Far-from-Equilibrium and Time-Dependent Phenomena: Theory Avraham - - PowerPoint PPT Presentation

Avraham Schiller / QIMP11

Time-Dependent Phenomena:

Avraham Schiller

Far-from-Equilibrium and

Racah Institute of Physics, The Hebrew University

Theory

Avraham Schiller / QIMP11

Correlated systems

• ut of equilibrium

Avraham Schiller / QIMP11

Correlated systems

• ut of equilibrium

Femtosecond spectroscopy Femtosecond spectroscopy

Avraham Schiller / QIMP11

Correlated systems

• ut of equilibrium

Femtosecond spectroscopy Bulk materials particles

Avraham Schiller / QIMP11

Correlated systems

• ut of equilibrium

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics

Avraham Schiller / QIMP11

Correlated systems

• ut of equilibrium

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles

Correlated systems

• ut of equilibrium

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles Quench dynamics Thermalization?

Correlated systems

• ut of equilibrium

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles Quench dynamics Thermalization?

Correlated systems

• ut of equilibrium

Avraham Schiller / QIMP11

t = 0 t = 100µs t = 150µs t = 250µs t = 350µs t = 400µs t = 550µs

Collapse and revival of coherent matter waves

• M. Greiner et al., Nature 2002

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles Quench dynamics Thermalization?

Correlated systems

• ut of equilibrium

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles Quench dynamics

Correlated systems

• ut of equilibrium

Thermalization?

Avraham Schiller / QIMP11

• T. Kinoshita et al., Nature 440, 900 (2006).

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 distribution

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles Quench dynamics Thermalization?

Correlated systems

• ut of equilibrium

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles Quench dynamics Thermalization?

Correlated systems

• ut of equilibrium

Confined nanostructures

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles Quench dynamics Thermalization?

Correlated systems

• ut of equilibrium

Confined nanostructures Few interacting degrees of freedom coupled to environment

Avraham Schiller / QIMP11

Femtosecond spectroscopy Bulk materials particles Relaxation dynamics Cold atoms particles Quench dynamics Thermalization?

Correlated systems

• ut of equilibrium

Confined nanostructures Few interacting degrees of freedom coupled to environment Steady state/quench dynamics

Avraham Schiller / QIMP11

Correlated systems

• ut 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

Quantum dot

Kondo effect in ultra-small quantum dots

Avraham Schiller / QIMP11

Leads

Kondo effect in ultra-small quantum dots

Avraham Schiller / QIMP11

Lead Lead

Kondo effect in ultra-small quantum dots

Avraham Schiller / QIMP11

Lead Lead

Kondo effect in ultra-small quantum dots

Avraham Schiller / QIMP11

U Lead Lead

Kondo effect in ultra-small quantum dots

Avraham Schiller / QIMP11

U Lead Lead

Conductance vs gate voltage

Kondo effect in ultra-small quantum dots

Avraham Schiller / QIMP11

U Lead Lead

Conductance vs gate voltage

Kondo effect in ultra-small quantum dots

Avraham Schiller / QIMP11

U Lead Lead

Conductance vs gate voltage

dI/dV (e2/h)

Kondo effect in ultra-small quantum dots

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

dI/dV (e2/h)

Differential conductance in two-terminal devices

Steady state

van der Wiel et al.,Science 2000

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

dI/dV (e2/h)

Differential conductance in two-terminal devices

Steady state ac drive

Photon-assisted side peaks

Kogan et al.,Science 2004 van der Wiel et al.,Science 2000

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

dI/dV (e2/h)

Differential conductance in two-terminal devices

Steady state ac drive

Photon-assisted side peaks

Kogan et al.,Science 2004 van der Wiel et al.,Science 2000

+hω −hω

Avraham Schiller / QIMP11

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) (b)

Elzerman et al., Nature (2005)

Abrupt change

• f gate voltage

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

Response to pulsed bias

(a) (b)

Elzerman et al., Nature (2005)

Abrupt change

• f gate voltage

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

Response to pulsed bias

(a) (b)

Elzerman et al., Nature (2005)

