)
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
Avraham Schiller / Seattle 09
T = 0 Ed (t < 0) = -10Ed (t > 0) = = 2
Source of inaccuracies
Avraham Schiller / Seattle 09
T = 0 Ed (t < 0) = -10Ed (t > 0) = = 2
Source of inaccuracies
Avraham Schiller / Seattle 09
T = 0 Ed (t < 0) = -10Ed (t > 0) = = 2
Source of inaccuracies
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
Avraham Schiller / Seattle 09
Analysis of discretization effects
Ed (t < 0) = -10Ed (t > 0) =
Avraham Schiller / Seattle 09
Analysis of discretization effects
Ed (t < 0) = -10Ed (t > 0) =
Ed (t < 0) = Ed (t > 0) = -10
Avraham Schiller / Seattle 09
Bosonic benchmark: Spin-boson model
iiii
zx
iiii bbbbH )(
22
ssc
iiicJ 12 2)()(
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
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
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)
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
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
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
Avraham Schiller / Seattle 09
Anderson impurity model
ddtHtEccH dk
kkk )(2
)(,
dddUdcddctVk
kk
,
)()(
t < 0
02 V
t > 0
12 V
2/UEd
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
Avraham Schiller / Seattle 09
Anderson impurity model: Spin relaxation
1t
Avraham Schiller / Seattle 09
Anderson impurity model: Spin relaxation
KTt 1t
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
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