Real-time dynamics in Quantum Impurity Systems: A Time-dependent - - PowerPoint PPT Presentation

▶
real time dynamics in quantum impurity systems a time
SMART_READER_LITE
LIVE PREVIEW

Real-time dynamics in Quantum Impurity Systems: A Time-dependent - - PowerPoint PPT Presentation

Aug 10, 2023 348 likes •626 views

Real-time dynamics in Quantum Impurity Systems: A Time-dependent Numerical Renormalization Group Approach Frithjof B Anders Institut fr theoretische Physik, Universitt Bremen Concepts in Electron Correlation, Hvar, 30. September 2005

slide-1
SLIDE 1

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Real-time dynamics in Quantum Impurity Systems: A Time-dependent Numerical Renormalization Group Approach

Frithjof B Anders

Institut für theoretische Physik, Universität Bremen

Concepts in Electron Correlation, Hvar, 30. September 2005 Collaborator:

  • A. Schiller, Hebrew University, Jerusalem, Israel
  • R. Bulla, S. Tornow, University of Augsburg, Germany
  • M. Vojta, University of Karlsruhe, Germany

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.1/18

slide-2
SLIDE 2

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

What is a Quantum-Impurity System (QIS)?

Quantum Impurity |α>

(metallic) host

|γ>

bosonic bath

quantum-impurity: embedded in a (metallic) host interacting with the environment of non-interacting particles (Bosons/Fermions) Problem: infrared divergence due to local degeneracy

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.2/18

slide-3
SLIDE 3

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

What is a Quantum-Impurity System (QIS)?

Quantum Impurity |α>

(metallic) host

|γ>

bosonic bath

Examples: transition metal ion Cu, Mn, Ce in a metal two-level system (Qubit) in a bosonic bath Quantum dot coupled to leads donor-acceptor centers of a large bio-molecule · · ·

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.2/18

slide-4
SLIDE 4

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Goal of the Talk

Our new Approach to Non-Equilibrium of QIS: based on the non-perturbative NRG uses the complete basis of the many body Fock space takes into account all energy scales describes short and long time scales does not accummulate an error ∝ t as the TD-DMRG ☞ breakthrough in the description of real time dynamics of non-equilibrium quantum systems:

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.3/18

slide-5
SLIDE 5

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Contents

  • 1. Introduction

Modelling of quantum dots Charge transfer in molecules (spin-boson model)

  • 2. Non-equilibrium dynamics

Time evolution of quantum systems New approach to quantum impurity problems

  • 3. Results

Dissipation and decoherence in a two level system Spin- and charge dynamics in ulta-small quantum dots AF-Kondo model spin precession

  • 4. Summary and outlook

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.4/18

slide-6
SLIDE 6

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Modelling of a Quantum Dot

H =

  • kσ

ǫkσc†

kσckσ

+

  • σ

[Ed(t) − σH(t)] d†

σdσ + Und ↑nd ↓

+

  • kσ

V (t)

  • c†

kσdσ + d† σckσ

  • Single Impurity Anderson Model (SIAM)

charge fluctuation scale: Γi = V 2

i πρF

infrared problem ☞ low temperature scale: TK ∝ exp(−πU/8Γ)

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.5/18

slide-7
SLIDE 7

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Spin-Boson Model

qubit plus environment (Unruh) electron transfer in (bio)-molecules (Marcus, Schulten) | ↑ = |A, | ↓ = |D H = ǫσz − ∆ 2 σx +

  • q

ωqb†

qbq + σz

  • q

Mq

  • b†

q + bq

  • J(ω)

=

  • q

|Mq|2δ(ω − ωq) ∝ 2παω1−s

c

ωs

Leggett et. al. (RMP 1987), Xu and Schulten 1994, Bulla et. al. (2003) · · ·

Questions: influence of the bosonic spectrum J(ω) on the real time dynamics critical slowdown of the charge transfer process for large coupling

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.6/18

slide-8
SLIDE 8

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Where do we stand in the description of non-equilibrium, dissipation and decoherence in quantum systems?

