nonequilibrium quasiparticles 1 anděla kalvov institute of physics, v.v.i. academy of sciences of...

Nonequilibrium Quasiparticles 1

Nonequilibrium Quasiparticles

Anděla Kalvová Institute of Physics, v.v.i.Academy of Sciences of the Czech Republic




Quasi-Particle States of Electron Systems out of EquilibriumB. Velický, A. Kalvová, V. Špička; PRB 75, 195125 (2007)




Quasi-Particle States of Electron Systems out of EquilibriumB. Velický, A. Kalvová, V. Špička; PRB 75, 195125 (2007)

Transients in Quantum Transport B. Velický: Semigroup Property of Propagators ... A. Kalvová: Correlated Initial Condition ...


- motivation

- definition



motivation

First slide from: Transients in Quantum Transport A. Kalvová: Correlated Initial Condition …

6

(Non-linear) quantum transport non-equilibrium problem many-body Hamiltonian

many-body density matrix

additive operator

Many-body system

Initial state

External disturbance

H

0 0 0at ( )t t t P P0 ( ) for U t t t ( )tU

7

0( ) for t t t

(Non-linear) quantum transport non-equilibrium problem Many-body system

Initial state


Response

many-body Hamiltonian


additive operator

one-particle density matrix

H


8

(Non-linear) quantum transport non-equilibrium problem

Quantum Transport Equation a closed equation for ( )t

drift [ ( ); ]tt

generalized collision term

Many-body system

Initial state


Response



additive operator


H

0 0 0at ( )t t t P P

0( ) for t t t 0 ( ) for U t t t ( )tU

9


Quantum Transport Equation a closed equation for ( )t

drift [ ( ); ]tt

Many-body system

Initial state


Response



additive operator


H

0 0 0at ( )t t t P P

0( ) for t t t

QUESTIONS existence, construction of incorporation of the initial

condition

0P

interaction term

0 ( ) for U t t t ( )tU

10

a closed equation for ( )t


Quantum Transport Equation

drift [ ( ); ]tt

Many-body system

Initial state


Response



additive operator


H

0 0 0at ( )t t t P P

0( ) for t t t

QUESTIONS existence, construction of incorporation of the initial

condition

0P

interaction term

0 ( ) for U t t t ( )tU

not quasiparticle


quasiparticle way .... Landau theory for Fermi liquids

motivation


Landau theory for Fermi liquids

the original system of quantum particles with strong interactions transformed

(purely phenomenologically in physical sense) to an effective

system of quasiparticles - forming a weakly non-ideal gas

- governed by a (slightly generalized) Boltzmann equation

motivation



Initial state

Weak extern. disturb.


m.-b. equilibrium state

additive operator

H


motivation



Initial state


Response


mb equilibrium state

additive operator

distribution function for QP

H

0 0 0at ( )t t t P P

0( ) for f t t t0 ( ) for U t t t ( )tU

motivation



Quantum Transport Eq. Quantum Boltzmann Eq.

Many-body system

Initial state


Response



additive operator


H

0 0 0at ( )t t t P P


motivation

d

/ d current + influence of external fields

+ collisions

f t

Newtonchange macroscopically understandable molecular per time evolution ( ) chaos

16

d

/ d current + influence of external fields

+ collisions

f t


Quantum Transport Eq. Quantum Boltzmann Eq.

Many-body system

Initial state


Response



additive operator


H

0 0 0at ( )t t t P P


COLLISION INFLUENCE1. evolution to equilibrium2. time arrow = direction of the entropy growth3. forgetting of the past – after ONE collision, (during average collision time) from initial condition to “floating” initial condition

motivation

Newtonchange macroscopically understandable molecular per time evolution ( ) chaos


linear transport ... redistribution of the equilibrium QP (equilibrium GF)

motivation


motivation

non-linear transport ... redistribution of non-equilibrium QP (non-equilibrium GF) ??