Abrupt change

• f gate voltage

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

Response to pulsed bias

(a) (b) (c)

Elzerman et al., Nature (2005)

Abrupt change

• f gate voltage

Abrupt change

• f gate voltage

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

Response to pulsed bias

(a) (b) (c)

Elzerman et al., Nature (2005)

Abrupt change

• f gate voltage

Abrupt change

• f gate voltage

Avraham Schiller / QIMP11

Electronic correlations out of equilibrium

Response to pulsed bias

Elzerman et al., Nature (2005)

Single-shot readout of the spin state of a quantum dot from real-time dynamics

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, Required: Inherently nonperturbative treatment of nonequilibrium nonzero driving fields, and/or quench dynamics

Avraham Schiller / QIMP11

Nonequilibrium: A theoretical challenge

The Goal: The description of nanostructures at nonzero bias, Required: Inherently nonperturbative treatment of nonequilibrium Problem: Unlike equilibrium conditions, density operator is not known in the presence of interactions nonzero driving fields, and/or quench dynamics

Avraham Schiller / QIMP11

Nonequilibrium: A theoretical challenge

The Goal: The description of nanostructures at nonzero bias, Required: Inherently nonperturbative treatment of nonequilibrium Problem: Unlike equilibrium conditions, density operator is not Most nonperturbative approaches available in equilibrium known in the presence of interactions are simply inadequate nonzero driving fields, and/or quench dynamics

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: e.g., by constructing the many- particle scattering states 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: e.g., by constructing the many- particle scattering states Time-dependent 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: e.g., by constructing the many- particle scattering states Time-dependent formulation Work directly at steady state by imposing suitable boundary conditions Evolve the system in time to reach steady state

Avraham Schiller / QIMP11

Nonequilibrium: A theoretical challenge

Two possible strategies to treat steady state

Time-independent formulation: e.g., by constructing the many- particle scattering states Time-dependent formulation Work directly at steady state by imposing suitable boundary conditions Evolve the system in time to reach steady state

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 Scattering Bethe ansatz

(Andrei et al.)

Keldysh diagrammatics

Avraham Schiller / QIMP11

Brief division of theoretical approaches

Steady state Scattering Bethe ansatz

(Andrei et al.)

Keldysh diagrammatics 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 Scattering Bethe ansatz

(Andrei et al.)

Keldysh diagrammatics 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

Evolution in time

Brief division of theoretical approaches

Avraham Schiller / QIMP11

Evolution in time

Brief division of theoretical approaches

Keldysh diagrammatics

Avraham Schiller / QIMP11

Evolution in time Time-dependent DMRG

(White, Schollwoeck,…)

Brief division of theoretical approaches

Keldysh diagrammatics

Avraham Schiller / QIMP11

Evolution in time Time-dependent DMRG

(White, Schollwoeck,…)

Keldysh Quantum Monte Carlo

(Werner, Muehlbacher,…)

Brief division of theoretical approaches

Keldysh diagrammatics

Avraham Schiller / QIMP11

Evolution in time 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.)

Brief division of theoretical approaches

Keldysh diagrammatics

Avraham Schiller / QIMP11

Evolution in time 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.)

Brief division of theoretical approaches

Keldysh diagrammatics Time-dependent NRG

(Anders & AS)

Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Incoming state: decoupled leads Scattered state: entangled system

Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Incoming state: decoupled leads Scattered state: entangled system Divide H into H0+H1, where and Η1 contains all terms that drive the system out of equilibrium.

φ φ | | E H =

Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation Incoming state: decoupled leads Scattered state: entangled system Divide H into H0+H1, where and Η1 contains all terms that drive the system out of equilibrium.