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.7/18

slide-9
SLIDE 9

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Non-Equilibrium Dynamics of Quantum Systems

quantum dynamics single quantum state: Schrödinger equation i∂t|ψ >= H(t)|ψ > ensemble: density operator i ∂tˆ ρ(t) = [H, ˆ ρ] ; ρ(t) = e−iHt/ρ0eiHt/ finite size quantum system: only unitary dynamics, no dissipation dissipation and decoherence: infinitly large environment needed Size of Subsystem Size of environment − →

Environment

Subsystem

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.8/18

slide-10
SLIDE 10

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

NRG Approach to Quantum Impurity Problems

H = Himp + Hbath + Himp−bath

  • 1. discretizing the bath Hamiltonian on a logarithmic energy mesh

(Wilson 1975,Oliveira )

Λ −Λ −Λ −Λ

−z −(z+1)

Λ Λ

−(z+1) −z

  • 2. mapping onto a semi-infinite chain

R m,N Hm

tm−1 tm+1 tN−1 t1 t0 impurity

  • 3. diagonalizing the Hamiltonian HN+1 using the recursion

HN+1 = √ ΛHN +

  • α

ξNα

  • f †

N+1αfNα + f † NαfN+1α

  • 4. truncate the basis set, go back to step 3

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.9/18

slide-11
SLIDE 11

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Novel Many-Body Approach to NEQ of QIS

R m,N Hm

tm−1 tm+1 tN−1 t1 t0 impurity

use the NRG to generate a complete basis |l, e; m Hm|l = Em

l |l, l eliminated state

e ∈ Rm,N 1 =

  • m
  • l,e

|l, e; ml, e; m| Puls at t = 0: H(t) = HiΘ(−t) + HfΘ(t)

  • perator ˆ

O: property of the subsystem S

Environment

Subsystem

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.10/18

slide-12
SLIDE 12

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Novel Many-Body Approach to NEQ of QIS

time-dependent NRG (TD-NRG)

(FBA, A. Schiller, cond-mat/0505553, PRL 2005)

calculate ρred

NEQ(t)

ˆ O(t) = Tr

  • ρ(t) ˆ

O

  • =
  • m,αα′

α| ˆ O|α′ρred

αα′,m(t)

ρred

αα′,m(t)

= e−i(Eα−Eα′)t

e

α, e; m|ρeq|α′, e; m

Feynman 1972, White 1992, Hofstetter 2000, · · ·

Environment

Subsystem

mimic bath contiuum: use Oliveira’s z-trick evolves towards the new steady state: [H(t > 0), ρ(∞)] = 0 Trace over the environment: dissipation and decoherence!

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.10/18

slide-13
SLIDE 13

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Spin-Boson Model

H = ǫσz − ∆ 2 σx +

  • q

ωqb†

qbq + σz

  • q

Mq

  • b†

q + bq

  • Sx = 1

2 (| ↑↓ | + | ↓↑ |)

0.001 0.01 0.1 1 10

t*T

0.1 0.2 0.3 0.4 0.5

Sx(t)

s=1.5 s=1.5 (ana.) s=1.0 s=0.8 s=0.6 s=0.4 s=0.2 Ns=150, Nz=16, Nb=8, Niter=14, T=0.0078125, Λ=2

1/2, αdamp=0.1, α=0.1, ε=0, ∆ 0=0.,ωc=1

Mon Sep 19 11:46:59 2005

Decoherence QuBit state

1 √ 2 (| ↑ + | ↓)

exact solution

P(t) = e−Γ(t)

Leggett et al., Unruh, Palma et al. , Bulla et al.

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.11/18

slide-14
SLIDE 14

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Spin-Boson Model

H = ǫσz − ∆ 2 σx +

  • q

ωqb†

qbq + σz

  • q

Mq

  • b†

q + bq

  • J(ω) = 2παω1−s

c

ωs

for

0 < ω < ωc ; Ohmic case: s = 1

Fixed point: delocalized

  • verdamped
  • szillatory

c

Toulouse Point:

localized 0<α<1/2 α=1/2

c

α(∆) < α 1/2<α<α(∆)