( , ')RG t t QP formation

( , )RWZG t t

... Qt t

... Qt t

( , ) i exp[ i ( )]RW wG t t z t t

definition QP




( , )RWZG t t

... Qt t

... Qt t


phenomenological constants

( )R t

linear transport ... transfer equilibrium QP (equilibrium GF)

definition QP



( , )RWZG t t

... Qt t

... Qt t


( )R t

formation time



definition QP



( , )RWZG t t

... Qt t

... Qt t


( )R t

renormal. constant



formation time

definition QP



( , )RWZG t t

... Qt t

... Qt t


( )R t

complex energy


formation time

renormal. constant

definition QP




( , )RWZG t t

... Qt t

... Qt t


( )R t

1i(2 )w wz E QP condition

2Q


complex energy

formation time

renormal. constant

definition QP




( , )RWZG t t

... Qt t

... Qt t


( )R t

i

0

d ( )e wz tRw MFz t t


definition QP


formation time

renormal. constant complex energy

26 ... pole and residuum of the spectral representation of the equilibrium GF (one - el. propagator)


( , )RWZG t t

... Qt t

... Qt t


( )R t

i

0

d ( )e ( i 0)MF tR Rw MF MF MFz t t

... Wigner-Weisskopf (weak scatt.)

,wz Z


definition QP


formation time

renormal. constant complex energy


non-equilibrium initial state ... arbitrary ... full description is necessary

system evolves ... the processes… very fast, fast, slow

very fast processes... smoothing, coherence loss, decay of the many-body

correlation, chaotization characteristic time

fast processes ... collisions leading to the kinetic stage of transport

characteristic time

slow processes ... compensating macroskopic inhomogeneities

characteristic time

BOGOLJUBOV POSTULATES

initial stage, kinetic, hydrodynamicIn kinetic stage …(quantum) distribution function provides complete description satisfies quantum transport equation

C H

C

hierarchy of characteristic times

H

28

from non-equilibrium to „equilibrium“

KBA (Kadanoff Baym Ansatz) ... NGF ( )... in Wigner representation ...time dependent spectral density ... NGF quantum transport equation for QP distribution

restriction ... processes… quasiclassical in time

, ,R AG G G

motivation NeQP

29




, ,R AG G G

motivation NeQP

KB were inspired by equilibrium. Their ansatz leadsto the distribution function of QP. Their dynamical behavior is described by equation of Bolzmann type. Processes in such MB system have to be quasi-

classical in time

30




, ,R AG G G

GKBA (Generalized KBA)... quantum transport equation for real particles (density matrix) causal structure rapid transients ...quasiparticles

restriction ... renormalization (formation) …negligible during c

( , ) ( , ) ( )RG t t G t t t t t

motivation NeQP

In the opposite situation… in rapid transients, we construct the QTEfor real particles. The GKBA ansatz is formulated fully in time domain.

The using of GKBA is restricted too. It is possible to useit only if the formation processes during initial stage are negligible.

31




, ,R AG G G

GKBA (Generalized KBA)... quantum transport equation for real particles (density matrix) causal structure rapid transients ...quasiparticles

restriction ... renormalization (formation) …negligible during c

QKBA (Quasiparticle KBA) ... quantum transport equation for real particles dynamical (formation) processes …involved

( , ) ( , ) ( )RG t t G t t t t t

( , ) ( , ) ( )RwG t t G t t t t t

restriction ... if nonequilibrium QPs exist

motivation NeQP


definition NeQP

Any propagator … 1. should pass through a formation stage 2. then should represent a QP

In nonstationary (transient, nonequilibrium) conditions … two questions 1.? formation time ? 2.? is it possible to construct … as QP counterpart to

In transients, two processes compete permanently: formation and flight. Complex process results and we don’t know,

if it is a flight of independent quasiparticles or if this concept fails.

( , )RwG t t ( , )RG t t

??