φ φ | | E H =

Assume approach to steady state

Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation

∫

∞ − − →

+

=

) (

| lim | φ η ψ

η η t E H i t e

e dt

Because of the approach to steady state, one can “smear” the initial time:

Gell-Man and Goldberger, 1953

Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation

∫

∞ − − →

+

=

) (

| lim | φ η ψ

η η t E H i t e

e dt

φ η φ ψ | ) ( 1 | | E H i H E − + − + = Because of the approach to steady state, one can “smear” the initial time:

Gell-Man and Goldberger, 1953

Avraham Schiller / QIMP11

Time-independent formulation: the Lippmann-Schwinger equation

∫

∞ − − →

+

=

) (

| lim | φ η ψ

η η t E H i t e

e dt

φ η φ ψ | ) ( 1 | | E H i H E − + − + = Because of the approach to steady state, one can “smear” the initial time: Since , we arrive at the Lippmann-Schwinger equation

| ) ( = − φ E H

φ η φ ψ | 1 | |

1

H i H E + − + =

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 , where pi are typically equilibrium Boltzmann factors, one formally has that

∑

=

i i i i

p | | ˆ0 φ φ ρ

Avraham Schiller / QIMP11

The nonequilibrium steady-state density operator

Starting from , where pi are typically equilibrium Boltzmann factors, one formally has that

∑

=

i i i i

p | | ˆ0 φ φ ρ

∑

= →

i i i i

p | | ˆ ˆ0 ψ ψ ρ ρ

with

i i i i

H i H E φ η φ ψ | 1 | |

1

+ − + =

Avraham Schiller / QIMP11

The nonequilibrium steady-state density operator

Starting from , where pi are typically equilibrium Boltzmann factors, one formally has that

∑

=

i i i i

p | | ˆ0 φ φ ρ

∑

= →

i i i i

p | | ˆ ˆ0 ψ ψ ρ ρ

with

i i i i

H i H E φ η φ ψ | 1 | |

1

+ − + =

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

{ }

) ( ) (

ˆ

Y H Y H

e e

− − − −

=

β β

ρ Trace

with

[ ]

, = Y H

In practice, the initial density matrix generically takes the form

Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006

Avraham Schiller / QIMP11

Hershfield’s mapping onto equilibrium form

{ }

) ( ) (

ˆ

Y H Y H

e e

− − − −

=

β β

ρ Trace

with

[ ]

, = Y H

In practice, the initial density matrix generically takes the form

Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006

Indeed, in many cases one takes

∑ ∑

= +

+ =

R L k k k k

c c H

, ,

) (

α σ σ α σ α α α

µ ε

∑ ∑

= +

=

R L k k k c

c Y

, , α σ σ α σ α α

µ

Avraham Schiller / QIMP11

Hershfield’s mapping onto equilibrium form

{ }

) ( ) (

ˆ

Y H Y H

e e

− − − −

=

β β

ρ Trace

with

[ ]

) ( , → − = Y Y i H Y η

Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006

Avraham Schiller / QIMP11

Hershfield’s mapping onto equilibrium form

{ }

) ( ) (

ˆ

Y H Y H

e e

− − − −

=

β β

ρ Trace

with

[ ]

) ( , → − = Y Y i H Y η

Zubarev, 1960’s, Hershfield 1993, Doyon & Andrei 2006

The steady-state density operator takes an equilibrium- like form!

Avraham Schiller / QIMP11

Generalized fermionic scattering states

with The steady-state density operator can be represented in terms of generalized fermionic scattering states:

[ ]

) ( ,

+ + + +

− + − =

σ α σ α σ α α σ α

ψ η ψ ε ψ

k k k k k

c i H

Hershfield 1993, Han 2007

∑ ∑

= +

= −

R L k k k k

Y H

, , α σ σ α σ α α

ψ ψ ε

Avraham Schiller / QIMP11

with The steady-state density operator can be represented in terms of generalized fermionic scattering states:

[ ]

) ( ,

+ + + +

− + − =

σ α σ α σ α α σ α

ψ η ψ ε ψ

k k k k k

c i H

Hershfield 1993, Han 2007

∑ ∑

= +

= −

R L k k k k

Y H

, , α σ σ α σ α α

ψ ψ ε

In general is a complicated many-body operator:

 + + + =

∑ ∑

+ + + + + k j i l j i k ijl i i k i k k

c c c B c A c

, , α α σ α σ α

ψ

+ σ α

ψ

k

Generalized fermionic scattering states

Avraham Schiller / QIMP11

In the absence of interactions ψαkσ reduce to the familiar single-particle scattering states

