Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity...

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)

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)

Coulomb blockade in ultra-small quantum dots

Quantum dot

Lead Lead

ULead Lead

Conductance vs gate voltage

ULead Lead

e2 /h)

The Kondo effect in ultra-small quantum dots

RLdimp dtnUnnH

H.c.)0(

RLdimp dtnUnnH

H.c.)0(

Tunneling to leads

RLdimp dtnUnnH

H.c.)0(

Inter-configurational energies d and U+d

RLdimp dtnUnnH

H.c.)0(

RLdimp dtnUnnH

H.c.)0(

RLdimp dtnUnnH

H.c.)0(

Hybridization width )( 22RL tt

RLdimp dtnUnnH

H.c.)0(

RLdimp dtnUnnH

H.c.)0(

dd U ,Condition for formation of local moment:

RLdimp dtnUnnH

H.c.)0(

dd U ,Condition for formation of local moment:

RLdimp dtnUnnH

H.c.)0(

EFd d+U

RLdimp dtnUnnH

H.c.)0(

EFd d+U

A sharp resonance of width TK

develops at EF when T<TK

RLdimp dtnUnnH

H.c.)0(

EFd d+U

Abrikosov-Suhl resonance

RLdimp dtnUnnH

H.c.)0(

EFd d+U

Unitary scattering for T=0 and <n>=1

RLdimp dtnUnnH

H.c.)0(

EFd d+U

)(||exp

Nonperturbative scale:

RLdimp dtnUnnH

H.c.)0(

EFd d+U

)(||exp

Nonperturbative scale:

Perfect transmission for

symmetric structure

RLdimp dtnUnnH

H.c.)0(

Electronic correlations out of equilibrium

Electronic correlations out of equilibrium dI

Differential conductance intwo-terminal devices

Steady state

van der Wiel et al.,Science 2000

Steady state ac drive

Photon-assisted side peaks

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

Steady state ac drive

Photon-assisted side peaks

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

Nonequilibrium: A theoretical challenge

The Goal: The description of nano-structures at nonzero bias

and/or nonzero driving fields

Required: Inherently nonperturbative treatment of nonequilibrium

Problem: Unlike equilibrium conditions, density operator is not

known in the presence of interactions

Problem: Unlike equilibrium conditions, density operator is not

Most nonperturbative approaches available in equilibrium

known in the presence of interactions

are simply inadequate

Two possible strategies

Work directly atsteady state

e.g., construct the many-particle Scattering states

Evolve the system in time to reach steady

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

OeeOtO iHtiHt

ˆˆTraceˆ)(ˆTraceˆ0

Perturbed Hamiltonian

Initial density operator

Wilson’s numerical RG

-1 1--1 --2 --3 -1-2-3

Logarithmic discretization of band: 1

-1 1--1 --2 --3 -1-2-3

Logarithmic discretization of band: 1

After a unitary transformation the bath is represented by a semi-infinitechain

Why logarithmic discretization?

To properly account for the logarithmic infra-red divergences

Hopping decays exponentially along the chain: 1,2/ nn

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 chainby one site at a time. High-energy states are discarded at each step,refining the resolution as energy is decreased.

Exponentially small energy scales can be accessed, limited by T only

Equilibrium NRG:

Geared towards fine energy resolution at low energies

Discards high-energy states

Equilibrium NRG:

Problem: Real-time dynamics involves all energy scales

Equilibrium NRG:

Problem: Real-time dynamics involves all energy scales

Resolution: Combine information from all NRG iterations

Time-dependent NRG

Basis set for the “environment” statesNRG eigenstate of relevant iteration

Time-dependent NRG

Basis set for the “environment” statesNRG eigenstate of relevant iteration

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

Time-dependent NRG

tEEisr

msemOtO

)(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

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

Fermionic benchmark: Resonant-level model

kkk cddcVddtEccH )()(

0)0( tEd 0)0( 1 dd EtE

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

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

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

Source of inaccuracies

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

TD-NRG is essentially exact on the Wilson chain

Main source of inaccuracies is due to discretization

Analysis of discretization effects

Ed (t < 0) = -10Ed (t > 0) =

Analysis of discretization effects

Ed (t < 0) = -10Ed (t > 0) =

Ed (t < 0) = Ed (t > 0) = -10

Bosonic benchmark: Spin-boson model

iiii bbbbH )(

iiicJ 12 2)()(

iiii bbbbH )(

iiicJ 12 2)()(

Setting =0, we start from the pure spin state

BathThermalxxt ˆ11)0(ˆ

and compute 1)(ˆ1)(01 zBathz tTrt

)(01 t

Excellent agreement between TD-NRG (full lines) and theexact analytic solution (dashed lines) up to Tt /1

For nonzero and s = 1 (Ohmic bath), we prepare the system suchthat the spin is initially fully polarized (Sz = 1/2)

Damped oscillations

Monotonic decay

Localized phase

Anderson impurity model

ddtHtEccH dk

kkk )(2

dddUdcddctVk

Anderson impurity model: Charge relaxation

Charge relaxation is governed by tch=1/1

TD-NRG works better for interacting case!

Exact newEquilibrium

values

Anderson impurity model: Spin relaxation

KTt 1t

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

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

Avraham Schiller / Seattle 09 equilibrium: Real-time dynamics Avraham Schiller Quantum impurity...

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