( , )RWZG t t

... Qt t

... Qt t

functional relation nonequilibrium QP

( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t t t t

( , ) ( , ) ( , )R R Rw QG t t G t t G t t t t t t

composition rule SGR

QP comosition rule QCR

equilibrium:

definition NeQP



( , )RWZG t t

... Qt t

... Qt t

functional relation nonequilibrium QP

( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t t t t

( , ) ( , ) ( , )R R Rw QG t t G t t G t t t t t t

composition rule SGR

QP comosition rule QCR

equilibrium:

definition NeQP

i /R Rt w QG G z t t logarithmic derivative ... time independent


nonequilibrium propagator: .... double-time function (matrix)( , ')RG t t

( , )R t tselfenergy: .... 'M mt t t t

1[ ( , )] {i ( )} ( ) ( , )R Rt MFG t t H t t t t t

definition NeQP





definition NeQP

free-particle Ham.+ external fields+ mean-field





( , ) 0 ( )RQt t t t t t

definition NeQP

NeQP (formation time condition).....

It is natural to restrict our study to the case, when a NE extension of QP formation time exists. Therefore,

we assume the existence formation time such, that the selfenergy is zero outside a double time region, which has the strip adjoing the time diagonal.

Q





( , ) 0 ( )RQt t t t t t

1[ ( , )] {i ( )} ( ) ( , )R Rw t MF wG t t H t t t t t

definition NeQP

NeQP (formation time condition).....

NeQP propagator


NeQP propagator ( , ')RwG t t

SGR ...satisfied time local optical potential ( )w t

( , ) i ( , ) ( , )R R Rw w wG t t G t t G t t

1( , ) independent( , )

Rt w R

w

G t tG t t

t

time diagonal; ( ) ( i 0)R Rw w w t t t

( , ) i exp[ i d { ( ) ( )}t

Rw MF w

t

G t t T u H u u

definition NeQP






( , ) i exp[ i d { ( ) ( )}t

Rw MF

t

G t t T u H u u

definition NeQP


Rt w R

w

G t tG t t

t






( , ) i exp[ i d { ( ) ( )}t

Rw MF

t

G t t T u H u u

definition NeQP

mean-field propagator

unitary evolution independ. „particles“in external and internal fields


Rt w R

w

G t tG t t

t






Rt w R

w

G t t tG t t


( , ) i exp[ i d { ( ) ( )}t

Rw MF

t

G t t T u H u u

definition NeQP






Rt w R

w

G t t tG t t


( , ) i exp[ i d { ( ) ( )}t

Rw MF

t

G t t T u H u u

definition NeQP

NeQP propagator






Rt w R

w

G t t tG t t


( , ) i exp[ i d { ( ) ( )}t

Rw MF

t

G t t T u H u u

definition NeQP

NeQP propagator

not a unitary evolution ( eff. single particle. nonhermit. H)flight of independent NeQP


starting point Dyson equations for

to find RwG closed equation for ( )w t ... Phase Equation

,R RwG G

( , ) i ( , ) ( , )

d d ( , )( )( , ) ( , )

R R Rw

t tR R R Rw w

t t

t t t

G t t G t t G t t

t t G t t t t G t t

(subtraction)

renormalized quasiparticle composition rule (RQCR)

definition NeQP




,R RwG G

( , ) i ( , ) ( , )

d d ( , )( )( , ) ( , )

R R Rw

t tR R R Rw w

t t

t t t

G t t G t t G t t

t t G t t t t G t t

(subtraction)


definition NeQP




,R RwG G

( , ) i ( , ) ( , )

d d ( , )( )( , ) ( , )

R R Rw

t tR R R Rw w

t t

t t t

G t t G t t G t t

t t G t t t t G t t

(subtraction)


definition NeQP

QP composition rule (QCR)




,R RwG G

( , ) i ( , ) ( , )

d d ( , )( )( , ) ( , )

R R Rw

t tR R R Rw w

t t

t t t

G t t G t t G t t

t t G t t t t G t t

(subtraction)


definition NeQP

renorm. vertex correction




,R RwG G

( , ) i ( , ) ( , )

d d ( , )( )( , ) ( , )

R R Rw

t tR R R Rw w

t t

t t t

G t t G t t G t t

t t G t t t t G t t

(subtraction)


definition NeQP

slide Transients in Quantum Transport B. Velický: Semigroup Property of Propagators ...