∑

+ + +

+ =

i i k i k k

c A c

α σ α σ α

ψ

and one recovers the Landauer-Buttiker formulation

Hershfield 1993

Generalized fermionic scattering states

Avraham Schiller / QIMP11

Time-dependent formulation

Avraham Schiller / QIMP11

{ }

) , ( ˆ ) , ( ˆ ) ( ˆ t t U A t t U t A

+

= ρ Trace

Starting from at time t0, expectation values are explicitly propagated in time:

∑

=

i i i i

p | | ˆ0 φ φ ρ

Time-dependent formulation

Avraham Schiller / QIMP11

{ }

) , ( ˆ ) , ( ˆ ) ( ˆ t t U A t t U t A

+

= ρ Trace

Starting from at time t0, expectation values are explicitly propagated in time:

∑

=

i i i i

p | | ˆ0 φ φ ρ

t A(t) Steady state value Recurrence

Time-dependent formulation

Avraham Schiller / QIMP11

Time-dependent formulation

Avraham Schiller / QIMP11

Time-dependent formulation

Avraham Schiller / QIMP11

A challenge when the system features small energy Scales such as the Kondo temperature!

Time-dependent formulation

Avraham Schiller / QIMP11

Selected review of theoretical approaches

Scattering Bethe ansatz Time-dependent NRG Nonequilbrium variants of perturbative RG

Avraham Schiller / QIMP11

Selected review of theoretical approaches

Scattering Bethe ansatz Keldysh Quantum Monte Carlo Time-dependent NRG Will not addressed: Keldysh-based approaches (Hans Kroha’s talk) Theories based on Fermi-liquid theory (weak nonequilibrium) Time-dependent DMRG Nonequilbrium variants of perturbative RG

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

One uses the Bethe ansatz approach, adjusted to open boundary conditions, to explicitly construct the many-particle scattering states

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

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

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

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

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

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

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

General form of N-electron wave function:

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005 Impurity states

General form of N-electron wave function:

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

General form of N-electron wave function:

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

General form of N-electron wave function: where F obeys the Schroedinger-type equation

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

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

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

The Bethe ansatz then seeks solutions of the form

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

The Bethe ansatz then seeks solutions of the form

Projection onto the ordering affiliated with P

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

The Bethe ansatz then seeks solutions of the form

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

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

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

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

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

The main sources of difficulty:

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

The main sources of difficulty:

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

The main sources of difficulty: Expectation values of the current operators are extremely difficult to evaluate with respect to the Bethe ansatz wave function

Avraham Schiller / QIMP11

Scattering Bethe ansatz

Mehta & Andrei, 2005

The main sources of difficulty: 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 Expectation values of the current operators are extremely difficult to evaluate with respect to the Bethe ansatz wave function

Avraham Schiller / QIMP11

Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

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

• f Hamiltonians for each energy scale

Interactions correspond to irreducible vertices, whose evolution

• ne tracks

Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

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

• f Hamiltonians for each energy scale

Interactions correspond to irreducible vertices, whose evolution

• ne tracks

Generic example in equilibrium – the Kondo model

D D D - δD D - δD

Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

+ + … = δ J Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

+ + … = δ J

In equilibrium only J(ω = 0) is important for thermodynamics

Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

+ + … = δ J

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

Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

+ + … = δ J

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

• f equilibrium

Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

Rosch et al., 2003 D D D - δD D - δD D D Yields and RG-type differential equation for J(ω = 0)

Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

Rosch et al., 2003 D D D - δD D - δD D D 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

Nonequilibrium variants of perturbative RG

Avraham Schiller / QIMP11

Nonequilibrium variants of perturbative RG

Paaske et al., 2006 Example: The singlet-triplet transition in carbon nanotube quantum dots

Avraham Schiller / QIMP11

Nonequilibrium variants of perturbative RG

Paaske et al., 2006 Example: The singlet-triplet transition in carbon nanotube quantum dots

