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



Leads



Lead Lead



ULead Lead



ULead Lead


i+U

U


ULead Lead



ULead Lead

Conductance vs gate voltage



ULead Lead

Conductance vs gate voltage

dI/d

V (

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(



Hybridization width )( 22RL tt


RLdimp dtnUnnH

,

H.c.)0(



dd U ,Condition for formation of local moment:

Hybridization width )( 22RL tt



RLdimp dtnUnnH

,

H.c.)0(



EFd d+U

TK

RLdimp dtnUnnH

,

H.c.)0(



EFd d+U

TK

A sharp resonance of width TK

develops at EF when T<TK

RLdimp dtnUnnH

,

H.c.)0(



EFd d+U

Abrikosov-Suhl resonance

TK



RLdimp dtnUnnH

,

H.c.)0(



EFd d+U

TK



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

RLdimp dtnUnnH

,

H.c.)0(



EFd d+U

TK




U

UT dd

K 2

)(||exp

Nonperturbative scale:

RLdimp dtnUnnH

,

H.c.)0(



EFd d+U

TK




U

UT dd

K 2

)(||exp

Nonperturbative scale:

Perfect transmission for

symmetric structure

RLdimp dtnUnnH

,

H.c.)0(


Electronic correlations out of equilibrium


Electronic correlations out of equilibrium dI

/dV

(e2 /

h)

Differential conductance intwo-terminal devices

Steady state

van der Wiel et al.,Science 2000


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


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

state


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

LeadLead

Vg

t < 0




OeeOtO iHtiHt

t

ˆˆTraceˆ)(ˆTraceˆ0

0

Perturbed Hamiltonian

Initial density operator


Wilson’s numerical RG



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

/D

Logarithmic discretization of band: 1



-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


Why logarithmic discretization?





To properly account for the logarithmic infra-red divergences





imp

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




imp


Separation of energy scales along the chain





imp


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






imp


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

imp

r e

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


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”


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


Fermionic benchmark: Resonant-level model

k

kkdk

kkk cddcVddtEccH )()(



k

kkdk


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



k

kkdk


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



T = 0

Relaxed values(no runaway!)



T = 0

T > 0




T = 0

T > 0


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

Source of inaccuracies

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

zx

iiii bbbbH )(

22

ssc

iiicJ 12 2)()(



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



)(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

kk

,

)()(

t < 0

02 V

t > 0

12 V

2/UEd


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

1t



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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