50

Discussion of the RSG rule

1. universal vertex, derived with almost no effort and no specific properties of the GF

2. off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future

3. Similar to the Dyson Equation, but NO free GF

4. In fact, looks pretty much like a linear response

we would like to understand all these features

'

( , ') i ( , ) ( , ')

d d ( , ) ( , ) , ')

'

(

R R R

t tR R R

t t

t t t

G t t G t t G t t

t t G t t t t G t t

Semi-group property...FZÚ 14.2.2006

51

Discussion of the RSG rule

1. universal vertex, derived with almost no effort and no specific properties of the GF

2. off-diagonal vertex, linking in a smeared fashion propagation in the past and in the future

3. Similar to the Dyson Equation, but NO free GF

4. In fact, looks pretty much like a linear response

we would like to understand all these features

'

( , ') i ( , ) ( , ')

d d ( , ) ( , ) , ')

'

(

R R R

t tR R R

t t

t t t

G t t G t t G t t

t t G t t t t G t t

Semi-group property...FZÚ 14.2.2006




,R RwG G

( , ) i ( , ) ( , )

d d ( , )( )( , ) ( , )

R R Rw

t tR R R Rw w

t t

t t t

G t t G t t G t t

t t G t t t t G t t

(subtraction)


definition NeQP

self-consistent condition for cancellation of the vertex

explicit equation for ( )w t


ot

summary of notation of times :

,m Mt t ... minimum, maximum

,t t ...evolution from tto t ( , ,m Mt t t t )time- inhomogeneity

t

... time between ,t tNEW... time, where QCR is valid

... min t

( , ) ( , ) ( , )R R RwG t t G t t G t t

1i ( , ) ( ) ( )( , )

Rt MF wRG t t H t t

G t t

independent of ,t t

definition NeQP


ot

,m Mt t ... minimum, maximum

,t t ...evolution from tto t ( , ,m Mt t t t )time inhomogeneity

t

... time between ,t tNEW... time, where QCR is valid

... min t

( , ) ( , ) ( , )R R RwG t t G t t G t t

1i ( , ) ( ) ( )( , )

Rt MF wRG t t H t t

G t t

definition NeQP

implication: for fixed many exist... typically ... identical NeQP... ... independent

ot tmax ( )o o

mt t t t t

( , )R owG t t

t

phase equation

independent of ,t t

summary of notation of times :


cesta k fázové rovnici


how to get the phase equation for an optical potential ( )w t

definition NeQP phase equation


i ( , ) ( ) ( , ) d ( , ) ( , )t

R R R Rt MF

t

G t t H t G t t t t t G t t


definition NeQP

( )w t



i ( , ) ( ) ( , ) d ( , ) ( , )t

R R R Rt MF

t


i ( , ) 1( ) d ( , ) ( , )( , ) ( , )

tRR Rt

MFR Rt

G t t H t t t t G t tG t t G t t

( )t t definition...




i ( , ) ( ) ( , ) d ( , ) ( , )t

R R R Rt MF

t


i ( , ) 1( ) d ( , ) ( , )( , ) ( , )

tRR Rt

MFR Rt

G t t H t t t t G t tG t t G t t

( )t t definice...

i ( , ) ( ) ( )( , )

Rt

MF tR

G t t H t tG t t




1( ) d ( , ) ( , )( , )

tR R

t Rt

t t t t G t tG t t




? ( )w t ... ot t ?

( ) d ( , ) exp[ i d { ( ) ( )}]

1d ( , ) ( , )( , )

exp[ i d { ( ) ( )}]

o

o

t tR

t MF ttt

tR R

R ot

t

MF tt

t t t t T u H u u

t t t G t tG t t

T u H u u




? ( )w t ... ot t ?

( ) d ( , ) exp[ i d { ( ) ( )}]

1d ( , ) ( , )( , )

exp[ i d { ( ) ( )}]

o

o

t tR

t MF ttt

tR R

R ot

t

MF tt

t t t t T u H u u

t t t G t tG t t

T u H u u

( )oQt t t

phase equation


far enough from ot


( ) d ( , ) exp[ i d { ( ) ( )}]

1d ( , ) ( , )( , )

exp[ i d { ( ) ( )}]

o

o

t tR

t MF ttt

tR R

R ot

t

MF tt

t t t t T u H u u

t t t G t tG t t

T u H u u


Initialcondition

? ( )w t ... ot t ?