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

Lead Lead

Vg

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

Lead Lead

Vg

t > 0

Lead Lead

Vg

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

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

{ } { }

O e e O t O

iHt iHt t

ˆ ˆ Trace ˆ ) ( ˆ Trace ˆ ρ ρ

− >

= =

Perturbed Hamiltonian Initial density operator

Avraham Schiller / QIMP11

Wilson’s numerical RG

Avraham Schiller / QIMP11

Wilson’s numerical RG

• 1

1

• Λ-1
• Λ-2 -Λ-3

Λ-1 Λ-2 Λ-3

ε/D Logarithmic discretization of band:

1 > Λ

Avraham Schiller / QIMP11

Wilson’s numerical RG

• 1

1

• Λ-1
• Λ-2 -Λ-3

Λ-1 Λ-2 Λ-3

ε/D Logarithmic discretization of band:

1 > Λ

ξ0 ξ1 ξ2 ξ3

imp After a unitary transformation the bath is represented by a semi-infinite chain

Avraham Schiller / QIMP11

Why logarithmic discretization?

Wilson’s numerical RG

Avraham Schiller / QIMP11

Why logarithmic discretization?

Wilson’s numerical RG

To properly account for the logarithmic infra-red divergences

Avraham Schiller / QIMP11

Why logarithmic discretization?

Wilson’s numerical RG

To properly account for the logarithmic infra-red divergences

ξ0 ξ1 ξ2 ξ3

imp Hopping decays exponentially along the chain:

1 ,

2 /

> Λ Λ ∝

−n n

ξ

Avraham Schiller / QIMP11

Why logarithmic discretization?

Wilson’s numerical RG

ξ0 ξ1 ξ2 ξ3

imp Hopping decays exponentially along the chain:

1 ,

2 /

> Λ Λ ∝

−n n

ξ

Separation of energy scales along the chain To properly account for the logarithmic infra-red divergences

Avraham Schiller / QIMP11

Why logarithmic discretization?

Wilson’s numerical RG

ξ0 ξ1 ξ2 ξ3

imp Hopping decays exponentially along the chain:

1 ,

2 /

> Λ Λ ∝

−n n

ξ

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 / QIMP11

Why logarithmic discretization?

Wilson’s numerical RG

ξ0 ξ1 ξ2 ξ3

imp Hopping decays exponentially along the chain:

1 ,

2 /

> Λ Λ ∝

−n n

ξ

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. 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 / QIMP11

Equilibrium NRG: Geared towards fine energy resolution at low energies Discards high-energy states

Wilson’s numerical RG

Avraham Schiller / QIMP11

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 / QIMP11

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 / QIMP11

Time-dependent NRG ξ0 ξ1

imp

⊗

r e

Basis set for the “environment” states NRG eigenstate of relevant iteration (Anders & AS, PRL’05, PRB’06)

Avraham Schiller / QIMP11

Time-dependent NRG ξ0 ξ1

imp

⊗

r e

Basis set for the “environment” states NRG eigenstate of relevant iteration

For each NRG iteration, we trace over its “environment”

(Anders & AS, PRL’05, PRB’06)

Avraham Schiller / QIMP11

Time-dependent NRG

∑∑

= −

=

N m trun r s t E E i s r m r s

m r m s

e m O t O

1 , ) ( red , ,

) ( ) ( ρ

Sum over discarded NRG states

• f chain of length m

Matrix element of O

• n the m-site chain

Reduced density matrix for the m-site chain

∑

=

e s r

m e s m e r m ; , ; , ) (

red ,

ρ ρ

(Hostetter, PRL 2000) Sum over all chain lengths (all energy scales) Trace over the environment, i.e., sites not included in chain of length m

Avraham Schiller / QIMP11

Fermionic benchmark: Resonant-level model

∑ ∑

+ + + +

+ + + =

k k k d k k k k

c d d c V d d t E c c H ) ( ) ( ε

Avraham Schiller / QIMP11

Fermionic benchmark: Resonant-level model

∑ ∑

+ + + +

+ + + =

k k k d k k k k

c d d c V d d t E c c H ) ( ) ( ε

) ( = < t Ed ) (

1 <

= >

d d

E t E

Avraham Schiller / QIMP11

Fermionic benchmark: Resonant-level model

∑ ∑

+ + + +

+ + + =

k k k d k k k k

c d d c V d d t E c c H ) ( ) ( ε

) ( = < t Ed

We focus on

) ( ) ( t d d t nd

+

=

and compare the TD-NRG to exact analytic solution in the wide-band limit (i.e., for an infinite system) Basic energy scale:

2

V πρ = Γ

) (

1 <

= >

d d

E t E

Avraham Schiller / QIMP11

Fermionic benchmark: Resonant-level model T = 0

Relaxed values (no runaway!)

