Far-from-Equilibrium and
Time-Dependent Phenomena:
Theory
Avraham Schiller
Racah Institute of Physics,
The Hebrew University
Avraham Schiller / QIMP11
Correlated systems
out of equilibrium
Avraham Schiller / QIMP11
Femtosecond
spectroscopy
Correlated systems
out of equilibrium
Avraham Schiller / QIMP11
Femtosecond
spectroscopy
Bulk materials
particles
Correlated systems
out of equilibrium
Avraham Schiller / QIMP11
Femtosecond
spectroscopy
Bulk materials
particles
Relaxation dynamics
Avraham Schiller / QIMP11
Correlated systems
out of equilibrium
Femtosecond
spectroscopy
Bulk materials
particles
Relaxation dynamics
Avraham Schiller / QIMP11
Cold atoms
Correlated systems
out of equilibrium
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Avraham Schiller / QIMP11
Correlated systems
out of equilibrium
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Avraham Schiller / QIMP11
Quench dynamics
Correlated systems
out of equilibrium
Thermalization?
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Avraham Schiller / QIMP11
Quench dynamics
Correlated systems
out of equilibrium
Thermalization?
Collapse and revival of coherent matter waves
t=0
t = 100µs
t = 150µs
t = 350µs
t = 400µs
t = 550µs
t = 250µs
M. Greiner et al., Nature 2002
Avraham Schiller / QIMP11
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Avraham Schiller / QIMP11
Quench dynamics
Correlated systems
out of equilibrium
Thermalization?
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Avraham Schiller / QIMP11
Quench dynamics
Correlated systems
out of equilibrium
Thermalization?
distribution
Lack of thermalization in 1D traps
momentum
momentum
Exciting the system and allowing it to evolve in time, the momentum
distribution function remains non-Gaussian up to long time scales
T. Kinoshita et al., Nature 440, 900 (2006).
Avraham Schiller / QIMP11
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Avraham Schiller / QIMP11
Quench dynamics
Correlated systems
out of equilibrium
Thermalization?
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Quench dynamics
Correlated systems
out of equilibrium
Confined nanostructures
Avraham Schiller / QIMP11
Thermalization?
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Quench dynamics
Correlated systems
out of equilibrium
Thermalization?
Confined nanostructures
Few interacting degrees of freedom coupled to environment
Avraham Schiller / QIMP11
Femtosecond
spectroscopy
Cold atoms
particles
Bulk materials
particles
Relaxation dynamics
Quench dynamics
Correlated systems
out of equilibrium
Thermalization?
Confined nanostructures
Few interacting degrees of freedom coupled to environment
Steady state/quench dynamics
Avraham Schiller / QIMP11
Correlated systems
out of equilibrium
Confined nanostructures
Few interacting degrees of freedom coupled to environment
Steady state/quench dynamics
Avraham Schiller / QIMP11
Kondo effect in ultra-small quantum dots
Avraham Schiller / QIMP11
Kondo effect in ultra-small quantum dots
Quantum dot
Avraham Schiller / QIMP11
Kondo effect in ultra-small quantum dots
Leads
Avraham Schiller / QIMP11
Kondo effect in ultra-small quantum dots
Lead
Avraham Schiller / QIMP11
Lead
Kondo effect in ultra-small quantum dots
Lead
Avraham Schiller / QIMP11
Lead
Kondo effect in ultra-small quantum dots
Lead
Avraham Schiller / QIMP11
U
Lead
Kondo effect in ultra-small quantum dots
Conductance vs gate voltage
Lead
Avraham Schiller / QIMP11
U
Lead
Kondo effect in ultra-small quantum dots
Conductance vs gate voltage
Lead
Avraham Schiller / QIMP11
U
Lead
Kondo effect in ultra-small quantum dots
dI/dV (e2/h)
Conductance vs gate voltage
Avraham Schiller / QIMP11
Lead
U
Lead
Electronic correlations out of equilibrium
Avraham Schiller / QIMP11
Electronic correlations out of equilibrium
dI/dV (e2/h)
Steady state
Differential conductance in
two-terminal devices
van der Wiel et al.,Science 2000
Avraham Schiller / QIMP11
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
Avraham Schiller / QIMP11
Kogan et al.,Science 2004
Electronic correlations out of equilibrium
ac drive
dI/dV (e2/h)
Steady state
+hω
−hω
Differential conductance in
two-terminal devices
Photon-assisted side peaks
van der Wiel et al.,Science 2000
Avraham Schiller / QIMP11