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.11/18

slide-15
SLIDE 15

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Spin-Boson Model

H = ǫσz − ∆ 2 σx +

  • q

ωqb†

qbq + σz

  • q

Mq

  • b†

q + bq

  • 10
  • 1

10 10

1

10

2

10

3

10

4

10

5

t*ωc

  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4 0.5 Sz(t)

α=0.1 α=0.3 α=0.5 α=0.7 α=1.0 α=1.1 α=1.2 α=1.3 α=1.4 Ns=100, Nb=8, Niter=25, Nz=16, Λ=2, ∆

1=0.2, ε 1=0, α=0.1, ωc=1, s=1, T=3*10

  • 8

10

3

10

4

10

5

10

6

10

7

10

8

10

9

  • 0.2

0.2 0.4

Sz(t)

α=0.1 α=0.3 α=0.5 α=0.7 α=1.0 α=1.1 α=1.2 α=1.3 α=1.4

J(ω) = 2παω1−s

c

ωs

Ohmic Regime: s = 1 QPT at αc(∆) Toulouse point

α = 1/2

  • szillatory α < 1/2
  • verdamped

αc > α > 1/2

localize: α > αc

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.11/18

slide-16
SLIDE 16

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Charge Fluctuation in a Small Quantum Dot

µ Εd

H>0 H=0

time

impurity levels H =

  • kσ

ǫkσc†

kσckσ

+

  • σ

[Ed(t) − σH(t)] d†

σdσ + Und ↑nd ↓

+

  • kσ

V (t)

  • c†

kσdσ + d† σckσ

  • change of Ed: change dynamics

change of mag. field H: spin dynamics change of V : route to new equilibrium

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.12/18

slide-17
SLIDE 17

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Charge Fluctuation in a Small Quantum Dot

µ Εd

H>0 H=0

time

impurity levels

0.6 0.8 1

nd(t)

U/Γ1=2 U/Γ1=4 U/Γ1=6

0.01 0.1 1 10 100

t*Γ1

0.6 0.8 1

nd(t)

U/Γ1=8 U/Γ1=10 U/Γ1=12 U/Γ1=18

(a) Γ0 = Γ1 (b) Γ0 = 0

Charge relaxation time scale : tch = 1/Γ1

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.12/18

slide-18
SLIDE 18

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Spin Fluctuation in a Small Quantum Dot

µ Εd

H>0 H=0

time

impurity levels

0.1 0.2

sz(t)

U/Γ1=8 U/Γ1=10 U/Γ1=12 U/Γ1=18

10

  • 2

10

  • 1

10 10

1

10

2

10

3

t*Γ1

0.1 0.2

sz(t)

U/Γ1=2 U/Γ1=4 U/Γ1=6

(c) Γ0 = Γ1 (d) Γ0 = 0

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.13/18

slide-19
SLIDE 19

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Spin Fluctuation in a Small Quantum Dot

µ Εd

H>0 H=0

time

impurity levels

0.1 0.2

sz(t)

U/Γ1=2 U/Γ1=4 U/Γ1=6 U/Γ1=8 U/Γ1=10 U/Γ1=12 U/Γ1=18

10

  • 4

10

  • 3

10

  • 2

10

  • 1

10 10

1

10

2

t/tK

0.1 0.2

sz(t)

(a) Γ0 = Γ1 (b) Γ0 = 0

Spin relaxation time scale : tsp ∝ 1/TK

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.13/18

slide-20
SLIDE 20

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

SIAM Time-Evolution

start: high temperature Free Impurity V0 = 0 Mixed Valence V0 = 0 Kondo Regime T t e m p e r a t u r e T t e m p e r a t u r e T t i m e t t i m e t no universality time evolution depends on boundary conditions

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.14/18

slide-21
SLIDE 21

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Kondo Model

H = Hcb + Hz(t) 2 σz + Hx(t) 2 σx + J Sloc · scb Spin Precession: Hx = 0.1 → Hz

0.1 1 10 100 1000

t*D

  • 0.4
  • 0.2

0.2 0.4

<Sxyz>(t)