Avraham Schiller / QIMP11

Fermionic benchmark: Resonant-level model T = 0 T > 0

Relaxed values (no runaway!)

Avraham Schiller / QIMP11

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 value is a measure for the accuracy of the TD-NRG on all time scales For T > 0, the TD-NRG works well up to

T t / 1 ≈

Avraham Schiller / QIMP11

T = 0 Ed (t < 0) = -10Γ Ed (t > 0) = Γ Λ= 2 Source of inaccuracies

Avraham Schiller / QIMP11

T = 0 Ed (t < 0) = -10Γ Ed (t > 0) = Γ Λ= 2 Source of inaccuracies

Avraham Schiller / QIMP11

T = 0 Ed (t < 0) = -10Γ Ed (t > 0) = Γ Λ= 2 Source of inaccuracies

Avraham Schiller / QIMP11

T = 0 Ed (t < 0) = -10Γ Ed (t > 0) = Γ Λ= 2

TD-NRG is essentially exact on the Wilson chain

Source of inaccuracies

Main source of inaccuracies is due to discretization

Avraham Schiller / QIMP11

Bosonic benchmark: Spin-boson model

∑ ∑

+ + ∆ − =

+ + i i i i z x i i i i

b b b b H ) ( 2 2 λ σ σ ω

s s c i i i c

J ω ω πα ω ω δ λ π ω ω

1 2

2 ) ( ) (

−

= − = <

∑

Avraham Schiller / QIMP11

Bosonic benchmark: Spin-boson model

∑ ∑

+ + ∆ − =

+ + i i i i z x i i i i

b b b b H ) ( 2 2 λ σ σ ω

s s c i i i c

J ω ω πα ω ω δ λ π ω ω

1 2

2 ) ( ) (

−

= − = <

∑

Setting ∆=0, we start from the pure spin state

Bath Thermal x x

t

−

⊗ = = = = ρ σ σ ρ ˆ 1 1 ) ( ˆ

and compute

{ }

1 ) ( ˆ 1 ) (

01

− = = =

z Bath z

t Tr t σ ρ σ ρ

Avraham Schiller / QIMP11

Bosonic benchmark: Spin-boson model

) (

01 t

ρ

Excellent agreement between TD-NRG (full lines) and the exact analytic solution (dashed lines) up to

T t / 1 ≈

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 σ σ σ σ σ

σ ε d d t H t E c c H

d k k k k

) ( 2 ) (

,

↓ + ↓ ↑ + ↑ + +

+ + +

∑

d d d Ud c d d c t V

k k k σ σ σ σ σ ,

) ( ) ( t < 0

2

Γ = V πρ

t > 0

1 2

Γ = V πρ 2 / U Ed − =

Avraham Schiller / QIMP11

Anderson impurity model: Charge relaxation

Charge relaxation is governed by tch=1/Γ1 TD-NRG works better for interacting case! Exact new Equilibrium values

Avraham Schiller / QIMP11

Anderson impurity model: Spin relaxation

1

Γ ∗ t

Avraham Schiller / QIMP11

Anderson impurity model: Spin relaxation

K

T t ∗

1

Γ ∗ t

Avraham Schiller / QIMP11

Anderson impurity model: Spin relaxation

Spin relaxes on a much longer time scale Spin relaxation is sensitive to initial conditions!

ch sp

t t >>

Starting from a decoupled impurity, spin relaxation approaches a universal function of t/tsp with tsp=1/TK

K

T t ∗

1

Γ ∗ t

Avraham Schiller / QIMP11

On-going projects:

Eliminating discretization errors Extending approach to multiple switching events

New hybrid approach