Kogan et al.,Science 2004
Electronic correlations out of equilibrium
Response to pulsed bias
Elzerman et al., Nature (2005)
Avraham Schiller / QIMP11
Electronic correlations out of equilibrium
Response to pulsed bias
(a)
Elzerman et al., Nature (2005)
Avraham Schiller / QIMP11
Electronic correlations out of equilibrium
Response to pulsed bias
(a)
Abrupt change
of gate voltage
(b)
Elzerman et al., Nature (2005)
Avraham Schiller / QIMP11
Electronic correlations out of equilibrium
Response to pulsed bias
(a)
Abrupt change
of gate voltage
(b)
Elzerman et al., Nature (2005)
Avraham Schiller / QIMP11
Electronic correlations out of equilibrium
Response to pulsed bias
(a)
Abrupt change
of gate voltage
(b)
Elzerman et al., Nature (2005)
Avraham Schiller / QIMP11
Electronic correlations out of equilibrium
Response to pulsed bias
(a)
Abrupt change
of gate voltage
(b)
Abrupt change
of gate voltage
(c)
Elzerman et al., Nature (2005)
Avraham Schiller / QIMP11
Electronic correlations out of equilibrium
Response to pulsed bias
(a)
Abrupt change
of gate voltage
(b)
Abrupt change
of gate voltage
(c)
Elzerman et al., Nature (2005)
Avraham Schiller / QIMP11
Electronic correlations out of equilibrium
Response to pulsed bias
Single-shot readout of the spin state of a
quantum dot from real-time dynamics
Elzerman et al., Nature (2005)
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
The Goal: The description of nanostructures at nonzero bias,
nonzero driving fields, and/or quench dynamics
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
The Goal: The description of nanostructures at nonzero bias,
nonzero driving fields, and/or quench dynamics
Required: Inherently nonperturbative treatment of nonequilibrium
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
The Goal: The description of nanostructures at nonzero bias,
nonzero driving fields, and/or quench dynamics
Required: Inherently nonperturbative treatment of nonequilibrium
Problem: Unlike equilibrium conditions, density operator is not
known in the presence of interactions
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
The Goal: The description of nanostructures at nonzero bias,
nonzero driving fields, and/or quench dynamics
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
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
Two possible strategies to treat steady state
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
Two possible strategies to treat steady state
Time-independent formulation:
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
Two possible strategies to treat steady state
Time-independent formulation:
Work directly at steady state by
imposing suitable boundary conditions
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
Two possible strategies to treat steady state
Time-independent formulation:
Work directly at steady state by
imposing suitable boundary conditions
e.g., by constructing the manyparticle scattering states
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
Two possible strategies to treat steady state
Time-independent formulation:
Work directly at steady state by
imposing suitable boundary conditions
e.g., by constructing the manyparticle scattering states
Avraham Schiller / QIMP11
Time-dependent formulation
Nonequilibrium: A theoretical challenge
Two possible strategies to treat steady state
Time-independent formulation:
Time-dependent formulation
Work directly at steady state by
imposing suitable boundary conditions
Evolve the system in time
to reach steady state
e.g., by constructing the manyparticle scattering states
Avraham Schiller / QIMP11
Nonequilibrium: A theoretical challenge
Two possible strategies to treat steady state
Time-independent formulation:
Time-dependent formulation
Work directly at steady state by
imposing suitable boundary conditions
Evolve the system in time
to reach steady state
e.g., by constructing the manyparticle scattering states
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Steady state
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Steady state
Keldysh diagrammatics
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Steady state
Keldysh diagrammatics
Scattering Bethe ansatz
(Andrei et al.)