( ) d ( , ) exp[ i d { ( ) ( )}]

1d ( , ) ( , )( , )

exp[ i d { ( ) ( )}]

o

o

t tR

t MF ttt

tR R

R ot

t

MF tt

t t t t T u H u u

t t t G t tG t t

T u H u u




Initialcondition

Q


( )

( ) d ( , ) exp[ i d { ( ) ( )}]

1d ( , ) ( , )( , )

exp[ i d { ( ) ( )}]

o o

o

o

Qt t

t tR

t MF ttt

tR R

R o

t

MF tt

t t t t T u H u u

t t t G t tG t t

T u H u u




Initialcondition

Q


( )

( ) d ( , ) exp[ i d { ( ) ( )}]

1d ( , ) ( , )( , )

exp[ i d { ( ) ( )}]

o

o

t tR

t MF ttt

tR R

R o

t

MF tt

o oQt t

t t t t T u H u u

t t t G t tG t t

T u H u u




Initialcondition

Q


( )

( ) d ( , ) exp[ i d { ( ) ( )}]

1d ( , ) ( , )( , )

exp[ i d { ( ) ( )}]

o o

o

o

Qt t

t tR

t MF ttt

tR R

R o

t

MF tt

t t t t T u H u u

t t t G t tG t t

T u H u u




Initialcondition

Q


? ( )w t ... ot t ?

( )

( ) d ( , ) exp[ i d { ( ) ( )}]

d ( , )

e

1( , )

xp[ i d

(

{ ( ) ( )}]

, )o o

o

o

Q

t tR

t MF ttt

tR

R ott

R

F

t

t

M tt

t

G t tG t t

t t t T u H u u

t t t

T u H u u

initial condition



( )w t ... ot t

( )

( ) d ( , ) exp[ i d { ( ) ( )}]

1d ( , ) ( , )( , )

exp[ i d { ( ) ( )}]

o

o

o oQ

t tR

w MF wtt

tR R

R ott t

t

MF wt

t t t t T u H u u

t t t G t tG t t

T u H u u

closed phase equation



( ) d ( , ) exp[ i d { ( ) ( )}]o

t tR

w MF wtt

t t t t T u H u u

why the term... phase equation ?

why ... and not ? ( )w t

Qcondition for derivation of the phase equation... existence

( , ) 0 ( )RQt t t t t t ... „dominant strip“

... analogue to the dominant strip for ... existence ( , )t t C

„ times of forgetting“: ... decay of init. corr. (formation of the kinetic stage) ... phase eq. doesn´t depend on (formation of QP)

CQ t


( , )RwG t t

( )oQt t t


( ) d ( , ) exp[ i d { ( ) ( )}]o

t tR

w MF wtt

t t t t T u H u u

... semilocal in time ( ...effective time locality)( ) ( )MF wH t t


( )oQt t t


( ) d ( , ) exp[ i d { ( ) ( )}]o

t tR

w MF wtt

t t t t T u H u u

... effective time locality of ( )w t

- condensed influence of the past ...closed phase equation


( )oQt t t



( ) d ( , ) exp[ i d { ( ) ( )}]o

t tR

w MF wtt

t t t t T u H u u

... effective time locality of ( )w t

- condensed influence of the past ...closed phase equation


( )oQt t t


- the QP evolution operator with anti-causal time ordering

the past QP state evolves from the (near) future

sampling of the past by excursions backwards in time


Conclusion:

consistent definition of the NeQP

NeQP ... two properties (equivalent), used for definition-corresponding propagator satisfies QP composition rule-corresponding propagator contains optical potencial ( )w t

existence of the NeQP ... dependence on retard. selfenergy ( , ')R t t short formation time narrow „dominant strip“ in ( , ')R t t

optical potential ( )w t and NE retarded selfenergy ( , ')R t t ...... connected with the phase eq. For its solution ...necessity to know initial conditions... reflect time period of formation Q

QP composition rule ... „free term“ in (transf.) Dyson equationvanishing of the vertex correction validity of the phase equation


nonequilibrium quasiparticles 1 anděla kalvov institute of physics, v.v.i. academy of sciences of...

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