Sx Sy Sz Fit: -0.5*cos(Hz*t)*e

  • 0.015*t

Fri Jun 10 15:03:45 2005

external magnetic field: x → z-axis

−0.5 0.5 −0.5 0.5 0.1 0.2 0.3 0.4 0.5 Sx Spin Precession J=0.2 Sy Sz

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.15/18

slide-22
SLIDE 22

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Summary and Outlook

TD-NRG algorithm: complete basis set needed for the time evolution dissipation and decoherence due to the bath Spin-Boson Model benchmark: decoherence and non-ohmic baths

  • scillatory vs overdamped regime

SIAM: two relaxation time scale for spin and charge dynamics Kondo Model: spin relaxation and spin precession Outlook Bosonic QI models: charge transfer in bio-molecules, photosynthesis description of steady state currents

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.16/18

slide-23
SLIDE 23

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Benchmark: Resonant Level Model

2 4 6 8 10

t*Γ

0.4 0.6 0.8 1

nd(t)

Ed

1/Γ = -1

Ed

1/Γ = -2

Ed

1/Γ = -3

30 60

N - m

0.4 0.5 0.6

nd(t=0) 2 4 6 8 10

t*T

0.4 0.6 0.8

nd(t)

T/Γ = 0.1 T/Γ = 0.5 T/Γ = 1 T/Γ = 5

(a) (b)

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.17/18

slide-24
SLIDE 24

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Anisotropic Kondo Model

H = Hcb + Hz(t) 2 σz + Hx(t) 2 σx +

  • kk′αβ

Jz(t) 2 c†

kαc† k′βσzSz imp

+

  • kk′

Jperp(t) 2

  • c†

k↑c† k′↓S− imp + c† k↓c† k′↑S+ imp

  • Concepts in Electron Correlation, Hvar, 30/9/2005 – p.18/18
slide-25
SLIDE 25

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Anisotropic Kondo Model

spin relaxation: J0 = 0, J1

⊥ = 0.15D = const

1 2 3 4 5 6 7 8 9 10 0.44 0.46 0.48 0.5

Sz(t)

Jz=-0.1 J⊥=0.15 (analytic) J⊥=0.15(ana) O(t

2)

10 10

2

10

4

10

6

10

8

10

10

t*D

0,1 0,2 0,3 0,4 0,5

Sz(t)

Jz=0.15 Jz=0.1 Jz=0.05 Jz=0.0

(a) (b)

two regimes: short time scale analytical result:

Sz(t) − Sz(0) ∝ (2ρJ1

⊥)2

× [G(2Dt) − 2G(Dt)] G(x) =

∞

X

l=1

(−1)l+1 (2l)!2l(2l − 1) x2l

external magnetic field: z-axis

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.18/18

slide-26
SLIDE 26

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Anisotropic Kondo Model

spin relaxation: J0 = 0, J1

⊥ = 0.15D = const

10

  • 4

10

  • 2

10

t*TK

0,1 0,2 0,3 0,4 0,5

Sz(t)

Jz=0.15 Jz=0.1 Jz=0.05 Jz=0.0 Jz=-0.1

two regimes: short time scale analytical result:

Sz(t) − Sz(0) ∝ (2ρJ1

⊥)2

× [G(2Dt) − 2G(Dt)] G(x) =

∞

X

l=1

(−1)l+1 (2l)!2l(2l − 1) x2l

long time:

tlong ∝ 1/TK

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.18/18

slide-27
SLIDE 27

Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen

Anisotropic Kondo Model

ferromagnetic regime, spin relaxation: Hz = 0.01 → 0

10

  • 1

10 10

1

10

2

10

3

10

4

10

5

10

6

t*D

0.4 0.6 0.8 1

Sz(t)/Sz(0)

J=-0.1, T=0.13 J=-0.1, T=0.018 J=-0.1, T=0.0023 J=-0.1, T=4.0*10

  • 5

10

  • 3

10

  • 2

10

  • 1

10 10

1

t*T*J(T)

0.4 0.6 0.8 1

Sz(t)/Sz(0)

J=-0.2, T=0.13 J=-0.2, T=0.018 J=-0.2, T=0.0023 J=-0.2, T=4.0*10

  • 5

(a) (b)

external magnetic field: z-axis

Concepts in Electron Correlation, Hvar, 30/9/2005 – p.18/18