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Steady state
Keldysh diagrammatics
Scattering Bethe ansatz
(Andrei et al.)
Nonequilibrium variants of perturbative RG
Poor-man’s scaling (Rosch et al.)
Flow equations (Kehrein)
Real-time diagrammatics (Schoeller et al.)
Functional RG (Meden et al.)
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Steady state
Keldysh diagrammatics
Scattering Bethe ansatz
(Andrei et al.)
Nonequilibrium variants of perturbative RG
Poor-man’s scaling (Rosch et al.)
Flow equations (Kehrein)
Real-time diagrammatics (Schoeller et al.)
Functional RG (Meden et al.)
Exactly solvable models:
Toulouse limit (AS & Hershfield)
Extension to double dots (Sela & Affleck)
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Evolution in time
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Evolution in time
Keldysh diagrammatics
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Evolution in time
Keldysh diagrammatics
Time-dependent DMRG
(White, Schollwoeck,…)
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Evolution in time
Keldysh diagrammatics
Time-dependent DMRG
(White, Schollwoeck,…)
Keldysh Quantum Monte Carlo
(Werner, Muehlbacher,…)
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Evolution in time
Keldysh diagrammatics
Time-dependent DMRG
(White, Schollwoeck,…)
Keldysh Quantum Monte Carlo
(Werner, Muehlbacher,…)
Nonequilibrium variants of perturbative RG
Flow equations (Kehrein)
Real-time diagrammatics (Schoeller et al.)
Avraham Schiller / QIMP11
Brief division of theoretical approaches
Evolution in time
Keldysh diagrammatics
Time-dependent DMRG
(White, Schollwoeck,…)
Keldysh Quantum Monte Carlo
(Werner, Muehlbacher,…)
Nonequilibrium variants of perturbative RG
Flow equations (Kehrein)
Real-time diagrammatics (Schoeller et al.)
Time-dependent NRG
(Anders & AS)
Avraham Schiller / QIMP11
Time-independent formulation: the Lippmann-Schwinger equation
Incoming state: decoupled leads
Avraham Schiller / QIMP11
Scattered state: entangled system
Time-independent formulation: the Lippmann-Schwinger equation
Incoming state: decoupled leads
Scattered state: entangled system
Divide H into H0+H1, where H 0 | φ = E | φ and Η1 contains all terms
that drive the system out of equilibrium.
Avraham Schiller / QIMP11
Time-independent formulation: the Lippmann-Schwinger equation
Incoming state: decoupled leads
Scattered state: entangled system
Divide H into H0+H1, where H 0 | φ = E | φ and Η1 contains all terms
that drive the system out of equilibrium.
Assume approach to steady state
Avraham Schiller / QIMP11
Time-independent formulation: the Lippmann-Schwinger equation
Because of the approach to steady state, one can “smear” the initial time:
0
| ψ = limη →0+ η ∫ dt0 eηt0 e i ( H − E ) t0 | φ
−∞
Gell-Man and Goldberger, 1953
Avraham Schiller / QIMP11
Time-independent formulation: the Lippmann-Schwinger equation
Because of the approach to steady state, one can “smear” the initial time:
0
| ψ = limη →0+ η ∫ dt0 eηt0 e i ( H − E ) t0 | φ
−∞
| ψ =| φ +
1
(H − E) | φ
E − H + iη
Gell-Man and Goldberger, 1953
Avraham Schiller / QIMP11
Time-independent formulation: the Lippmann-Schwinger equation
Because of the approach to steady state, one can “smear” the initial time:
0
| ψ = limη →0+ η ∫ dt0 eηt0 e i ( H − E ) t0 | φ
−∞
| ψ =| φ +
1
(H − E) | φ
E − H + iη
Since ( H 0 − E ) | φ = 0 , we arrive at the Lippmann-Schwinger equation
| ψ =| φ +
1
H1 | φ
E − H + iη
Gell-Man and Goldberger, 1953
Avraham Schiller / QIMP11
Time-independent formulation: the Lippmann-Schwinger equation
Important points to take note of:
H and H0 must therefore have continuous overlapping spectra,
which implies the limit L → ∞
Avraham Schiller / QIMP11
The nonequilibrium steady-state density operator
Starting from ρˆ 0 = ∑ pi | φi φi | , where pi are typically equilibrium
i
Boltzmann factors, one formally has that
Avraham Schiller / QIMP11
The nonequilibrium steady-state density operator
Starting from ρˆ 0 = ∑ pi | φi φi | , where pi are typically equilibrium
i
Boltzmann factors, one formally has that
ρˆ 0 → ρˆ = ∑ pi | ψ i ψ i |
i
with
| ψ i =| φi +
Avraham Schiller / QIMP11
1
H1 | φi
Ei − H + iη
The nonequilibrium steady-state density operator
Starting from ρˆ 0 = ∑ pi | φi φi | , where pi are typically equilibrium
i
Boltzmann factors, one formally has that
ρˆ 0 → ρˆ = ∑ pi | ψ i ψ i |
i
with
| ψ i =| φi +
1
H1 | φi
Ei − H + iη
Assuming the approach to steady state, the form of the
nonequilibrium density operator is formally known
Avraham Schiller / QIMP11
Hershfield’s mapping onto equilibrium form
Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006
Avraham Schiller / QIMP11
Hershfield’s mapping onto equilibrium form
In practice, the initial density matrix generically takes the form
e − β ( H 0 −Y0 )
ρˆ 0 =
Trace e − β ( H 0 −Y0 )
{
}
with [H 0 , Y0 ] = 0
Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006
Avraham Schiller / QIMP11
Hershfield’s mapping onto equilibrium form
In practice, the initial density matrix generically takes the form
e − β ( H 0 −Y0 )
ρˆ 0 =
Trace e − β ( H 0 −Y0 )
{
}
with [H 0 , Y0 ] = 0
Indeed, in many cases one takes
H0 =
(ε α
∑
∑
α
σ
= L,R k ,
Y0 =
∑
α
k
+ µα )cα+kσ cαkσ
+
µ
c
∑ α αkσ cαkσ
= L , R k ,σ
Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006
Avraham Schiller / QIMP11
Hershfield’s mapping onto equilibrium form
e − β ( H −Y )
ρˆ =
Trace e − β ( H −Y )
{
}
with
[Y , H ] = iη (Y0 − Y )
→ 0
Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006
Avraham Schiller / QIMP11
Hershfield’s mapping onto equilibrium form
e − β ( H −Y )
ρˆ =
Trace e − β ( H −Y )
{
}
with
[Y , H ] = iη (Y0 − Y )
→ 0
The steady-state density operator takes an equilibriumlike form!
Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006
Avraham Schiller / QIMP11
Generalized fermionic scattering states
The steady-state density operator can be represented in terms of
generalized fermionic scattering states:
H −Y =
ε α ψ α σψ α σ
∑
∑
α
σ
= L,R k ,
k
+
k
k
with
[ψ
+
αkσ
]
, H = −ε αkψ α+kσ + iη (cα+kσ −ψ α+kσ )
Hershfield 1993, Han 2007
Avraham Schiller / QIMP11
Generalized fermionic scattering states
The steady-state density operator can be represented in terms of
generalized fermionic scattering states:
ε α ψ α σψ α σ
∑
∑
α
σ
H −Y =
= L,R k ,
k
+
k
k
with
[ψ
+
αkσ
]
, H = −ε αkψ α+kσ + iη (cα+kσ −ψ α+kσ )
In general ψ αkσ is a complicated many-body operator:
+
ψ α+kσ = cα+kσ + ∑ Aiαk ci+ + ∑ Bijlαk ci+ c +j cl + 
i
i , j ,k
Hershfield 1993, Han 2007
Avraham Schiller / QIMP11
Generalized fermionic scattering states
In the absence of interactions ψαkσ reduce to the familiar
single-particle scattering states
ψ α+kσ = cα+kσ + ∑ Aiαk ci+
i
and one recovers the Landauer-Buttiker formulation
Hershfield 1993
Avraham Schiller / QIMP11
Time-dependent formulation
Avraham Schiller / QIMP11
Time-dependent formulation
Starting from ρˆ 0 = ∑ pi | φi φi | at time t0, expectation values are
i
explicitly propagated in time:
{
}
Aˆ (t ) = Trace ρˆ 0U + (t , t0 ) Aˆ U (t , t0 )
Avraham Schiller / QIMP11
Time-dependent formulation
Starting from ρˆ 0 = ∑ pi | φi φi | at time t0, expectation values are
i
explicitly propagated in time:
{
}
Aˆ (t ) = Trace ρˆ 0U + (t , t0 ) Aˆ U (t , t0 )
A(t)
Recurrence
Steady state value
t
Avraham Schiller / QIMP11
Time-dependent formulation
Avraham Schiller / QIMP11
Time-dependent formulation
Avraham Schiller / QIMP11
Time-dependent formulation
A challenge when the system features small energy
Scales such as the Kondo temperature!
Avraham Schiller / QIMP11
Selected review of theoretical approaches
Scattering Bethe ansatz
Nonequilbrium variants of perturbative RG
Time-dependent NRG
Avraham Schiller / QIMP11
Selected review of theoretical approaches
Scattering Bethe ansatz
Nonequilbrium variants of perturbative RG
Time-dependent NRG
Will not addressed:
Keldysh-based approaches
(Hans Kroha’s talk)
Theories based on Fermi-liquid theory
Keldysh Quantum Monte Carlo
Time-dependent DMRG
Avraham Schiller / QIMP11
(weak nonequilibrium)
Scattering Bethe ansatz
One uses the Bethe ansatz approach, adjusted to open boundary
conditions, to explicitly construct the many-particle scattering states
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
One uses the Bethe ansatz approach, adjusted to open boundary
conditions, to explicitly construct the many-particle scattering states
Step 1: Conversion to continuum-limit Hamiltonian
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
One uses the Bethe ansatz approach, adjusted to open boundary
conditions, to explicitly construct the many-particle scattering states
Step 1: Conversion to continuum-limit Hamiltonian
Left lead
Right lead
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
One uses the Bethe ansatz approach, adjusted to open boundary
conditions, to explicitly construct the many-particle scattering states
Step 1: Conversion to continuum-limit Hamiltonian
Left lead
Right lead
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
General form of N-electron wave function:
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
General form of N-electron wave function:
Impurity states
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
General form of N-electron wave function:
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
General form of N-electron wave function:
where F obeys the Schroedinger-type equation
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
General form of N-electron wave function:
where F obeys the Schroedinger-type equation
Within each sector where x1,…, xN, and x0 = 0 obey some fixed ordering,
F has solutions in terms of plane waves
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The Bethe ansatz then seeks solutions of the form
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The Bethe ansatz then seeks solutions of the form
Projection onto the ordering affiliated with P
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The Bethe ansatz then seeks solutions of the form
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The Bethe ansatz then seeks solutions of the form
The existence of such solutions is highly nontrivial, and requires
special conditions known as the Yang-Baxter equations
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The Bethe ansatz then seeks solutions of the form
The existence of such solutions is highly nontrivial, and requires
special conditions known as the Yang-Baxter equations
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The main sources of difficulty:
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The main sources of difficulty:
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The main sources of difficulty:
Expectation values of the current operators are extremely difficult
to evaluate with respect to the Bethe ansatz wave function
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Scattering Bethe ansatz
The main sources of difficulty:
Expectation values of the current operators are extremely difficult
to evaluate with respect to the Bethe ansatz wave function
Remarkably
Mehta and Andrei succeeded in obtaining an exact solution for
the interacting resonant-level model, though even there the limit
was not worked out analytically
Mehta & Andrei, 2005
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
Perturbative RG has become one of the key concepts in analyzing
correlated electron systems in thermal equilibrium
The basic idea is to systematically reduce the energy scale by
integrating high-energy excitations, thus generating a sequence
of Hamiltonians for each energy scale
Interactions correspond to irreducible vertices, whose evolution
one tracks
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
Perturbative RG has become one of the key concepts in analyzing
correlated electron systems in thermal equilibrium
The basic idea is to systematically reduce the energy scale by
integrating high-energy excitations, thus generating a sequence
of Hamiltonians for each energy scale
Interactions correspond to irreducible vertices, whose evolution
one tracks
Generic example in equilibrium – the Kondo model
D
D - δD
D
D - δD
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
δJ
=
Avraham Schiller / QIMP11
+
+ …
Nonequilibrium variants of perturbative RG
δJ
=
+
In equilibrium only J(ω = 0) is important for thermodynamics
Avraham Schiller / QIMP11
+ …
Nonequilibrium variants of perturbative RG
δJ
=
+
+ …
In equilibrium only J(ω = 0) is important for thermodynamics
Out of equilibrium one needs to keep track of J(ω) , as transport properties
are determined by a window of energies
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
δJ
=
+
+ …
In equilibrium only J(ω = 0) is important for thermodynamics
Out of equilibrium one needs to keep track of J(ω) , as transport properties
are determined by a window of energies
Different strategies have been put forward to implement these RG ideas out
of equilibrium
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
D
D
D - δD
D
D
D - δD
Yields and RG-type differential equation for J(ω = 0)
Rosch et al., 2003
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
D
D
D - δD
D
D
D - δD
Yields and RG-type differential equation for J(ω = 0)
The logarithmic singularities at µL and µR are cut off by the effective
spin-flip rate, which is inserted by hand
Rosch et al., 2003
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
Example: The singlet-triplet transition in carbon nanotube quantum dots
Paaske et al., 2006
Avraham Schiller / QIMP11
Nonequilibrium variants of perturbative RG
Example: The singlet-triplet transition in carbon nanotube quantum dots
Paaske et al., 2006
Avraham Schiller / QIMP11
Time-dependent numerical RG
Avraham Schiller / QIMP11
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
Avraham Schiller / QIMP11
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
Vg
Avraham Schiller / QIMP11
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
t <0
Lead
Vg
Avraham Schiller / QIMP11
t >0
Lead
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
Avraham Schiller / QIMP11
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 ρˆ (t )Oˆ = Trace e − iHt ρˆ 0 e iHt Oˆ
Perturbed Hamiltonian
Avraham Schiller / QIMP11
Wilson’s numerical RG
Avraham Schiller / QIMP11
Wilson’s numerical RG
Λ >1
Logarithmic discretization of band:
ε/D
-1
Avraham Schiller / QIMP11
-Λ-1
-Λ-2 -Λ-3
Λ-3 Λ-2
Λ-1
1
Wilson’s numerical RG
Λ >1
Logarithmic discretization of band:
ε/D
-1
-Λ-1
-Λ-2 -Λ-3
Λ-3 Λ-2
Λ-1
1
After a unitary transformation the bath is represented by a semi-infinite
chain
imp
Avraham Schiller / QIMP11
ξ0
ξ1
ξ2
ξ3
Wilson’s numerical RG
Why logarithmic discretization?
Avraham Schiller / QIMP11
Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Avraham Schiller / QIMP11
Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Hopping decays exponentially along the chain:
imp
Avraham Schiller / QIMP11
ξ0
ξ1
ξ2
ξ n ∝ Λ−n / 2 , Λ > 1
ξ3
Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Hopping decays exponentially along the chain:
ξ0
imp
ξ1
ξ2
Separation of energy scales along the chain
Avraham Schiller / QIMP11
ξ n ∝ Λ−n / 2 , Λ > 1
ξ3
Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Hopping decays exponentially along the chain:
ξ0
imp
ξ1
ξ2
ξ n ∝ Λ−n / 2 , Λ > 1
ξ3
Separation of energy scales along the chain
Exponentially small energy scales can be accessed, limited by T only
Avraham Schiller / QIMP11
Wilson’s numerical RG
Why logarithmic discretization?
To properly account for the logarithmic infra-red divergences
Hopping decays exponentially along the chain:
ξ0
imp
ξ1
ξ2
ξ n ∝ Λ−n / 2 , Λ > 1
ξ3
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.
Avraham Schiller / QIMP11
Wilson’s numerical RG
Equilibrium NRG:
Geared towards fine energy resolution at low energies
Discards high-energy states
Avraham Schiller / QIMP11
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
Avraham Schiller / QIMP11
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:
Avraham Schiller / QIMP11
Combine information from all NRG iterations
Time-dependent NRG
imp
ξ0
(Anders & AS, PRL’05, PRB’06)
ξ1
r
NRG eigenstate of relevant iteration
Avraham Schiller / QIMP11
⊗
e
Basis set for the “environment” states
Time-dependent NRG
imp
ξ0
(Anders & AS, PRL’05, PRB’06)
ξ1
r
NRG eigenstate of relevant iteration
⊗
e
Basis set for the “environment” states
For each NRG iteration, we trace over its “environment”
Avraham Schiller / QIMP11
Time-dependent NRG
Sum over all chain lengths
(all energy scales)
N trun
O(t ) = ∑∑ O ρ (m)e
m =1 s , r
Sum over discarded NRG states
of chain of length m
m
s ,r
red
r ,s
i ( E sm − Erm ) t
Reduced density
matrix for the
m-site chain
Matrix element of O
on the m-site chain
ρ (m) = ∑ r , e; m ρ 0 s, e; m
red
r ,s
e
Trace over the environment, i.e., sites
not included in chain of length m
Avraham Schiller / QIMP11
(Hostetter, PRL 2000)
Fermionic benchmark: Resonant-level model
H = ∑ ε k ck+ ck + Ed (t )d + d + V ∑ (ck+ d + d + ck )
k
Avraham Schiller / QIMP11
k
Fermionic benchmark: Resonant-level model
H = ∑ ε k ck+ ck + Ed (t )d + d + V ∑ (ck+ d + d + ck )
k
Ed (t < 0) = 0
Avraham Schiller / QIMP11
k
Ed (t > 0) = Ed1 < 0
Fermionic benchmark: Resonant-level model
H = ∑ ε k ck+ ck + Ed (t )d + d + V ∑ (ck+ d + d + ck )
k
k
Ed (t < 0) = 0
We focus on
nd (t ) = d + d (t )
Ed (t > 0) = Ed1 < 0
and compare the TD-NRG to exact
analytic solution in the wide-band limit (i.e., for an infinite system)
Basic energy scale:
Avraham Schiller / QIMP11
Γ = πρV 2
Fermionic benchmark: Resonant-level model
T=0
Avraham Schiller / QIMP11
Relaxed values
(no runaway!)
Fermionic benchmark: Resonant-level model
T=0
T>0
Avraham Schiller / QIMP11
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
Avraham Schiller / QIMP11
Source of inaccuracies
T=0
Avraham Schiller / QIMP11
Ed (t < 0) = -10Γ
Ed (t > 0) = Γ
Λ= 2
Source of inaccuracies
T=0
Avraham Schiller / QIMP11
Ed (t < 0) = -10Γ
Ed (t > 0) = Γ
Λ= 2
Source of inaccuracies
T=0
Avraham Schiller / QIMP11
Ed (t < 0) = -10Γ
Ed (t > 0) = Γ
Λ= 2
Source of inaccuracies
Ed (t < 0) = -10Γ
T=0
Ed (t > 0) = Γ
TD-NRG is essentially exact on the Wilson chain
Main source of inaccuracies is due to discretization
Avraham Schiller / QIMP11
Λ= 2
Bosonic benchmark: Spin-boson model
H = ∑ ωi bi+ bi −
i
σ
∆
σx + z
2
2
+
λ
(
b
∑ i i + bi )
i
J (ω < ωc ) = π ∑ λi2 δ (ω − ωi ) = 2πα ωcs −1ω s
i
Avraham Schiller / QIMP11
Bosonic benchmark: Spin-boson model
H = ∑ ωi bi+ bi −
i
σ
∆
σx + z
2
2
+
λ
(
b
∑ i i + bi )
i
J (ω < ωc ) = π ∑ λi2 δ (ω − ωi ) = 2πα ωcs −1ω s
i
Setting ∆=0, we start from the pure spin state
ρˆ (t = 0) = σ x = 1 σ x = 1 ⊗ ρˆThermal − Bath
and compute
Avraham Schiller / QIMP11
ρ 01 (t ) = σ z = 1 TrBath {ρˆ (t )} σ z = −1
Bosonic benchmark: Spin-boson model
ρ 01 (t )
Excellent agreement between TD-NRG (full lines) and the
exact analytic solution (dashed lines) up to t ≈ 1 / T
Avraham Schiller / QIMP11
Bosonic benchmark: Spin-boson model
For nonzero ∆ and s = 1 (Ohmic bath), we prepare the system such
that the spin is initially fully polarized (Sz = 1/2)
Avraham Schiller / QIMP11
Bosonic benchmark: Spin-boson model
For nonzero ∆ and s = 1 (Ohmic bath), we prepare the system such
that the spin is initially fully polarized (Sz = 1/2)
Damped oscillations
Avraham Schiller / QIMP11
Bosonic benchmark: Spin-boson model
For nonzero ∆ and s = 1 (Ohmic bath), we prepare the system such
that the spin is initially fully polarized (Sz = 1/2)
Monotonic decay
Avraham Schiller / QIMP11
Bosonic benchmark: Spin-boson model
For nonzero ∆ and s = 1 (Ohmic bath), we prepare the system such
that the spin is initially fully polarized (Sz = 1/2)
Localized phase
Avraham Schiller / QIMP11
Anderson impurity model
σ


H = ∑ ε c ckσ + ∑  Ed (t ) − H (t ) dσ+ dσ
2

σ 
k ,σ
+
k kσ
+ V (t )∑ (ck+σ dσ + dσ+ ckσ ) + Ud ↑+ d ↑ d ↓+ d ↓
k ,σ
t<0
t>0
πρV 2 = Γ0
Avraham Schiller / QIMP11
Ed = −U / 2
πρV 2 = Γ1
Anderson impurity model: Charge relaxation
Exact new
Equilibrium
values
Charge relaxation is governed by tch=1/Γ1
TD-NRG works better for interacting case!
Avraham Schiller / QIMP11
Anderson impurity model: Spin relaxation
t ∗ Γ1
Avraham Schiller / QIMP11
Anderson impurity model: Spin relaxation
t ∗ Γ1
Avraham Schiller / QIMP11
t ∗ TK
Anderson impurity model: Spin relaxation
t ∗ Γ1
t ∗ TK
Spin relaxes on a much longer time scale tsp >> tch
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
Avraham Schiller / QIMP11
On-going projects:
Eliminating discretization errors
Extending approach to multiple switching events
Avraham Schiller / QIMP11
New hybrid
approach